diff git a/books/bookvol10.2.pamphlet b/books/bookvol10.2.pamphlet
index 3e48804..7727b56 100644
 a/books/bookvol10.2.pamphlet
+++ b/books/bookvol10.2.pamphlet
@@ 350,6 +350,8 @@ digraph pic {
ArcHyperbolicFunctionCategory examples
====================================================================
+This is the Category for the inverse hyperbolic trigonometric functions
+
See Also:
o )show ArcHyperbolicFunctionCategory
@@ 383,9 +385,6 @@ These are directly exported but not implemented:
\begin{chunk}{category AHYP ArcHyperbolicFunctionCategory}
)abbrev category AHYP ArcHyperbolicFunctionCategory
++ Category for the inverse hyperbolic trigonometric functions
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the inverse hyperbolic trigonometric functions;
@@ 461,6 +460,8 @@ intermediate test to check that the argument has a reciprocal values.
ArcTrigonometricFunctionCategory examples
====================================================================
+This is the Category for the inverse trigonometric functions
+
See Also:
o )show ArcTrigonometricFunctionCategory
@@ 497,9 +498,6 @@ These are implemented by this category:
\begin{chunk}{category ATRIG ArcTrigonometricFunctionCategory}
)abbrev category ATRIG ArcTrigonometricFunctionCategory
++ Category for the inverse trigonometric functions
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the inverse trigonometric functions;
@@ 549,6 +547,17 @@ digraph pic {
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagehead{AttributeRegistry}{ATTREG}
\pagepic{ps/v102attributeregistry.ps}{ATTREG}{1.00}
+\begin{chunk}{AttributeRegistry.help}
+====================================================================
+AttributeRegistry examples
+====================================================================
+
+This category exports the attributes in the AXIOM Library.
+
+See Also:
+o )show BasicType
+
+\end{chunk}
{\bf See:}
@@ 748,6 +757,9 @@ digraph pic {
BasicType examples
====================================================================
+BasicType is the basic category for describing a collection
+of elements with = (equality).
+
See Also:
o )show BasicType
@@ 778,18 +790,9 @@ These are implemented by this category:
\begin{chunk}{category BASTYPE BasicType}
)abbrev category BASTYPE BasicType
% BasicType
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ \spadtype{BasicType} is the basic category for describing a collection
++ of elements with \spadop{=} (equality).
+++ BasicType is the basic category for describing a collection
+++ of elements with = (equality).
BasicType(): Category == with
"=": (%,%) > Boolean ++ x=y tests if x and y are equal.
@@ 856,6 +859,9 @@ digraph pic {
CoercibleTo examples
====================================================================
+A is coercible to B means any element of A can automatically be
+converted into an element of B by the interpreter.
+
See Also:
o )show CoercibleTo
@@ 882,9 +888,7 @@ This is directly exported but not implemented:
\begin{chunk}{category KOERCE CoercibleTo}
)abbrev category KOERCE CoercibleTo
++ Category for coerce
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ A is coercible to B means any element of A can automatically be
@@ 963,6 +967,8 @@ digraph pic {
CombinatorialFunctionCategory examples
====================================================================
+This is the Category for the usual combinatorial functions
+
See Also:
o )show CombinatorialFunctionCategory
@@ 990,9 +996,7 @@ These are directly exported but not implemented:
\begin{chunk}{category CFCAT CombinatorialFunctionCategory}
)abbrev category CFCAT CombinatorialFunctionCategory
++ Category for the usual combinatorial functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the usual combinatorial functions;
@@ 1072,6 +1076,9 @@ digraph pic {
ConvertibleTo examples
====================================================================
+A is convertible to B means any element of A can be converted into
+an element of B, but not automatically by the interpreter.
+
See Also:
o )show ConvertibleTo
@@ 1098,9 +1105,7 @@ This is directly exported but not implemented:
\begin{chunk}{category KONVERT ConvertibleTo}
)abbrev category KONVERT ConvertibleTo
++ Category for convert
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ A is convertible to B means any element of A
@@ 1231,6 +1236,8 @@ digraph pic {
ElementaryFunctionCategory examples
====================================================================
+This is the Category for the elementary functions.
+
See Also:
o )show ElementaryFunctionCategory
@@ 1263,7 +1270,6 @@ These are implemented by this category:
)abbrev category ELEMFUN ElementaryFunctionCategory
++ Category for the elementary functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the elementary functions;
@@ 1334,6 +1340,11 @@ digraph pic {
Eltable examples
====================================================================
+An eltable over domains D and I is a structure which can be viewed
+as a function from D to I. Examples of eltable structures range from
+data structures, e.g. those of type List, to algebraic structures like
+Polynomial.
+
See Also:
o )show Eltable
@@ 1361,18 +1372,11 @@ This is directly exported but not implemented:
++ Author: Michael Monagan; revised by Manuel Bronstein and Manuel Bronstein
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An eltable over domains D and I is a structure which can be viewed
++ as a function from D to I.
++ Examples of eltable structures range from data structures, e.g. those
++ of type \spadtype{List}, to algebraic structures like
++ \spadtype{Polynomial}.
+++ of type List, to algebraic structures like Polynomial.
Eltable(S:SetCategory, Index:Type): Category == with
elt : (%, S) > Index
@@ 1456,6 +1460,8 @@ intermediate test to check that the argument has a reciprocal values.
HyperbolicFunctionCategory examples
====================================================================
+This is the Category for the hyperbolic trigonometric functions.
+
See Also:
o )show HyperbolicFunctionCategory
@@ 1488,11 +1494,7 @@ These are implemented by this category:
\begin{chunk}{category HYPCAT HyperbolicFunctionCategory}
)abbrev category HYPCAT HyperbolicFunctionCategory
++ Category for the hyperbolic trigonometric functions
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the hyperbolic trigonometric functions;
HyperbolicFunctionCategory(): Category == with
@@ 1579,6 +1581,12 @@ digraph pic {
InnerEvalable examples
====================================================================
+This category provides eval operations. A domain may belong to this
+category if it is possible to make "evaluation" substitutions. The
+difference between this and Evalable is that the operations in this
+category specify the substitution as a pair of arguments rather than
+as an equation.
+
See Also:
o )show InnerEvalable
@@ 1611,16 +1619,7 @@ These are implemented by this category:
\begin{chunk}{category IEVALAB InnerEvalable}
)abbrev category IEVALAB InnerEvalable
 FOR THE BENEFIT OF LIBAX0 GENERATION
++ Author:
++ Date Created:
++ Date Last Updated: June 3, 1991
++ Basic Operations:
++ Related Domains:
++ Also See: Evalable
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This category provides \spadfun{eval} operations.
++ A domain may belong to this category if it is possible to make
@@ 1870,6 +1869,8 @@ digraph pic {
OpenMath examples
====================================================================
+OpenMath provides operations for exporting an object in OpenMath format.
+
See Also:
o )show OpenMath
@@ 1897,11 +1898,6 @@ These are directly exported but not implemented:
)abbrev category OM OpenMath
++ Author: Mike Dewar & Vilya Harvey
++ Basic Functions: OMwrite
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ \spadtype{OpenMath} provides operations for exporting an object
++ in OpenMath format.
@@ 1997,6 +1993,9 @@ digraph pic {
PartialTranscendentalFunctions examples
====================================================================
+This is the description of any package which provides partial
+functions on a domain belonging to TranscendentalFunctionCategory.
+
See Also:
o )show PartialTranscendentalFunctions
@@ 2070,15 +2069,12 @@ These are directly exported but not implemented:
\begin{chunk}{category PTRANFN PartialTranscendentalFunctions}
)abbrev category PTRANFN PartialTranscendentalFunctions
++ Description of a package which provides partial transcendental
++ functions, i.e. functions which return an answer or "failed"
++ Author: Clifton J. Williamson
++ Date Created: 12 February 1990
++ Date Last Updated: 14 February 1990
++ Keywords:
++ Examples:
++ References:
++ Description:
+++ A package which provides partial transcendental
+++ functions, i.e. functions which return an answer or "failed"
++ This is the description of any package which provides partial
++ functions on a domain belonging to TranscendentalFunctionCategory.
@@ 2222,6 +2218,10 @@ digraph pic {
Patternable examples
====================================================================
+Category of sets that can be converted to useful patterns. An object
+S is Patternable over an object R if S can lift the conversions from
+R into Pattern(Integer) and Pattern(Float) to itself.
+
See Also:
o )show Patternable
@@ 2252,12 +2252,11 @@ These exports come from \refto{ConvertibleTo}(Pattern(Float)):
\begin{chunk}{category PATAB Patternable}
)abbrev category PATAB Patternable
++ Category of sets that can be converted to useful patterns
++ Author: Manuel Bronstein
++ Date Created: 29 Nov 1989
++ Date Last Updated: 29 Nov 1989
++ Keywords: pattern, matching.
++ Description:
+++ Category of sets that can be converted to useful patterns
++ An object S is Patternable over an object R if S can
++ lift the conversions from R into \spadtype{Pattern(Integer)} and
++ \spadtype{Pattern(Float)} to itself;
@@ 2340,6 +2339,8 @@ digraph pic {
PrimitiveFunctionCategory examples
====================================================================
+This is the Category for the functions defined by integrals.
+
See Also:
o )show PrimitiveFunctionCategory
@@ 2363,9 +2364,7 @@ These are directly exported but not implemented:
\begin{chunk}{category PRIMCAT PrimitiveFunctionCategory}
)abbrev category PRIMCAT PrimitiveFunctionCategory
++ Category for the integral functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the functions defined by integrals;
@@ 2437,6 +2436,8 @@ digraph pic {
RadicalCategory examples
====================================================================
+The RadicalCategory is a model for the rational numbers.
+
See Also:
o )show RadicalCategory
@@ 2473,14 +2474,7 @@ These are implemented by this category:
\begin{chunk}{category RADCAT RadicalCategory}
)abbrev category RADCAT RadicalCategory
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations: nthRoot, sqrt, **
++ Related Constructors:
++ Keywords: rational numbers
++ Description:
++ The \spad{RadicalCategory} is a model for the rational numbers.
RadicalCategory(): Category == with
sqrt : % > %
@@ 2553,6 +2547,10 @@ digraph pic {
RetractableTo examples
====================================================================
+A is retractable to B means that some elementsif A can be converted
+into elements of B and any element of B can be converted into an
+element of A.
+
See Also:
o )show RetractableTo
@@ 2601,12 +2599,9 @@ These are implemented by this category:
\begin{chunk}{category RETRACT RetractableTo}
)abbrev category RETRACT RetractableTo
++ Category for retract
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ A is retractable to B means that some elementsif A can be converted
+++ A is retractable to B means that some elements if A can be converted
++ into elements of B and any element of B can be converted into an
++ element of A.
@@ 2740,6 +2735,8 @@ digraph pic {
SpecialFunctionCategory examples
====================================================================
+This is the Category for the other special functions.
+
See Also:
o )show SpecialFunctionCategory
@@ 2782,9 +2779,7 @@ These are directly exported but not implemented:
\begin{chunk}{category SPFCAT SpecialFunctionCategory}
)abbrev category SPFCAT SpecialFunctionCategory
++ Category for the other special functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 11 May 1993
++ Description:
++ Category for the other special functions;
@@ 2880,6 +2875,8 @@ intermediate test to check that the argument has a reciprocal values.
TrigonometricFunctionCategory examples
====================================================================
+This is the Category for the trigonometric functions.
+
See Also:
o )show TrigonometricFunctionCategory
@@ 2916,9 +2913,6 @@ These are implemented by this category:
\begin{chunk}{category TRIGCAT TrigonometricFunctionCategory}
)abbrev category TRIGCAT TrigonometricFunctionCategory
++ Category for the trigonometric functions
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the trigonometric functions;
@@ 2971,6 +2965,17 @@ digraph pic {
\pagehead{Type}{TYPE}
\pagepic{ps/v102type.ps}{TYPE}{1.00}
+\begin{chunk}{Type.help}
+====================================================================
+Type examples
+====================================================================
+
+The fundamental Type.
+
+See Also:
+o )show Type
+
+\end{chunk}
{\bf See:}
\pageto{Aggregate}{AGG}
@@ 2986,7 +2991,6 @@ digraph pic {
\begin{chunk}{category TYPE Type}
)abbrev category TYPE Type
++ The new fundamental Type (keeping Object for 1.5 as well)
++ Author: Richard Jenks
++ Date Created: 14 May 1992
++ Date Last Updated: 14 May 1992
@@ 3058,6 +3062,16 @@ digraph pic {
Aggregate examples
====================================================================
+The notion of aggregate serves to model any data structure aggregate,
+designating any collection of objects, with heterogenous or homogeneous
+members, with a finite or infinite number of members, explicitly or
+implicitly represented. An aggregate can in principle represent
+everything from a string of characters to abstract sets such
+as "the set of x satisfying relation r(x)"
+
+An attribute "finiteAggregate" is used to assert that a domain
+has a finite number of elements.
+
See Also:
o )show Aggregate
@@ 3104,12 +3118,6 @@ These are implemented by this category:
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The notion of aggregate serves to model any data structure aggregate,
++ designating any collection of objects, with heterogenous or homogeneous
@@ 3117,7 +3125,7 @@ These are implemented by this category:
++ implicitly represented. An aggregate can in principle represent
++ everything from a string of characters to abstract sets such
++ as "the set of x satisfying relation r(x)"
++ An attribute \spadatt{finiteAggregate} is used to assert that a domain
+++ An attribute "finiteAggregate" is used to assert that a domain
++ has a finite number of elements.
Aggregate: Category == Type with
@@ 3218,6 +3226,9 @@ digraph pic {
CombinatorialOpsCategory examples
====================================================================
+CombinatorialOpsCategory is the category obtaining by adjoining
+summations and products to the usual combinatorial operations;
+
See Also:
o )show CombinatorialOpsCategory
@@ 3256,9 +3267,7 @@ These exports come from \refto{CombinatorialFunctionCategory}():
\begin{chunk}{category COMBOPC CombinatorialOpsCategory}
)abbrev category COMBOPC CombinatorialOpsCategory
++ Category for summations and products
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 22 February 1993 (JHD/BMT)
++ Description:
++ CombinatorialOpsCategory is the category obtaining by adjoining
@@ 3349,6 +3358,12 @@ digraph pic {
EltableAggregate examples
====================================================================
+An eltable aggregate is one which can be viewed as a function.
+For example, the list [1,7,4] can applied to 0,1, and 2 respectively
+will return the integers 1, 7, and 4; thus this list may be viewed
+as mapping 0 to 1, 1 to 7 and 2 to 4. In general, an aggregate
+can map members of a domain Dom to an image domain Im.
+
See Also:
o )show EltableAggregate
@@ 3398,17 +3413,11 @@ These exports come from \refto{Eltable}():
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An eltable aggregate is one which can be viewed as a function.
++ For example, the list \axiom{[1,7,4]} can applied to 0,1, and 2
++ respectively will return the integers 1,7, and 4; thus this list may
++ be viewed as mapping 0 to 1, 1 to 7 and 2 to 4. In general, an aggregate
+++ For example, the list [1,7,4] can applied to 0,1, and 2 respectively
+++ will return the integers 1, 7, and 4; thus this list may be viewed as
+++ mapping 0 to 1, 1 to 7 and 2 to 4. In general, an aggregate
++ can map members of a domain Dom to an image domain Im.
EltableAggregate(Dom:SetCategory, Im:Type): Category ==
@@ 3507,6 +3516,9 @@ digraph pic {
Evalable examples
====================================================================
+This category provides eval operations. A domain may belong to this
+category if it is possible to make "evaluation" substitutions.
+
See Also:
o )show Evalable
@@ 3542,16 +3554,7 @@ These exports come from \refto{InnerEvalable}(R:SetCategory,R:SetCategory):
\begin{chunk}{category EVALAB Evalable}
)abbrev category EVALAB Evalable
++ Author:
++ Date Created:
++ Date Last Updated: June 3, 1991
++ Basic Operations:
++ Related Domains:
++ Also See: FullyEvalable
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This category provides \spadfun{eval} operations.
++ A domain may belong to this category if it is possible to make
@@ 3645,6 +3648,9 @@ digraph pic {
FortranProgramCategory examples
====================================================================
+FortranProgramCategory provides various models of FORTRAN subprograms.
+These can be transformed into actual FORTRAN code.
+
See Also:
o )show FortranProgramCategory
@@ 3685,16 +3691,9 @@ These exports come from \refto{CoercibleTo}(OutputForm):
)abbrev category FORTCAT FortranProgramCategory
++ Author: Mike Dewar
++ Date Created: November 1992
++ Date Last Updated:
++ Basic Operations:
++ Related Constructors: FortranType, FortranCode, Switch
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ \axiomType{FortranProgramCategory} provides various models of
++ FORTRAN subprograms. These can be transformed into actual FORTRAN code.
+++ FortranProgramCategory provides various models of FORTRAN subprograms.
+++ These can be transformed into actual FORTRAN code.
FortranProgramCategory():Category == Join(Type,CoercibleTo OutputForm) with
outputAsFortran : $ > Void
@@ 3779,6 +3778,11 @@ digraph pic {
FullyRetractableTo examples
====================================================================
+A is fully retractable to B means that A is retractable to B and
+if B is retractable to the integers or rational numbers then so is A.
+In particular, what we are asserting is that there are no integers
+(rationals) in A which don't retract into B.
+
See Also:
o )show FullyRetractableTo
@@ 3946,6 +3950,10 @@ digraph pic {
FullyPatternMatchable examples
====================================================================
+A set S is PatternMatchable over R if S can lift the patternmatching
+functions of S over the integers and float to itself (necessary for
+matching in towers).
+
See Also:
o )show FullyPatternMatchable
@@ 4008,11 +4016,9 @@ These exports come from \refto{Type}():
\begin{chunk}{category FPATMAB FullyPatternMatchable}
)abbrev category FPATMAB FullyPatternMatchable
++ Category of sets that can be patternmatched on
++ Author: Manuel Bronstein
++ Date Created: 28 Nov 1989
++ Date Last Updated: 29 Nov 1989
++ Keywords: pattern, matching.
++ Description:
++ A set S is PatternMatchable over R if S can lift the
++ patternmatching functions of S over the integers and float
@@ 4100,6 +4106,8 @@ digraph pic {
Logic examples
====================================================================
+Logic provides the basic operations for lattices, e.g., boolean algebra.
+
See Also:
o )show Logic
@@ 4138,12 +4146,6 @@ These exports come from \refto{BasicType}():
\begin{chunk}{category LOGIC Logic}
)abbrev category LOGIC Logic
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations: ~, /\, \/
++ Related Constructors:
++ Keywords: boolean
++ Description:
++ `Logic' provides the basic operations for lattices, e.g., boolean algebra.
@@ 4222,6 +4224,12 @@ digraph pic {
PlottablePlaneCurveCategory examples
====================================================================
+PlottablePlaneCurveCategory is the category of curves in the plane
+which may be plotted via the graphics facilities. Functions are
+provided for obtaining lists of lists of points, representing the
+branches of the curve, and for determining the ranges of the
+xcoordinates and ycoordinates of the points on the curve.
+
See Also:
o )show PlottablePlaneCurveCategory
@@ 4256,18 +4264,7 @@ These exports come from \refto{CoercibleTo}(OutputForm):
++ Author: Clifton J. Williamson
++ Date Created: 11 January 1990
++ Date Last Updated: 15 June 1990
++ Basic Operations: listBranches, xRange, yRange
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: plot, graphics
++ References:
++ Description:
++ PlottablePlaneCurveCategory is the category of curves in the
++ plane which may be plotted via the graphics facilities. Functions are
++ provided for obtaining lists of lists of points, representing the
++ branches of the curve, and for determining the ranges of the
++ xcoordinates and ycoordinates of the points on the curve.
PlottablePlaneCurveCategory(): Category == Definition where
L ==> List
@@ 4356,6 +4353,12 @@ digraph pic {
PlottableSpaceCurveCategory examples
====================================================================
+PlottableSpaceCurveCategory is the category of curves in 3space which
+may be plotted via the graphics facilities. Functions are provided for
+obtaining lists of lists of points, representing the branches of the
+curve, and for determining the ranges of the x, y, and zcoordinates
+of the points on the curve.
+
See Also:
o )show PlottableSpaceCurveCategory
@@ 4392,12 +4395,6 @@ These exports come from \refto{CoercibleTo}(OutputForm):
++ Author: Clifton J. Williamson
++ Date Created: 11 January 1990
++ Date Last Updated: 15 June 1990
++ Basic Operations: listBranches, xRange, yRange, zRange
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: plot, graphics
++ References:
++ Description:
++ PlottableSpaceCurveCategory is the category of curves in
++ 3space which may be plotted via the graphics facilities. Functions are
@@ 4490,6 +4487,8 @@ digraph pic {
RealConstant examples
====================================================================
+The category of real numeric domains, i.e. convertible to floats.
+
See Also:
o )show RealConstant
@@ 4519,15 +4518,6 @@ These exports come from \refto{ConvertibleTo}(Float):
\begin{chunk}{category REAL RealConstant}
)abbrev category REAL RealConstant
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of real numeric domains, i.e. convertible to floats.
@@ 4608,6 +4598,8 @@ digraph pic {
SegmentCategory examples
====================================================================
+This category provides operations on ranges, or segments as they are called.
+
See Also:
o )show SegmentCategory
@@ 4654,13 +4646,6 @@ These are directly exported but not implemented:
++ Author: Stephen M. Watt
++ Date Created: December 1986
++ Date Last Updated: June 3, 1991
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: range, segment
++ Examples:
++ References:
++ Description:
++ This category provides operations on ranges, or segments
++ as they are called.
@@ 4760,6 +4745,11 @@ digraph pic {
SetCategory examples
====================================================================
+SetCategory is the basic category for describing a collection
+of elements with = (equality) and coerce to output form.
+
+Conditional Attributes canonical data structure equality is the same as =
+
See Also:
o )show SetCategory
@@ 4820,15 +4810,7 @@ These exports come from \refto{CoercibleTo}(OutputForm):
\begin{chunk}{category SETCAT SetCategory}
)abbrev category SETCAT SetCategory
++ Author:
++ Date Created:
++ Date Last Updated: November 10, 2009 tpd happy birthday
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ \spadtype{SetCategory} is the basic category for describing a collection
++ of elements with \spadop{=} (equality) and \spadfun{coerce} to
@@ 4931,6 +4913,8 @@ digraph pic {
TranscendentalFunctionCategory examples
====================================================================
+This is the Category for the transcendental elementary functions.
+
See Also:
o )show TranscendentalFunctionCategory
@@ 5039,9 +5023,7 @@ These exports come from \refto{ElementaryFunctionCategory}():
\begin{chunk}{category TRANFUN TranscendentalFunctionCategory}
)abbrev category TRANFUN TranscendentalFunctionCategory
++ Category for the transcendental elementary functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the transcendental elementary functions;
@@ 5170,6 +5152,13 @@ digraph pic {
AbelianSemiGroup examples
====================================================================
+This is the class of all additive (commutative) semigroups, i.e.
+a set with a commutative and associative operation +.
+
+Axioms:
+ associative("+":(%,%)>%) (x+y)+z = x+(y+z)
+ commutative("+":(%,%)>%) x+y = y+x
+
See Also:
o )show AbelianSemiGroup
@@ 5213,15 +5202,6 @@ These exports come from \refto{SetCategory}():
\begin{chunk}{category ABELSG AbelianSemiGroup}
)abbrev category ABELSG AbelianSemiGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of all additive (commutative) semigroups, i.e.
++ a set with a commutative and associative operation \spadop{+}.
@@ 5510,6 +5490,8 @@ digraph pic {
DesingTreeCategory examples
====================================================================
+This category is part of the PAFF package.
+
See Also:
o )show DesingTreeCategory
@@ 5728,6 +5710,9 @@ digraph pic {
FortranFunctionCategory examples
====================================================================
+FortranFunctionCategory is the category of arguments to NAG Library
+routines which return (sets of) function values.
+
See Also:
o )show FortranFunctionCategory
@@ 5780,7 +5765,6 @@ These exports come from \refto{FortranProgramCategory}():
++ Author: Mike Dewar
++ Date Created: 13 January 1994
++ Date Last Updated: 18 March 1994
++ Related Constructors: FortranProgramCategory.
++ Description:
++ \axiomType{FortranFunctionCategory} is the category of arguments to
++ NAG Library routines which return (sets of) function values.
@@ 5911,6 +5895,10 @@ digraph pic {
FortranMatrixCategory examples
====================================================================
+FortranMatrixCategory provides support for producing Functions and
+Subroutines when the input to these is an AXIOM object of type Matrix
+or in domains involving FortranCode.
+
See Also:
o )show FortranMatrixCategory
@@ 5949,8 +5937,6 @@ These exports come from \refto{FortranProgramCategory}():
)abbrev category FMC FortranMatrixCategory
++ Author: Mike Dewar
++ Date Created: 21 March 1994
++ Date Last Updated:
++ Related Constructors: FortranProgramCategory.
++ Description:
++ \axiomType{FortranMatrixCategory} provides support for
++ producing Functions and Subroutines when the input to these
@@ 6057,6 +6043,9 @@ digraph pic {
FortranMatrixFunctionCategory examples
====================================================================
+FortranMatrixFunctionCategory provides support for producing Functions
+and Subroutines representing matrices of expressions.
+
See Also:
o )show FortranMatrixFunctionCategory
@@ 6108,8 +6097,6 @@ These exports come from \refto{FortranProgramCategory}():
)abbrev category FMFUN FortranMatrixFunctionCategory
++ Author: Mike Dewar
++ Date Created: March 18 1994
++ Date Last Updated:
++ Related Constructors: FortranProgramCategory.
++ Description:
++ \axiomType{FortranMatrixFunctionCategory} provides support for
++ producing Functions and Subroutines representing matrices of
@@ 6241,6 +6228,10 @@ digraph pic {
FortranVectorCategory examples
====================================================================
+FortranVectorCategory provides support for producing Functions and
+Subroutines when the input to these is an AXIOM object of type
+Vector or in domains involving FortranCode.
+
See Also:
o )show FortranVectorCategory
@@ 6278,7 +6269,6 @@ These exports come from \refto{FortranProgramCategory}():
++ Author: Mike Dewar
++ Date Created: October 1993
++ Date Last Updated: 18 March 1994
++ Related Constructors: FortranProgramCategory.
++ Description:
++ \axiomType{FortranVectorCategory} provides support for
++ producing Functions and Subroutines when the input to these
@@ 6385,6 +6375,9 @@ digraph pic {
FortranVectorFunctionCategory examples
====================================================================
+FortranVectorFunctionCategory is the catagory of arguments
+to NAG Library routines which return the values of vectors of functions.
+
See Also:
o )show FortranVectorFunctionCategory
@@ 6437,7 +6430,6 @@ These exports come from \refto{FortranProgramCategory}():
++ Author: Mike Dewar
++ Date Created: 11 March 1994
++ Date Last Updated: 18 March 1994
++ Related Constructors: FortranProgramCategory.
++ Description:
++ \axiomType{FortranVectorFunctionCategory} is the catagory of arguments
++ to NAG Library routines which return the values of vectors of functions.
@@ 6572,6 +6564,9 @@ digraph pic {
FullyEvalableOver examples
====================================================================
+This category provides a selection of evaluation operations depending
+on what the argument type R provides.
+
See Also:
o )show FullyEvalableOver
@@ 6618,16 +6613,7 @@ These exports come from \refto{InnerEvalable}(a:Symbol,b:SetCategory):
\begin{chunk}{category FEVALAB FullyEvalableOver}
)abbrev category FEVALAB FullyEvalableOver
++ Author:
++ Date Created:
++ Date Last Updated: June 3, 1991
++ Basic Operations:
++ Related Domains: Equation
++ Also See:
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This category provides a selection of evaluation operations
++ depending on what the argument type R provides.
@@ 6748,6 +6734,11 @@ digraph pic {
FileCategory examples
====================================================================
+This category provides an interface to operate on files in the
+computer's file system. The precise method of naming files
+is determined by the Name parameter. The type of the contents
+of the file is determined by S.
+
See Also:
o )show FileCategory
@@ 6797,15 +6788,7 @@ These exports come from SetCategory():
\begin{chunk}{category FILECAT FileCategory}
)abbrev category FILECAT FileCategory
++ Author: Stephen M. Watt, Victor Miller
++ Date Created:
++ Date Last Updated: June 4, 1991
++ Basic Operations:
++ Related Domains: File
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ This category provides an interface to operate on files in the
++ computer's file system. The precise method of naming files
@@ 6929,6 +6912,14 @@ digraph pic {
Finite examples
====================================================================
+The category of domains composed of a finite set of elements. We include
+the functions lookup and index to give a bijection between the finite set
+and an initial segment of positive integers.
+
+Axioms:
+ lookup(index(n)) = n
+ index(lookup(s)) = s
+
See Also:
o )show Finite
@@ 6973,15 +6964,6 @@ These exports come from \refto{SetCategory}():
\begin{chunk}{category FINITE Finite}
)abbrev category FINITE Finite
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of domains composed of a finite set of elements.
++ We include the functions \spadfun{lookup} and \spadfun{index}
@@ 7086,6 +7068,8 @@ digraph pic {
FileNameCategory examples
====================================================================
+This category provides an interface to names in the file system.
+
See Also:
o )show FileNameCategory
@@ 7140,13 +7124,6 @@ These exports come from \refto{SetCategory}():
++ Author: Stephen M. Watt
++ Date Created: 1985
++ Date Last Updated: June 20, 1991
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ This category provides an interface to names in the file system.
@@ 7262,6 +7239,15 @@ digraph pic {
GradedModule examples
====================================================================
+GradedModule(R,E) denotes "Egraded Rmodule", i.e. collection of
+Rmodules indexed by an abelian monoid E. An element g of G[s] for
+some specific s in E is said to be an element of G with degree s.
+Sums are defined in each module G[s] so two elements of G have a
+sum if they have the same degree.
+
+Morphisms can be defined and composed by degree to give the mathematical
+category of graded modules.
+
See Also:
o )show GradedModule
@@ 7316,12 +7302,6 @@ These exports come from \refto{SetCategory}():
++ Author: Stephen M. Watt
++ Date Created: May 20, 1991
++ Date Last Updated: May 20, 1991
++ Basic Operations: +, *, degree
++ Related Domains: CartesianTensor(n,dim,R)
++ Also See:
++ AMS Classifications:
++ Keywords: graded module, tensor, multilinear algebra
++ Examples:
++ References: Algebra 2d Edition, MacLane and Birkhoff, MacMillan 1979
++ Description:
++ GradedModule(R,E) denotes ``Egraded Rmodule'', i.e. collection of
@@ 7456,6 +7436,14 @@ digraph pic {
HomogeneousAggregate examples
====================================================================
+A homogeneous aggregate is an aggregate of elements all of the same type.
+
+In the current system, all aggregates are homogeneous. Two attributes
+characterize classes of aggregates. Aggregates from domains with
+attribute "finiteAggregate" have a finite number of members. Those
+with attribute "shallowlyMutable" allow an element to be modified
+or updated without changing its overall value.
+
See Also:
o )show HomogeneousAggregate
@@ 7575,12 +7563,6 @@ These exports come from \refto{SetCategory}():
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991, May 1995
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A homogeneous aggregate is an aggregate of elements all of the
++ same type.
@@ 7737,6 +7719,9 @@ digraph pic {
IndexedDirectProductCategory examples
====================================================================
+This category represents the direct product of some set with respect
+to an ordered indexing set.
+
See Also:
o )show IndexedDirectProductCategory
@@ 7781,14 +7766,6 @@ These exports come from \refto{SetCategory}():
\begin{chunk}{category IDPC IndexedDirectProductCategory}
)abbrev category IDPC IndexedDirectProductCategory
++ Author: James Davenport
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category represents the direct product of some set with
++ respect to an ordered indexing set.
@@ 7909,6 +7886,8 @@ digraph pic {
LiouvillianFunctionCategory examples
====================================================================
+This is the Category for the transcendental Liouvillian functions.
+
See Also:
o )show LiouvillianFunctionCategory
@@ 8014,7 +7993,6 @@ These exports come from \refto{TranscendentalFunctionCategory}():
)abbrev category LFCAT LiouvillianFunctionCategory
++ Category for the transcendental Liouvillian functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the transcendental Liouvillian functions;
@@ 8142,6 +8120,9 @@ digraph pic {
Monad examples
====================================================================
+Monad is the class of all multiplicative monads, i.e. sets
+with a binary operation.
+
See Also:
o )show Monad
@@ 8191,11 +8172,6 @@ These exports come from \refto{SetCategory}():
++ Authors: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: *, **
++ Related Constructors: SemiGroup, Monoid, MonadWithUnit
++ Also See:
++ AMS Classifications:
++ Keywords: Monad, binary operation
++ Reference:
++ N. Jacobson: Structure and Representations of Jordan Algebras
++ AMS, Providence, 1968
@@ 8323,6 +8299,9 @@ digraph pic {
NumericalIntegrationCategory examples
====================================================================
+NumericalIntegrationCategory is the category for describing the set of
+Numerical Integration domains with measure and numericalIntegration.
+
See Also:
o )show NumericalIntegrationCategory
@@ 8520,6 +8499,9 @@ digraph pic {
NumericalOptimizationCategory examples
====================================================================
+NumericalOptimizationCategory is the category for describing the set of
+Numerical Optimization domains with measure and optimize.
+
See Also:
o )show NumericalOptimizationCategory
@@ 8709,6 +8691,9 @@ digraph pic {
OrdinaryDifferentialEquationsSolverCategory examples
====================================================================
+OrdinaryDifferentialEquationsSolverCategory is the category for describing
+the set of ODE solver domains with measure and ODEsolve.
+
See Also:
o )show OrdinaryDifferentialEquationsSolverCategory
@@ 8768,7 +8753,6 @@ These exports come from \refto{SetCategory}():
++ Author: Brian Dupee
++ Date Created: February 1995
++ Date Last Updated: June 1995
++ Basic Operations:
++ Description:
++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} is the
++ \axiom{category} for describing the set of ODE solver \axiom{domains}
@@ 8879,6 +8863,10 @@ digraph pic {
OrderedSet examples
====================================================================
+The class of totally ordered sets, that is, sets such that for each
+pair of elements (a,b) exactly one of the following relations holds
+a a%) (x*y)*z = x*(y*z)
+
+Conditional attributes:
+ commutative("*":(%,%)>%) x*y = y*x
+
See Also:
o )show SemiGroup
@@ 9836,15 +9819,6 @@ These exports come from \refto{SetCategory}():
\begin{chunk}{category SGROUP SemiGroup}
)abbrev category SGROUP SemiGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ the class of all multiplicative semigroups, i.e. a set
++ with an associative operation \spadop{*}.
@@ 9950,6 +9924,8 @@ digraph pic {
SetCategoryWithDegree examples
====================================================================
+This is part of the PAFF package, related to projective space.
+
See Also:
o )show SetCategoryWithDegree
@@ 9992,7 +9968,6 @@ These exports come from \refto{SetCategory}():
++ Author: Gaetan Hache
++ Date Created: 17 nov 1992
++ Date Last Updated: May 2010 by Tim Daly
++ Keywords:
++ Description:
++ This is part of the PAFF package, related to projective space.
SetCategoryWithDegree:Category == SetCategory with
@@ 10088,6 +10063,9 @@ digraph pic {
SExpressionCategory examples
====================================================================
+This category allows the manipulation of Lisp values while keeping
+the grunge fairly localized.
+
See Also:
o )show SExpressionCategory
@@ 10167,7 +10145,6 @@ These exports come from \refto{SetCategory}():
\begin{chunk}{category SEXCAT SExpressionCategory}
)abbrev category SEXCAT SExpressionCategory
++ Category for Lisp values
++ Author: S.M.Watt
++ Date Created: July 1987
++ Date Last Modified: 23 May 1991
@@ 10320,6 +10297,18 @@ digraph pic {
StepThrough examples
====================================================================
+A class of objects which can be 'stepped through'.
+
+Repeated applications of nextItem is guaranteed never to return
+duplicate items and only return "failed" after exhausting all
+elements of the domain. This assumes that the sequence starts
+with init(). For infinite domains, repeated application of nextItem
+is not required to reach all possible domain elements starting from
+any initial element.
+
+Conditional attributes:
+ infinite  repeated nextItem's are never "failed".
+
See Also:
o )show StepThrough
@@ 10361,15 +10350,6 @@ These exports come from \refto{SetCategory}():
\begin{chunk}{category STEP StepThrough}
)abbrev category STEP StepThrough
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A class of objects which can be 'stepped through'.
++ Repeated applications of \spadfun{nextItem} is guaranteed never to
@@ 10497,6 +10477,10 @@ digraph pic {
ThreeSpaceCategory examples
====================================================================
+The category ThreeSpaceCategory is used for creating three dimensional
+objects using functions for defining points, curves, polygons,
+constructs and the subspaces containing them.
+
See Also:
o )show ThreeSpaceCategory
@@ 10615,19 +10599,6 @@ These exports come from \refto{SetCategory}():
\begin{chunk}{category SPACEC ThreeSpaceCategory}
)abbrev category SPACEC ThreeSpaceCategory
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Operations: create3Space, numberOfComponents, numberOfComposites,
++ merge, composite, components, copy, enterPointData, modifyPointData,
++ point, point?, curve, curve?, closedCurve, closedCurve?, polygon,
++ polygon? mesh, mesh?, lp, lllip, lllp, llprop, lprop, objects,
++ check, subspace, coerce
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category ThreeSpaceCategory is used for creating
++ three dimensional objects using functions for defining points, curves,
@@ 10986,6 +10957,13 @@ digraph pic {
AbelianMonoid examples
====================================================================
+The class of multiplicative monoids, i.e. semigroups with an
+additive identity element.
+
+Axioms:
+ leftIdentity("+":(%,%)>%,0) 0+x=x
+ rightIdentity("+":(%,%)>%,0) x+0=x
+
See Also:
o )show AbelianMonoid
@@ 11038,15 +11016,6 @@ These exports come from \refto{AbelianSemiGroup}():
\begin{chunk}{category ABELMON AbelianMonoid}
)abbrev category ABELMON AbelianMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of multiplicative monoids, i.e. semigroups with an
++ additive identity element.
@@ 11174,6 +11143,9 @@ digraph pic {
AffineSpaceCategory examples
====================================================================
+The following is all the categories and domains related to projective
+space and part of the PAFF package
+
See Also:
o )show AffineSpaceCategory
@@ 11415,6 +11387,10 @@ digraph pic {
BagAggregate examples
====================================================================
+A bag aggregate is an aggregate for which one can insert and extract
+objects, and where the order in which objects are inserted determines the
+order of extraction. Examples of bags are stacks, queues, and dequeues.
+
See Also:
o )show BagAggregate
@@ 11524,12 +11500,6 @@ These exports come from \refto{HomogeneousAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A bag aggregate is an aggregate for which one can insert and extract
++ objects, and where the order in which objects are inserted determines
@@ 11631,6 +11601,9 @@ digraph pic {
CachableSet examples
====================================================================
+A cachable set is a set whose elements keep an integer as part
+of their structure.
+
See Also:
o )show CachableSet
@@ 11680,7 +11653,6 @@ These exports come from \refto{OrderedSet}():
\begin{chunk}{category CACHSET CachableSet}
)abbrev category CACHSET CachableSet
++ Sets whose elements can cache an integer
++ Author: Manuel Bronstein
++ Date Created: 31 Oct 1988
++ Date Last Updated: 14 May 1991
@@ 11802,6 +11774,13 @@ digraph pic {
Collection examples
====================================================================
+A collection is a homogeneous aggregate which can built from
+list of members. The operation used to build the aggregate is
+generically named construct. However, each collection provides
+its own special function with the same name as the data type,
+except with an initial lower case letter, e.g.
+list for List, flexibleArray for FlexibleArray, and so on.
+
See Also:
o )show Collection
@@ 11930,12 +11909,6 @@ These exports come from \refto{ConvertibleTo}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A collection is a homogeneous aggregate which can built from
++ list of members. The operation used to build the aggregate is
@@ 12109,6 +12082,42 @@ digraph pic {
DifferentialVariableCategory examples
====================================================================
+DifferentialVariableCategory constructs the set of derivatives of a
+given set of (ordinary) differential indeterminates. If x,...,y is
+an ordered set of differential indeterminates, and the prime notation
+is used for differentiation, then the set of derivatives (including
+zeroth order) of the differential indeterminates is
+ x, x', x'',..., y, y', y'',...
+(Note that in the interpreter, the nth derivative of y is displayed as
+y with a subscript n.) This set is viewed as a set of algebraic
+indeterminates, totally ordered in a way compatible with differentiation
+and the given order on the differential indeterminates. Such a total
+order is called a ranking of the differential indeterminates.
+
+A domain in this category is needed to construct a differential
+polynomial domain. Differential polynomials are ordered by a ranking
+on the derivatives, and by an order (extending the ranking) on the set
+of differential monomials. One may thus associate a domain in this
+category with a ranking of the differential indeterminates, just as
+one associates a domain in the category OrderedAbelianMonoidSup with
+an ordering of the set of monomials in a set of algebraic indeterminates.
+The ranking is specified through the binary relation <. For example, one
+may define one derivative to be less than another by lexicographically
+comparing first the order, then the given order of the differential
+indeterminates appearing in the derivatives. This is the default
+implementation.
+
+The notion of weight generalizes that of degree. A polynomial domain
+may be made into a graded ring if a weight function is given on the set
+of indeterminates. Very often, a grading is the first step in ordering
+the set of monomials. For differential polynomial domains, this
+constructor provides a function \spadfun{weight}, which allows the
+assignment of a nonnegative number to each derivative of a differential
+indeterminate. For example, one may define the weight of a derivative
+to be simply its order (this is the default assignment). This weight
+function can then be extended to the set of all differential polynomials,
+providing a graded ring structure.
+
See Also:
o )show DifferentialVariableCategory
@@ 12182,13 +12191,6 @@ These exports come from \refto{RetractableTo}(S:OrderedSet):
++ Author: William Sit
++ Date Created: 19 July 1990
++ Date Last Updated: 13 September 1991
++ Basic Operations:
++ Related Constructors:DifferentialPolynomialCategory
++ See Also:OrderedDifferentialVariable,
++ SequentialDifferentialVariable,
++ DifferentialSparseMultivariatePolynomial.
++ AMS Classifications:12H05
++ Keywords: differential indeterminates, ranking, order, weight
++ References:Ritt, J.F. "Differential Algebra" (Dover, 1950).
++ Description:
++ \spadtype{DifferentialVariableCategory} constructs the
@@ 12422,6 +12424,8 @@ digraph pic {
ExpressionSpace examples
====================================================================
+An expression space is a set which is closed under certain operators.
+
See Also:
o )show ExpressionSpace
@@ 12563,7 +12567,6 @@ These exports come from \refto{Evalable}(a:SetCategory):
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Date Last Updated: 27 May 1994
++ Keywords: operator, kernel, expression, space.
++ Description:
++ An expression space is a set which is closed under certain operators;
@@ 12991,6 +12994,13 @@ digraph pic {
GradedAlgebra examples
====================================================================
+GradedAlgebra(R,E) denotes "Egraded Ralgebra". A graded algebra is a
+graded module together with a degree preserving Rlinear map, called
+the product.
+
+The name "product" is written out in full so inner and outer products
+with the same mapping type can be distinguished by name.
+
See Also:
o )show GradedAlgebra
@@ 13059,12 +13069,6 @@ These exports come from \refto{RetractableTo}(R:CommutativeRing):
++ Author: Stephen M. Watt
++ Date Created: May 20, 1991
++ Date Last Updated: May 20, 1991
++ Basic Operations: +, *, degree
++ Related Domains: CartesianTensor(n,dim,R)
++ Also See:
++ AMS Classifications:
++ Keywords: graded module, tensor, multilinear algebra
++ Examples:
++ References: Encyclopedic Dictionary of Mathematics, MIT Press, 1977
++ Description:
++ GradedAlgebra(R,E) denotes ``Egraded Ralgebra''.
@@ 13215,6 +13219,11 @@ digraph pic {
IndexedAggregate examples
====================================================================
+An indexed aggregate is a manytoone mapping of indices to entries.
+For example, a onedimensionalarray is an indexed aggregate where
+the index is an integer. Also, a table is an indexed aggregate
+where the indices and entries may have any type.
+
See Also:
o )show IndexedAggregate
@@ 13356,12 +13365,6 @@ These exports come from \refto{EltableAggregate}(Index:SetCategory,Entry:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An indexed aggregate is a manytoone mapping of indices to entries.
++ For example, a onedimensionalarray is an indexed aggregate where
@@ 13540,6 +13543,16 @@ digraph pic {
MonadWithUnit examples
====================================================================
+MonadWithUnit is the class of multiplicative monads with unit,
+i.e. sets with a binary operation and a unit element.
+
+Axioms:
+ leftIdentity("*":(%,%)>%,1) 1*x=x
+ rightIdentity("*":(%,%)>%,1) x*1=x
+
+Common Additional Axioms:
+ unitsKnown  if "recip" says "failed", it PROVES input wasn't a unit
+
See Also:
o )show MonadWithUnit
@@ 13602,12 +13615,6 @@ These exports come from \refto{Monad}():
++ Authors: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: *, **, 1
++ Related Constructors: SemiGroup, Monoid, Monad
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Keywords: Monad with unit, binary operation
++ Reference:
++ N. Jacobson: Structure and Representations of Jordan Algebras
++ AMS, Providence, 1968
@@ 13759,6 +13766,16 @@ digraph pic {
Monoid examples
====================================================================
+The class of multiplicative monoids, i.e. semigroups with a
+multiplicative identity element.
+
+Axioms:
+ leftIdentity("*":(%,%)>%,1) 1*x=x
+ rightIdentity("*":(%,%)>%,1) x*1=x
+
+Conditional attributes:
+ unitsKnown  \spadfun{recip} only returns "failed" on nonunits
+
See Also:
o )show Monoid
@@ 13817,15 +13834,6 @@ These exports come from \refto{SemiGroup}():
\begin{chunk}{category MONOID Monoid}
)abbrev category MONOID Monoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of multiplicative monoids, i.e. semigroups with a
++ multiplicative identity element.
@@ 13956,6 +13964,8 @@ digraph pic {
OrderedFinite examples
====================================================================
+This is the category of Ordered finite sets.
+
See Also:
o )show OrderedFinite
@@ 14010,15 +14020,6 @@ These exports come from \refto{Finite}():
\begin{chunk}{category ORDFIN OrderedFinite}
)abbrev category ORDFIN OrderedFinite
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered finite sets.
@@ 14120,6 +14121,8 @@ digraph pic {
PlacesCategory examples
====================================================================
+This is part of the PAFF package, related to projective space.
+
See Also:
o )show PlacesCategory
@@ 14340,6 +14343,8 @@ digraph pic {
ProjectiveSpaceCategory examples
====================================================================
+This is part of the PAFF package, related to projective space.
+
See Also:
o )show ProjectiveSpaceCategory
@@ 14606,6 +14611,13 @@ digraph pic {
RecursiveAggregate examples
====================================================================
+A recursive aggregate over a type S is a model for a a directed graph
+containing values of type S. Recursively, a recursive aggregate is a node
+consisting of a value from S and 0 or more children which are recursive
+aggregates. A node with no children is called a leaf node. A recursive
+aggregate may be cyclic for which some operations as noted may go into
+an infinite loop.
+
See Also:
o )show RecursiveAggregate
@@ 14736,12 +14748,6 @@ These exports come from \refto{HomogeneousAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A recursive aggregate over a type S is a model for a
++ a directed graph containing values of type S.
@@ 14906,6 +14912,8 @@ first column in an array and vice versa.
TwoDimensionalArrayCategory examples
====================================================================
+This is the category of two dimensional array categories and domains.
+
See Also:
o )show TwoDimensionalArrayCategory
@@ 15039,12 +15047,9 @@ These exports come from \refto{HomogeneousAggregate}(R:Type)
\begin{chunk}{category ARR2CAT TwoDimensionalArrayCategory}
)abbrev category ARR2CAT TwoDimensionalArrayCategory
++ Author:
++ Date Created: 27 October 1989
++ Date Last Updated: 27 June 1990
++ Keywords: array, data structure
++ Examples:
++ References:
++ Description:
++ Two dimensional array categories and domains
@@ 15521,6 +15526,9 @@ digraph pic {
BinaryRecursiveAggregate examples
====================================================================
+A binaryrecursive aggregate has 0, 1 or 2 children and serves
+as a model for a binary tree or a doublylinked aggregate structure
+
See Also:
o )show BinaryRecursiveAggregate
@@ 15665,12 +15673,6 @@ These exports come from \refto{RecursiveAggregate}(S:Type)
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A binaryrecursive aggregate has 0, 1 or 2 children and serves
++ as a model for a binary tree or a doublylinked aggregate structure
@@ 15854,6 +15856,12 @@ digraph pic {
CancellationAbelianMonoid examples
====================================================================
+This is an AbelianMonoid with the cancellation property, i.e.
+ a+b = a+c => b=c
+This is formalised by the partial subtraction operator, which satisfies
+the Axiom
+ c = a+b <=> cb = a
+
See Also:
o )show CancellationAbelianMonoid
@@ 15903,15 +15911,6 @@ These exports come from \refto{AbelianMonoid}():
\begin{chunk}{category CABMON CancellationAbelianMonoid}
)abbrev category CABMON CancellationAbelianMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References: Davenport & Trager I
++ Description:
++ This is an \spadtype{AbelianMonoid} with the cancellation property, i.e.\br
++ \tab{5}\spad{ a+b = a+c => b=c }.\br
@@ 16051,6 +16050,9 @@ digraph pic {
DictionaryOperations examples
====================================================================
+This category is a collection of operations common to both
+categories Dictionary and MultiDictionary.
+
See Also:
o )show DictionaryOperations
@@ 16196,12 +16198,6 @@ These exports come from \refto{Collection}(S:SetCategory)
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category is a collection of operations common to both
++ categories \spadtype{Dictionary} and \spadtype{MultiDictionary}
@@ 16360,6 +16356,10 @@ digraph pic {
DoublyLinkedAggregate examples
====================================================================
+A doublylinked aggregate serves as a model for a doublylinked
+list, that is, a list which can has links to both next and previous
+nodes and thus can be efficiently traversed in both directions.
+
See Also:
o )show DoublyLinkedAggregate
@@ 16499,12 +16499,6 @@ These exports come from \refto{RecursiveAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A doublylinked aggregate serves as a model for a doublylinked
++ list, that is, a list which can has links to both next and previous
@@ 16621,6 +16615,12 @@ digraph pic {
Group examples
====================================================================
+The class of multiplicative groups, i.e. monoids with multiplicative inverses.
+
+Axioms:
+ leftInverse("*":(%,%)>%,inv) inv(x)*x = 1
+ rightInverse("*":(%,%)>%,inv) x*inv(x) = 1
+
See Also:
o )show Group
@@ 16694,15 +16694,6 @@ These exports come from \refto{Monoid}():
\begin{chunk}{category GROUP Group}
)abbrev category GROUP Group
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of multiplicative groups, i.e. monoids with
++ multiplicative inverses.
@@ 16883,6 +16874,17 @@ digraph pic {
LinearAggregate examples
====================================================================
+A linear aggregate is an aggregate whose elements are indexed by integers.
+Examples of linear aggregates are strings, lists, and arrays.
+
+Most of the exported operations for linear aggregates are nondestructive
+but are not always efficient for a particular aggregate.
+
+For example, concat of two lists needs only to copy its first argument,
+whereas concat of two arrays needs to copy both arguments. Most of the
+operations exported here apply to infinite objects (e.g. streams) as well
+to finite ones. For finite linear aggregates, see FiniteLinearAggregate.
+
See Also:
o )show LinearAggregate
@@ 17058,12 +17060,6 @@ These exports come from \refto{Collection}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A linear aggregate is an aggregate whose elements are indexed by integers.
++ Examples of linear aggregates are strings, lists, and
@@ 18093,6 +18089,16 @@ inverse matrix [[j**i for i in 0..4] for j in 1..5]
MatrixCategory examples
====================================================================
+MatrixCategory is a general matrix category which allows different
+representations and indexing schemes. Rows and columns may be
+extracted with rows returned as objects of type Row and colums
+returned as objects of type Col. A domain belonging to this category
+will be shallowly mutable. The index of the 'first' row may be
+obtained by calling the function minRowIndex. The index of the
+'first' column may be obtained by calling the function minColIndex.
+The index of the first element of a Row is the same as the index of the
+first column in a matrix and vice versa.
+
Predicates:
square?(m) returns true if m is a square matrix
@@ 18595,13 +18601,6 @@ Col:FiniteLinearAggregate(R):
++ Authors: Grabmeier, Gschnitzer, Williamson, Gabriel Dos Reis
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: Matrix(R)
++ Also See:
++ AMS Classifications:
++ Keywords: matrix, linear algebra
++ Examples:
++ References:
++ Description:
++ \spadtype{MatrixCategory} is a general matrix category which allows
++ different representations and indexing schemes. Rows and
@@ 19456,6 +19455,12 @@ digraph pic {
OrderedAbelianSemiGroup examples
====================================================================
+Ordered sets which are also abelian semigroups, such that the addition
+preserves the ordering.
+
+Axiom:
+ x < y => x+z < y+z
+
See Also:
o )show OrderedAbelianSemiGroup
@@ 19514,15 +19519,6 @@ These exports come from \refto{AbelianMonoid}():
\begin{chunk}{category OASGP OrderedAbelianSemiGroup}
)abbrev category OASGP OrderedAbelianSemiGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also abelian semigroups, such that the addition
++ preserves the ordering.\br
@@ 19635,6 +19631,13 @@ digraph pic {
OrderedMonoid examples
====================================================================
+Ordered sets which are also monoids, such that multiplication
+preserves the ordering.
+
+Axioms:
+ x < y => x*z < y*z
+ x < y => z*x < z*y
+
See Also:
o )show OrderedMonoid
@@ 19697,15 +19700,6 @@ These exports come from \refto{OrderedSet}():
\begin{chunk}{category ORDMON OrderedMonoid}
)abbrev category ORDMON OrderedMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also monoids, such that multiplication
++ preserves the ordering.
@@ 19859,6 +19853,16 @@ digraph pic {
PolynomialSetCategory examples
====================================================================
+A category for finite subsets of a polynomial ring. Such a set is
+only regarded as a set of polynomials and not identified to the ideal
+it generates. So two distinct sets may generate the same the ideal.
+Furthermore, for R being an integral domain, a set of polynomials may
+be viewed as a representation of the ideal it generates in the polynomial
+ring (R)^(1) P, or the set of its zeros (described for instance by the
+radical of the previous ideal, or a split of the associated affine
+variety) and so on. So this category provides operations about
+those different notions.
+
See Also:
o )show PolynomialSetCategory
@@ 20023,12 +20027,6 @@ These exports come from \refto{IntegralDomain}():
++ Author: Marc Moreno Maza
++ Date Created: 04/26/1994
++ Date Last Updated: 12/15/1998
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References:
++ Description:
++ A category for finite subsets of a polynomial ring.
++ Such a set is only regarded as a set of polynomials and not
@@ 20523,6 +20521,9 @@ digraph pic {
PriorityQueueAggregate examples
====================================================================
+A priority queue is a bag of items from an ordered set where the item
+extracted is always the maximum element.
+
See Also:
o )show PriorityQueueAggregate
@@ 20632,12 +20633,6 @@ These exports come from \refto{BagAggregate}(S:OrderedSet):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A priority queue is a bag of items from an ordered set where the item
++ extracted is always the maximum element.
@@ 20761,6 +20756,8 @@ digraph pic {
QueueAggregate examples
====================================================================
+A queue is a bag where the first item inserted is the first item extracted.
+
See Also:
o )show QueueAggregate
@@ 20876,12 +20873,6 @@ These exports come from \refto{BagAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A queue is a bag where the first item inserted is the first
++ item extracted.
@@ 21021,6 +21012,12 @@ digraph pic {
SetAggregate examples
====================================================================
+A set category lists a collection of settheoretic operations useful
+for both finite sets and multisets. Note however that finite sets are
+distinct from multisets. Although the operations defined for set
+categories are common to both, the relationship between the two cannot
+be described by inclusion or inheritance.
+
See Also:
o )show SetAggregate
@@ 21179,12 +21176,6 @@ These exports come from \refto{Collection}(S:SetCategory):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: 14 Oct, 1993 by RSS
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A set category lists a collection of settheoretic operations
++ useful for both finite sets and multisets.
@@ 21371,6 +21362,8 @@ digraph pic {
StackAggregate examples
====================================================================
+A stack is a bag where the last item inserted is the first item extracted.
+
See Also:
o )show StackAggregate
@@ 21482,12 +21475,6 @@ These exports come from \refto{BagAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A stack is a bag where the last item inserted is the first item extracted.
@@ 21650,6 +21637,16 @@ digraph pic {
UnaryRecursiveAggregate examples
====================================================================
+A unaryrecursive aggregate is a one where nodes may have either
+0 or 1 children. This aggregate models, though not precisely, a linked
+list possibly with a single cycle.
+
+A node with one children models a nonempty list, with the value of the
+list designating the head, or first, of the list, and the child
+designating the tail, or rest, of the list. A node with no child then
+designates the empty list. Since these aggregates are recursive aggregates,
+they may be cyclic.
+
See Also:
o )show UnaryRecursiveAggregate
@@ 21826,12 +21823,6 @@ These exports come from \refto{RecursiveAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A unaryrecursive aggregate is a one where nodes may have either
++ 0 or 1 children.
@@ 22163,6 +22154,13 @@ digraph pic {
AbelianGroup examples
====================================================================
+The class of abelian groups, i.e. additive monoids where each element
+has an additive inverse.
+
+Axioms:
+ (x) = x
+ x+(x) = 0
+
See Also:
o )show AbelianGroup
@@ 22225,15 +22223,6 @@ These exports come from \refto{CancellationAbelianMonoid}():
\begin{chunk}{category ABELGRP AbelianGroup}
)abbrev category ABELGRP AbelianGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of abelian groups, i.e. additive monoids where
++ each element has an additive inverse.
@@ 22384,6 +22373,10 @@ digraph pic {
BinaryTreeCategory examples
====================================================================
+BinaryTreeCategory(S) is the category of binary trees: a tree which
+is either empty or else is a node consisting of a value and a left and
+right, both binary trees.
+
See Also:
o )show BinaryTreeCategory
@@ 22514,14 +22507,6 @@ These exports come from \refto{BinaryRecursiveAggregate}(S:SetCategory):
)abbrev category BTCAT BinaryTreeCategory
++ Author:W. H. Burge
++ Date Created:17 Feb 1992
++ Date Last Updated:
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ \spadtype{BinaryTreeCategory(S)} is the category of
++ binary trees: a tree which is either empty or else is a
@@ 22685,6 +22670,13 @@ digraph pic {
Dictionary examples
====================================================================
+A dictionary is an aggregate in which entries can be inserted,
+searched for and removed. Duplicates are thrown away on insertion.
+This category models the usual notion of dictionary which involves
+large amounts of data where copying is impractical.
+
+Principal operations are thus destructive (noncopying) ones.
+
See Also:
o )show Dictionary
@@ 22821,12 +22813,6 @@ These exports come from \refto{DictionaryOperations}(S:SetCategory):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A dictionary is an aggregate in which entries can be inserted,
++ searched for and removed. Duplicates are thrown away on insertion.
@@ 22983,6 +22969,10 @@ digraph pic {
DequeueAggregate examples
====================================================================
+A dequeue is a doubly ended stack, that is, a bag where first items
+inserted are the first items extracted, at either the front or
+the back end of the data structure.
+
See Also:
o )show DequeueAggregate
@@ 23130,12 +23120,6 @@ These exports come from \refto{QueueAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A dequeue is a doubly ended stack, that is, a bag where first items
++ inserted are the first items extracted, at either the front or
@@ 23313,8 +23297,16 @@ digraph pic {
ExtensibleLinearAggregate examples
====================================================================
+An extensible aggregate is one which allows insertion and deletion of
+entries. These aggregates are models of lists and streams which are
+represented by linked structures so as to make insertion, deletion, and
+concatenation efficient. However, access to elements of these
+extensible aggregates is generally slow since access is made from the end.
+See FlexibleArray for an exception.
+
See Also:
o )show ExtensibleLinearAggregate
+o )show FlexibleArray
\end{chunk}
{\bf See:}
@@ 23500,12 +23492,6 @@ These exports come from \refto{LinearAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An extensible aggregate is one which allows insertion and deletion of
++ entries. These aggregates are models of lists and streams which are
@@ 23715,6 +23701,10 @@ digraph pic {
FiniteLinearAggregate examples
====================================================================
+A finite linear aggregate is a linear aggregate of finite length.
+The finite property of the aggregate adds several exports to the
+list of exports from LinearAggregate such as reverse, sort, and so on.
+
See Also:
o )show FiniteLinearAggregate
@@ 23937,12 +23927,6 @@ These exports come from \refto{OrderedSet}:
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A finite linear aggregate is a linear aggregate of finite length.
++ The finite property of the aggregate adds several exports to the
@@ 24111,6 +24095,10 @@ digraph pic {
FreeAbelianMonoidCategory examples
====================================================================
+A free abelian monoid on a set S is the monoid of finite sums of
+the form reduce(+,[ni * si]) where the si's are in S, and the ni's
+are in a given abelian monoid. The operation is commutative.
+
See Also:
o )show FreeAbelianMonoidCategory
@@ 24372,6 +24360,10 @@ digraph pic {
MultiDictionary examples
====================================================================
+A multidictionary is a dictionary which may contain duplicates.
+As for any dictionary, its size is assumed large so that
+copying (nondestructive) operations are generally to be avoided.
+
See Also:
o )show MultiDictionary
@@ 24509,12 +24501,6 @@ These exports come from \refto{DictionaryOperations}(S:SetCategory):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A multidictionary is a dictionary which may contain duplicates.
++ As for any dictionary, its size is assumed large so that
@@ 24619,6 +24605,9 @@ digraph pic {
OrderedAbelianMonoid examples
====================================================================
+Ordered sets which are also abelian monoids, such that the addition
+preserves the ordering.
+
See Also:
o )show OrderedAbelianMonoid
@@ 24677,15 +24666,6 @@ These exports come from \refto{AbelianMonoid}():
\begin{chunk}{category OAMON OrderedAbelianMonoid}
)abbrev category OAMON OrderedAbelianMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also abelian monoids, such that the addition
++ preserves the ordering.
@@ 24783,6 +24763,9 @@ digraph pic {
PermutationCategory examples
====================================================================
+PermutationCategory provides a categorial environment for subgroups
+of bijections of a set (i.e. permutations)
+
See Also:
o )show PermutationCategory
@@ 24879,12 +24862,6 @@ These exports come from \refto{OrderedSet}():
++ Authors: Holger Gollan, Johannes Grabmeier, Gerhard Schneider
++ Date Created: 27 July 1989
++ Date Last Updated: 29 March 1990
++ Basic Operations: cycle, cycles, eval, orbit
++ Related Constructors: PermutationGroup, PermutationGroupExamples
++ Also See: RepresentationTheoryPackage1
++ AMS Classifications:
++ Keywords: permutation, symmetric group
++ References:
++ Description:
++ PermutationCategory provides a categorial environment
++ for subgroups of bijections of a set (i.e. permutations)
@@ 25094,8 +25071,15 @@ digraph pic {
StreamAggregate examples
====================================================================
+A stream aggregate is a linear aggregate which possibly has an infinite
+number of elements. A basic domain constructor which builds stream
+aggregates is Stream. From streams, a number of infinite structures
+such power series can be built. A stream aggregate may also be infinite
+since it may be cyclic. For example, see DecimalExpansion.
+
See Also:
o )show StreamAggregate
+o )show DecimalExpansion
\end{chunk}
{\bf See:}
@@ 25350,12 +25334,6 @@ These exports come from \refto{LinearAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A stream aggregate is a linear aggregate which possibly has an infinite
++ number of elements. A basic domain constructor which builds stream
@@ 25595,6 +25573,28 @@ digraph pic {
TriangularSetCategory examples
====================================================================
+The category of triangular sets of multivariate polynomials with
+coefficients in an integral domain.
+
+Let R be an integral domain and V a finite ordered set of variables,
+ X1 < X2 < ... < Xn
+
+A set S of polynomials in R[X1,X2,...,Xn] is triangular if no elements
+of S lies in R, and if two distinct elements of S have distinct main
+variables.
+
+Note that the empty set is a triangular set. A triangular set is not
+necessarily a (lexicographical) Groebner basis and the notion of
+reduction related to triangular sets is based on the recursive view
+of polynomials. We recall this notion here. For details see
+ P. AUBRY, D. LAZARD and M. MORENO MAZA "On the Theories
+ of Triangular Sets" Journal of Symbol. Comp.
+
+A polynomial P is reduced with respect to a nonconstant polynomial Q
+if the degree of P in the main variable of Q is less than the main
+degree of Q. A polynomial P is reduced with respect to a triangular
+set T if it is reduced with respect to every polynomial of T.
+
See Also:
o )show TriangularSetCategory
@@ 25825,11 +25825,6 @@ V:OrderedSet, P:RecursivePolynomialCategory(R,E,V)):
++ Author: Marc Moreno Maza (marc@nag.co.uk)
++ Date Created: 04/26/1994
++ Date Last Updated: 12/15/1998
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References :
++ [1] P. AUBRY, D. LAZARD and M. MORENO MAZA "On the Theories
++ of Triangular Sets" Journal of Symbol. Comp. (to appear)
@@ 26457,6 +26452,10 @@ digraph pic {
FiniteDivisorCategory examples
====================================================================
+This category describes finite rational divisors on a curve, that
+is finite formal sums SUM(n * P) where the n's are integers and the
+P's are finite rational points on the curve.
+
See Also:
o )show FiniteDivisorCategory
@@ 26533,7 +26532,6 @@ These exports come from \refto{AbelianGroup}():
++ Author: Manuel Bronstein
++ Date Created: 19 May 1993
++ Date Last Updated: 19 May 1993
++ Keywords: divisor, algebraic, curve.
++ Description:
++ This category describes finite rational divisors on a curve, that
++ is finite formal sums SUM(n * P) where the n's are integers and the
@@ 26716,8 +26714,13 @@ digraph pic {
FiniteSetAggregate examples
====================================================================
+A finiteset aggregate models the notion of a finite set, that is,
+a collection of elements characterized by membership, but not
+by order or multiplicity. See Set for an example.
+
See Also:
o )show FiniteSetAggregate
+o )show Set
\end{chunk}
{\bf See:}
@@ 26910,12 +26913,6 @@ These exports come from \refto{SetAggregate}(S:SetCategory):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: 14 Oct, 1993 by RSS
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A finiteset aggregate models the notion of a finite set, that is,
++ a collection of elements characterized by membership, but not
@@ 27123,6 +27120,9 @@ digraph pic {
KeyedDictionary examples
====================================================================
+A keyed dictionary is a dictionary of keyentry pairs for which there is
+a unique entry for each key.
+
See Also:
o )show KeyedDictionary
@@ 27282,12 +27282,6 @@ and S=Record(key: Key,entry: Entry)
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A keyed dictionary is a dictionary of keyentry pairs for which there is
++ a unique entry for each key.
@@ 27484,6 +27478,12 @@ digraph pic {
LazyStreamAggregate examples
====================================================================
+LazyStreamAggregate is the category of streams with lazy
+evaluation. It is understood that the function 'empty?' will
+cause lazy evaluation if necessary to determine if there are
+entries. Functions which call 'empty?', e.g. 'first' and 'rest',
+will also cause lazy evaluation if necessary.
+
See Also:
o )show LazyStreamAggregate
@@ 27737,11 +27737,9 @@ These exports come from \refto{StreamAggregate}(S:Type):
\begin{chunk}{category LZSTAGG LazyStreamAggregate}
)abbrev category LZSTAGG LazyStreamAggregate
++ Category of streams with lazy evaluation
++ Author: Clifton J. Williamson
++ Date Created: 22 November 1989
++ Date Last Updated: 20 July 1990
++ Keywords: stream, infinite list, infinite sequence
++ Description:
++ LazyStreamAggregate is the category of streams with lazy
++ evaluation. It is understood that the function 'empty?' will
@@ 28347,6 +28345,10 @@ digraph pic {
LeftModule examples
====================================================================
+The category of left modules over an rng (ring not necessarily with unit).
+This is an abelian group which supports left multiplation by elements of
+the rng.
+
See Also:
o )show LeftModule
@@ 28402,15 +28404,6 @@ These exports come from \refto{AbelianGroup}():
\begin{chunk}{category LMODULE LeftModule}
)abbrev category LMODULE LeftModule
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of left modules over an rng (ring not necessarily with unit).
++ This is an abelian group which supports left multiplation by elements of
@@ 28611,6 +28604,10 @@ digraph pic {
ListAggregate examples
====================================================================
+A list aggregate is a model for a linked list data structure. A linked
+list is a versatile data structure. Insertion and deletion are efficient
+and searching is a linear operation.
+
See Also:
o )show ListAggregate
@@ 28919,12 +28916,6 @@ These exports come from \refto{ExtensibleLinearAggregate}(S:Type):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A list aggregate is a model for a linked list data structure.
++ A linked list is a versatile
@@ 29267,6 +29258,9 @@ digraph pic {
MultisetAggregate examples
====================================================================
+A multiset aggregate is a set which keeps track of the multiplicity
+of its elements.
+
See Also:
o )show MultisetAggregate
@@ 29428,12 +29422,6 @@ These exports come from \refto{SetAggregate}(S:SetCategory):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A multiset aggregate is a set which keeps track of the multiplicity
++ of its elements.
@@ 29527,6 +29515,16 @@ digraph pic {
NonAssociativeRng examples
====================================================================
+NonAssociativeRng is a basic ringtype structure, not necessarily
+commutative or associative, and not necessarily with unit.
+
+Axioms:
+ x*(y+z) = x*y + x*z
+ (x+y)*z = x*z + y*z
+
+Common Additional Axioms
+ noZeroDivisors ab = 0 => a=0 or b=0
+
See Also:
o )show NonAssociativeRng
@@ 29601,11 +29599,6 @@ These exports come from \refto{Monad}():
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 03 July 1991
++ Basic Operations: +, *, , **
++ Related Constructors: Rng, Ring, NonAssociativeRing
++ Also See:
++ AMS Classifications:
++ Keywords: not associative ring
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
@@ 29784,8 +29777,24 @@ digraph pic {
OneDimensionalArrayAggregate examples
====================================================================
+Onedimensionalarray aggregates serves as models for onedimensional
+arrays. Categorically, these aggregates are finite linear aggregates
+with the shallowlyMutable property, that is, any component of the array
+may be changed without affecting the identity of the overall array.
+Array data structures are typically represented by a fixed area in
+storage and cannot efficiently grow or shrink on demand as can list
+structures (see however FlexibleArray for a data structure
+which is a cross between a list and an array).
+
+Iteration over, and access to, elements of arrays is extremely fast
+(and often can be optimized to opencode).
+
+Insertion and deletion however is generally slow since an entirely new
+data structure must be created for the result.
+
See Also:
o )show OneDimensionalArrayAggregate
+o )show FlexibleArray
\end{chunk}
{\bf See:}
@@ 30323,6 +30332,9 @@ digraph pic {
OrderedCancellationAbelianMonoid examples
====================================================================
+Ordered sets which are also abelian cancellation monoids, such that
+the addition preserves the ordering.
+
See Also:
o )show OrderedCancellationAbelianMonoid
@@ 30384,15 +30396,6 @@ These exports come from \refto{CancellationAbelianMonoid}():
\begin{chunk}{category OCAMON OrderedCancellationAbelianMonoid}
)abbrev category OCAMON OrderedCancellationAbelianMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also abelian cancellation monoids,
++ such that the addition preserves the ordering.
@@ 30572,6 +30575,47 @@ digraph pic {
RegularTriangularSetCategory examples
====================================================================
+The category of regular triangular sets was introduced under the name
+regular chains in M. KALKBRENER "Three contributions to elimination theory".
+
+In P. AUBRY, D. LAZARD and M. MORENO MAZA "On the Theories of Triangular Sets" it is proved that regular triangular sets and towers of simple
+extensions of a field are equivalent notions.
+
+In the following definitions, all polynomials and ideals are taken from
+the polynomial ring k[x1,...,xn] where k is the fraction field of R.
+
+The triangular set [t1,...,tm] is regular iff for every i the initial
+of ti+1 is invertible in the tower of simple extensions associated
+with [t1,...,ti].
+
+A family [T1,...,Ts] of regular triangular sets is a split of
+Kalkbrener of a given ideal I iff the radical of I is equal to the
+intersection of the radical ideals generated by the saturated ideals
+of the [T1,...,Ti].
+
+A family [T1,...,Ts] of regular triangular sets is a split of Kalkbrener
+of a given triangular set T iff it is a split of Kalkbrener of the
+saturated ideal of T. Let K be an algebraic closure of k.
+
+Assume that V is finite with cardinality n and let A be the affine
+space K^n.
+
+For a regular triangular set T let denote by W(T) the set of regular
+zeros of T. A family [T1,...,Ts] of regular triangular sets is a split
+of Lazard of a given subset S of A iff the union of the W(Ti) contains
+S and is contained in the closure of S (w.r.t. Zariski topology).
+
+A family [T1,...,Ts] of regular triangular sets is a split of Lazard
+of a given triangular set T if it is a split of Lazard of W(T).
+Note that if [T1,...,Ts] is a split of Lazard of T then it is also a
+split of Kalkbrener of T. The converse is false.
+
+This category provides operations related to both kinds of splits, the
+former being related to ideals decomposition whereas the latter deals
+with varieties decomposition. See the example illustrating the
+RegularTriangularSet constructor for more explanations about
+decompositions by means of regular triangular sets.
+
See Also:
o )show RegularTriangularSetCategory
@@ 30838,11 +30882,6 @@ P:RecursivePolynomialCategory(R,E,V)):
++ Author: Marc Moreno Maza
++ Date Created: 09/03/1998
++ Date Last Updated: 12/15/1998
++ Basic Functions:
++ Related Constructors:
++ Also See: essai Graphisme
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References :
++ [1] M. KALKBRENER "Three contributions to elimination theory"
++ Phd Thesis, University of Linz, Austria, 1991.
@@ 31273,6 +31312,15 @@ digraph pic {
RightModule examples
====================================================================
+The category of right modules over an rng (ring not necessarily with unit).
+This is an abelian group which supports right multiplication by elements
+of the rng.
+
+Axioms:
+ x*(a*b) = (x*a)*b
+ x*(a+b) = (x*a)+(x*b)
+ (x+y)*x = (x*a)+(y*a)
+
See Also:
o )show RightModule
@@ 31326,15 +31374,6 @@ These exports come from \refto{AbelianGroup}():
\begin{chunk}{category RMODULE RightModule}
)abbrev category RMODULE RightModule
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of right modules over an rng (ring not necessarily
++ with unit). This is an abelian group which supports right
@@ 31431,6 +31470,17 @@ Rng is a Ring that does not necessarily have a unit.
Rng examples
====================================================================
+The category of associative rings, not necessarily commutative, and not
+necessarily with a 1. This is a combination of an abelian group
+and a semigroup, with multiplication distributing over addition.
+
+Axioms:
+ x*(y+z) = x*y + x*z
+ (x+y)*z = x*z + y*z
+
+Conditional attributes:
+ noZeroDivisors ab = 0 => a=0 or b=0
+
See Also:
o )show Rng
@@ 31490,15 +31540,6 @@ These exports come from \refto{SemiGroup}():
\begin{chunk}{category RNG Rng}
)abbrev category RNG Rng
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of associative rings, not necessarily commutative, and not
++ necessarily with a 1. This is a combination of an abelian group
@@ 31602,6 +31643,12 @@ digraph pic {
BiModule examples
====================================================================
+A BiModule is both a left and right module with respect to potentially
+different rings.
+
+Axiom:
+ r*(x*s) = (r*x)*s
+
See Also:
o )show BiModule
@@ 31675,15 +31722,6 @@ These exports come from \refto{RightModule}(S:Ring):
\begin{chunk}{category BMODULE BiModule}
)abbrev category BMODULE BiModule
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A \spadtype{BiModule} is both a left and right module with respect
++ to potentially different rings.
@@ 31861,6 +31899,9 @@ digraph pic {
BitAggregate examples
====================================================================
+The bit aggregate category models aggregates representing large
+quantities of Boolean data.
+
See Also:
o )show BitAggregate
@@ 32105,12 +32146,6 @@ These exports come from \refto{OneDimensionalArrayAggregate}(Boolean):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The bit aggregate category models aggregates representing large
++ quantities of Boolean data.
@@ 32269,6 +32304,9 @@ digraph pic {
NonAssociativeRing examples
====================================================================
+A NonAssociativeRing is a non associative rng which has a unit,
+the multiplication is not necessarily commutative or associative.
+
See Also:
o )show NonAssociativeRing
@@ 32361,11 +32399,6 @@ These exports come from \refto{MonadWithUnit}():
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +, *, , **
++ Related Constructors: NonAssociativeRng, Rng, Ring
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative ring with unit
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
@@ 32565,6 +32598,15 @@ digraph pic {
NormalizedTriangularSetCategory examples
====================================================================
+The category of normalized triangular sets. A triangular set ts is said
+normalized if for every algebraic variable v of ts the polynomial
+select(ts,v) is normalized w.r.t. every polynomial in collectUnder(ts,v).
+
+A polynomial p is said normalized w.r.t. a nonconstant polynomial q
+if p is constant or degree(p,mdeg(q)) = 0 and init(p) is normalized
+w.r.t. q. One of the important features of normalized triangular sets
+is that they are regular sets.
+
See Also:
o )show NormalizedTriangularSetCategory
@@ 32824,11 +32866,6 @@ P:RecursivePolynomialCategory(R,E,V)):
++ Author: Marc Moreno Maza
++ Date Created: 10/07/1998
++ Date Last Updated: 12/12/1998
++ Basic Functions:
++ Related Constructors:
++ Also See: essai Graphisme
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References :
++ [1] D. LAZARD "A new method for solving algebraic systems of
++ positive dimension" Discr. App. Math. 33:147160,1991
@@ 32960,6 +32997,9 @@ digraph pic {
OrderedAbelianGroup examples
====================================================================
+Ordered sets which are also abelian groups, such that the
+addition preserves the ordering.
+
See Also:
o )show OrderedAbelianGroup
@@ 33025,15 +33065,6 @@ These exports come from \refto{AbelianGroup}():
\begin{chunk}{category OAGROUP OrderedAbelianGroup}
)abbrev category OAGROUP OrderedAbelianGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also abelian groups, such that the
++ addition preserves the ordering.
@@ 33122,6 +33153,16 @@ digraph pic {
OrderedAbelianMonoidSup examples
====================================================================
+This domain is an OrderedAbelianMonoid with a sup operation added.
+The purpose of the sup operator in this domain is to act as a
+supremum with respect to the partial order imposed by ``, rather
+than with respect to the total > order (since that is "max").
+
+Axioms:
+ sup(a,b)a ~= "failed"
+ sup(a,b)b ~= "failed"
+ xa ~= "failed" and xb ~= "failed" => x >= sup(a,b)
+
See Also:
o )show OrderedAbelianMonoidSup
@@ 33184,15 +33225,6 @@ These exports come from \refto{OrderedCancellationAbelianMonoid}():
\begin{chunk}{category OAMONS OrderedAbelianMonoidSup}
)abbrev category OAMONS OrderedAbelianMonoidSup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This domain is an OrderedAbelianMonoid with a sup
++ operation added. The purpose of the sup operator
@@ 33339,6 +33371,10 @@ digraph pic {
OrderedMultisetAggregate examples
====================================================================
+An orderedmultiset aggregate is a multiset built over an ordered set S
+so that the relative sizes of its entries can be assessed.
+These aggregates serve as models for priority queues.
+
See Also:
o )show OrderedMultisetAggregate
@@ 33518,12 +33554,6 @@ These exports come from \refto{PriorityQueueAggregate}(S:OrderedSet):
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An orderedmultiset aggregate is a multiset built over an ordered set S
++ so that the relative sizes of its entries can be assessed.
@@ 33640,6 +33670,9 @@ digraph pic {
Ring examples
====================================================================
+The category of rings with unity, always associative, but not
+necessarily commutative.
+
See Also:
o )show Ring
@@ 33742,15 +33775,6 @@ These exports come from \refto{Monoid}():
\begin{chunk}{category RING Ring}
)abbrev category RING Ring
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of rings with unity, always associative, but
++ not necessarily commutative.
@@ 33954,6 +33978,12 @@ digraph pic {
SquareFreeRegularTriangularSetCategory examples
====================================================================
+The category of squarefree regular triangular sets. A regular
+triangular set ts is squarefree if the gcd of any polynomial p in ts
+and differentiate(p,mvar(p)) w.r.t. collectUnder(ts,mvar(p))
+has degree zero w.r.t. mvar(p). Thus any squarefree regular
+set defines a tower of squarefree simple extensions.
+
See Also:
o )show SquareFreeRegularTriangularSetCategory
@@ 34213,11 +34243,6 @@ P:RecursivePolynomialCategory(R,E,V)):
++ Author: Marc Moreno Maza
++ Date Created: 09/03/1996
++ Date Last Updated: 09/10/1998
++ Basic Functions:
++ Related Constructors:
++ Also See: essai Graphisme
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References :
++ [1] D. LAZARD "A new method for solving algebraic systems of
++ positive dimension" Discr. App. Math. 33:147160,1991
@@ 34423,6 +34448,9 @@ digraph pic {
StringAggregate examples
====================================================================
+A string aggregate is a category for strings, that is, one dimensional
+arrays of characters.
+
See Also:
o )show StringAggregate
@@ 34691,12 +34719,6 @@ These exports come from \refto{OneDimensionalArrayAggregate}(Character):
++ revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A string aggregate is a category for strings, that is,
++ one dimensional arrays of characters.
@@ 34941,6 +34963,9 @@ digraph pic {
TableAggregate examples
====================================================================
+A table aggregate is a model of a table, i.e. a discrete manytoone
+mapping from keys to entries.
+
See Also:
o )show TableAggregate
@@ 35197,12 +35222,6 @@ and RecKE=Record(key: Key,entry: Entry):
++ revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A table aggregate is a model of a table, i.e. a discrete manytoone
++ mapping from keys to entries.
@@ 35505,6 +35524,13 @@ digraph pic {
VectorCategory examples
====================================================================
+VectorCategory represents the type of vector like objects,
+i.e. finite sequences indexed by some finite segment of the
+integers. The operations available on vectors depend on the structure
+of the underlying components. Many operations from the component domain
+are defined for vectors componentwise. It can by assumed that extraction or
+updating components can be done in constant time.
+
See Also:
o )show VectorCategory
@@ 35706,15 +35732,6 @@ These exports come from \refto{OneDimensionalArrayAggregate}(R:Type):
\begin{chunk}{category VECTCAT VectorCategory}
)abbrev category VECTCAT VectorCategory
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: DirectProductCategory, Vector, IndexedVector
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ \spadtype{VectorCategory} represents the type of vector like objects,
++ i.e. finite sequences indexed by some finite segment of the
@@ 36059,6 +36076,10 @@ digraph pic {
AssociationListAggregate examples
====================================================================
+An association list is a list of key entry pairs which may be viewed
+as a table. It is a poor mans version of a table; searching for a key
+is a linear operation.
+
See Also:
o )show AssociationListAggregate
@@ 36501,12 +36522,6 @@ and RecKE=Record(key: Key,entry: Entry)
++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: April 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An association list is a list of key entry pairs which may be viewed
++ as a table. It is a poor mans version of a table:
@@ 36622,6 +36637,8 @@ digraph pic {
CharacteristicNonZero examples
====================================================================
+The category of Rings of Characteristic Non Zero
+
See Also:
o )show CharacteristicNonZero
@@ 36703,15 +36720,6 @@ These exports come from \refto{Ring}():
\begin{chunk}{category CHARNZ CharacteristicNonZero}
)abbrev category CHARNZ CharacteristicNonZero
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Rings of Characteristic Non Zero
@@ 36821,6 +36829,8 @@ digraph pic {
CharacteristicZero examples
====================================================================
+The category of Rings of Characteristic Zero.
+
See Also:
o )show CharacteristicZero
@@ 36899,15 +36909,6 @@ These exports come from \refto{Ring}():
\begin{chunk}{category CHARZ CharacteristicZero}
)abbrev category CHARZ CharacteristicZero
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Rings of Characteristic Zero.
@@ 37014,6 +37015,9 @@ digraph pic {
CommutativeRing examples
====================================================================
+The category of commutative rings with unity, i.e. rings where * is
+commutative, and which have a multiplicative identity element.
+
See Also:
o )show CommutativeRing
@@ 37102,15 +37106,6 @@ These exports come from \refto{Ring}():
\begin{chunk}{category COMRING CommutativeRing}
)abbrev category COMRING CommutativeRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of commutative rings with unity, i.e. rings where
++ \spadop{*} is commutative, and which have a multiplicative identity
@@ 37237,6 +37232,13 @@ digraph pic {
DifferentialRing examples
====================================================================
+An ordinary differential ring, that is, a ring with an operation
+differentiate.
+
+Axioms:
+ differentiate(x+y) = differentiate(x)+differentiate(y)
+ differentiate(x*y) = x*differentiate(y) + differentiate(x)*y
+
See Also:
o )show DifferentialRing
@@ 37325,15 +37327,6 @@ These exports come from \refto{Ring}():
\begin{chunk}{category DIFRING DifferentialRing}
)abbrev category DIFRING DifferentialRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An ordinary differential ring, that is, a ring with an operation
++ \spadfun{differentiate}.
@@ 37463,6 +37456,13 @@ digraph pic {
EntireRing examples
====================================================================
+Entire Rings (noncommutative Integral Domains), i.e. a ring
+not necessarily commutative which has no zero divisors.
+
+Axioms:
+ ab=0 => a=0 or b=0  known as noZeroDivisors
+ not(1=0)
+
See Also:
o )show EntireRing
@@ 37543,15 +37543,6 @@ These exports come from \refto{Ring}():
\begin{chunk}{category ENTIRER EntireRing}
)abbrev category ENTIRER EntireRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Entire Rings (noncommutative Integral Domains), i.e. a ring
++ not necessarily commutative which has no zero divisors.
@@ 37680,8 +37671,16 @@ digraph pic {
FreeModuleCat examples
====================================================================
+A domain of this category implements formal linear combinations
+of elements from a domain Basis with coefficients in a domain R.
+The domain Basis needs only to belong to the category SetCategory
+and R to the category Ring. Thus the coefficient ring may be
+noncommutative. See the XDistributedPolynomial constructor for
+examples of domains built with the FreeModuleCat category constructor.
+
See Also:
o )show FreeModuleCat
+o )show XDistributedPolynomial
\end{chunk}
{\bf See:}
@@ 37780,12 +37779,6 @@ These exports come from \refto{RetractableTo}(Basis:SetCategory):
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Fix History: compilation v 2.1 le 13 dec 98
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A domain of this category
++ implements formal linear combinations
@@ 37946,6 +37939,8 @@ digraph pic {
LeftAlgebra examples
====================================================================
+The category of all left algebras over an arbitrary ring.
+
See Also:
o )show LeftAlgebra
@@ 38137,6 +38132,8 @@ digraph pic {
LinearlyExplicitRingOver examples
====================================================================
+An extension ring with an explicit linear dependence test.
+
See Also:
o )show LinearlyExplicitRingOver
@@ 38218,15 +38215,6 @@ These exports come from \refto{Ring}():
\begin{chunk}{category LINEXP LinearlyExplicitRingOver}
)abbrev category LINEXP LinearlyExplicitRingOver
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An extension ring with an explicit linear dependence test.
@@ 38344,6 +38332,14 @@ digraph pic {
Module examples
====================================================================
+The category of modules over a commutative ring.
+
+Axioms:
+ 1*x = x
+ (a*b)*x = a*(b*x)
+ (a+b)*x = (a*x)+(b*x)
+ a*(x+y) = (a*x)+(a*y)
+
See Also:
o )show Module
@@ 38410,15 +38406,6 @@ These exports come from \refto{BiModule}(a:Ring,b:Ring):
\begin{chunk}{category MODULE Module}
)abbrev category MODULE Module
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of modules over a commutative ring.
++
@@ 38534,6 +38521,12 @@ digraph pic {
OrderedRing examples
====================================================================
+Ordered sets which are also rings, that is, domains where the ring
+operations are compatible with the ordering.
+
+Axiom:
+ 0 ab < ac
+
See Also:
o )show OrderedRing
@@ 38639,15 +38632,6 @@ These exports come from \refto{Ring}():
\begin{chunk}{category ORDRING OrderedRing}
)abbrev category ORDRING OrderedRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also rings, that is, domains where the ring
++ operations are compatible with the ordering.
@@ 38793,6 +38777,13 @@ digraph pic {
PartialDifferentialRing examples
====================================================================
+A partial differential ring with differentiations indexed by a
+parameter type S.
+
+Axioms:
+ differentiate(x+y,e)=differentiate(x,e)+differentiate(y,e)
+ differentiate(x*y,e)=x*differentiate(y,e)+differentiate(x,e)*y
+
See Also:
o )show PartialDifferentialRing
@@ 38886,15 +38877,6 @@ These exports come from \refto{Ring}():
\begin{chunk}{category PDRING PartialDifferentialRing}
)abbrev category PDRING PartialDifferentialRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A partial differential ring with differentiations indexed by a
++ parameter type S.
@@ 39130,6 +39112,10 @@ digraph pic {
PointCategory examples
====================================================================
+PointCategory is the category of points in space which may be plotted
+via the graphics facilities. Functions are provided for defining
+points and handling elements of points.
+
See Also:
o )show PointCategory
@@ 39338,16 +39324,6 @@ These exports come from \refto{VectorCategory}(R:Ring):
\begin{chunk}{category PTCAT PointCategory}
)abbrev category PTCAT PointCategory
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Operations: point, elt, setelt, copy, dimension, minIndex, maxIndex,
++ convert
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ PointCategory is the category of points in space which
++ may be plotted via the graphics facilities. Functions are provided for
@@ 39505,6 +39481,10 @@ The RectangularMatrix domain is matrices of fixed dimension.
RectangularMatrixCategory examples
====================================================================
+RectangularMatrixCategory is a category of matrices of fixed dimensions.
+The dimensions of the matrix will be parameters of the domain.
+Domains in this category will be Rmodules and will be nonmutable.
+
See Also:
o )show RectangularMatrixCategory
@@ 39668,13 +39648,6 @@ These exports come from \refto{HomogeneousAggregate}(Ring)"
++ Authors: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: RectangularMatrix(m,n,R)
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ \spadtype{RectangularMatrixCategory} is a category of matrices of fixed
++ dimensions. The dimensions of the matrix will be parameters of the
@@ 40003,6 +39976,11 @@ digraph pic {
SquareFreeNormalizedTriangularSetCategory examples
====================================================================
+The category of squarefree and normalized triangular sets.
+Thus, up to the primitivity axiom of D. LAZARD
+"A new method for solving algebraic systems of positive dimension",
+these sets are Lazard triangular sets.
+
See Also:
o )show SquareFreeNormalizedTriangularSetCategory
@@ 40262,11 +40240,6 @@ P:RecursivePolynomialCategory(R,E,V)):
++ Author: Marc Moreno Maza
++ Date Created: 10/07/1998
++ Date Last Updated: 12/16/1998
++ Basic Functions:
++ Related Constructors:
++ Also See: essai Graphisme
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References :
++ [1] D. LAZARD "A new method for solving algebraic systems of
++ positive dimension" Discr. App. Math. 33:147160,1991
@@ 40446,6 +40419,8 @@ digraph pic {
StringCategory examples
====================================================================
+A category for stringlike objects
+
See Also:
o )show StringCategory
@@ 40725,8 +40700,6 @@ These exports come from \refto{OpenMath}():
\begin{chunk}{category STRICAT StringCategory}
)abbrev category STRICAT StringCategory
 Note that StringCategory is built into the old compiler
 redundant SetCategory added to help A# compiler
++ Description:
++ A category for stringlike objects
@@ 40869,6 +40842,12 @@ digraph pic {
UnivariateSkewPolynomialCategory examples
====================================================================
+This is the category of univariate skew polynomials over an Ore
+coefficient ring. The multiplication is given by
+ x a = \sigma(a) x + \delta a
+This category is an evolution of the types MonogenicLinearOperator,
+OppositeMonogenicLinearOperator, and NonCommutativeOperatorDivision
+
See Also:
o )show UnivariateSkewPolynomialCategory
@@ 41414,6 +41393,10 @@ digraph pic {
XAlgebra examples
====================================================================
+This is the category of algebras over noncommutative rings.
+It is used by constructors of noncommutative algebras such as
+XPolynomialRing and XFreeAlgebra
+
See Also:
o )show XAlgebra
@@ 41508,12 +41491,6 @@ These exports come from \refto{BiModule}(R:Ring,R:Ring):
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Fix History: compilation v 2.1 le 13 dec 98
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is the category of algebras over noncommutative rings.
++ It is used by constructors of noncommutative algebras such as
@@ 41639,6 +41616,15 @@ digraph pic {
Algebra examples
====================================================================
+The category of associative algebras (modules which are themselves rings).
+
+Axioms:
+ (b+c)::% = (b::%) + (c::%)
+ (b*c)::% = (b::%) * (c::%)
+ (1::R)::% = 1::%
+ b*x = (b::%)*x
+ r*(a*b) = (r*a)*b = a*(r*b)
+
See Also:
o )show Algebra
@@ 41735,15 +41721,6 @@ These exports come from \refto{Module}(R:CommutativeRing):
\begin{chunk}{category ALGEBRA Algebra}
)abbrev category ALGEBRA Algebra
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of associative algebras (modules which are themselves rings).
++
@@ 41913,6 +41890,9 @@ digraph pic {
DifferentialExtension examples
====================================================================
+Differential extensions of a ring R. Given a differentiation on R,
+extend it to a differentiation on %.
+
See Also:
o )show DifferentialExtension
@@ 42032,15 +42012,6 @@ These exports come from \refto{PartialDifferentialRing}(Symbol):
\begin{chunk}{category DIFEXT DifferentialExtension}
)abbrev category DIFEXT DifferentialExtension
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Differential extensions of a ring R.
++ Given a differentiation on R, extend it to a differentiation on %.
@@ 42197,6 +42168,10 @@ digraph pic {
FullyLinearlyExplicitRingOver examples
====================================================================
+S is FullyLinearlyExplicitRingOver R means that S is a
+LinearlyExplicitRingOver R and, in addition, if R is a
+LinearlyExplicitRingOver Integer, then so is S
+
See Also:
o )show FullyLinearlyExplicitRingOver
@@ 42289,15 +42264,6 @@ These exports come from \refto{LinearlyExplicitRingOver}(a:Ring):
\begin{chunk}{category FLINEXP FullyLinearlyExplicitRingOver}
)abbrev category FLINEXP FullyLinearlyExplicitRingOver
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ S is \spadtype{FullyLinearlyExplicitRingOver R} means that S is a
++ \spadtype{LinearlyExplicitRingOver R} and, in addition, if R is a
@@ 42431,6 +42397,9 @@ digraph pic {
LieAlgebra examples
====================================================================
+The category of Lie Algebras. It is used by the domains of noncommutative
+algebra, LiePolynomial and XPBWPolynomial.
+
See Also:
o )show LieAlgebra
@@ 42509,8 +42478,6 @@ These exports come from \refto{Module}(R:Ring):
++ Author: Michel Petitot (petitot@lifl.fr).
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Keywords:
++ References:
++ Description:
++ The category of Lie Algebras.
++ It is used by the domains of noncommutative algebra,
@@ 42665,6 +42632,13 @@ digraph pic {
LinearOrdinaryDifferentialOperatorCategory examples
====================================================================
+LinearOrdinaryDifferentialOperatorCategory is the category
+of differential operators with coefficients in a ring A with a given
+derivation.
+
+Multiplication of operators corresponds to functional composition:
+ (L1 * L2).(f) = L1 L2 f
+
See Also:
o )show LinearOrdinaryDifferentialOperatorCategory
@@ 42844,7 +42818,6 @@ These exports come from \refto{Eltable}(A:Ring,A:Ring):
++ Author: Manuel Bronstein
++ Date Created: 9 December 1993
++ Date Last Updated: 15 April 1994
++ Keywords: differential operator
++ Description:
++ LinearOrdinaryDifferentialOperatorCategory is the category
++ of differential operators with coefficients in a ring A with a given
@@ 43043,6 +43016,12 @@ digraph pic {
NonAssociativeAlgebra examples
====================================================================
+NonAssociativeAlgebra is the category of non associative algebras
+(modules which are themselves non associative rngs).\br
+
+Axiom:
+ r*(a*b) = (r*a)*b = a*(r*b)
+
See Also:
o )show NonAssociativeAlgebra
@@ 43128,11 +43107,6 @@ These exports come from \refto{Module}(R:CommutativeRing):
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +, , *, **
++ Related Constructors: Algebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
@@ 43261,6 +43235,8 @@ digraph pic {
VectorSpace examples
====================================================================
+Vector Spaces (not necessarily finite dimensional) over a field.
+
See Also:
o )show VectorSpace
@@ 43331,15 +43307,6 @@ These exports come from \refto{Module}():
\begin{chunk}{category VSPACE VectorSpace}
)abbrev category VSPACE VectorSpace
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Vector Spaces (not necessarily finite dimensional) over a field.
@@ 43470,6 +43437,9 @@ digraph pic {
XFreeAlgebra examples
====================================================================
+This category specifies opeations for polynomials and formal series
+with noncommutative variables.
+
See Also:
o )show XFreeAlgebra
@@ 43617,12 +43587,6 @@ where WORD:OrderedFreeMonoid(OrderedSet))
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Fix History: compilation v 2.1 le 13 dec 98
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category specifies opeations for polynomials
++ and formal series with noncommutative variables.
@@ 43926,6 +43890,9 @@ digraph pic {
DirectProductCategory examples
====================================================================
+This category represents a finite cartesian product of a given type.
+Many categorical properties are preserved under this construction.
+
See Also:
o )show DirectProductCategory
@@ 44252,18 +44219,6 @@ These exports come from \refto{OrderedAbelianMonoidSup}():
 all direct product category domains must be compiled
 without subsumption, set SourceLevelSubset to EQUAL
)bo $noSubsumption := true

% DirectProductCategory

++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: DirectProduct
++ Also See: VectorCategory
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category represents a finite cartesian product of a given type.
++ Many categorical properties are preserved under this construction.
@@ 44458,6 +44413,9 @@ digraph pic {
DivisionRing examples
====================================================================
+A division ring (sometimes called a skew field), i.e. a not necessarily
+commutative ring where all nonzero elements have multiplicative inverses.
+
See Also:
o )show DivisionRing
@@ 44557,15 +44515,6 @@ These exports come from \refto{Algebra}(Fraction(Integer)):
\begin{chunk}{category DIVRING DivisionRing}
)abbrev category DIVRING DivisionRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A division ring (sometimes called a skew field),
++ i.e. a not necessarily commutative ring where
@@ 44739,6 +44688,9 @@ digraph pic {
FiniteRankNonAssociativeAlgebra examples
====================================================================
+A FiniteRankNonAssociativeAlgebra is a non associative algebra over
+a commutative ring R which is a free Rmodule of finite rank.
+
See Also:
o )show FiniteRankNonAssociativeAlgebra
@@ 44921,12 +44873,6 @@ These exports come from \refto{NonAssociativeAlgebra}(R:CommutativeRing):
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 12 June 1991
++ Basic Operations: +,,*,**, someBasis
++ Related Constructors: FramedNonAssociativeAlgebra, FramedAlgebra,
++ FiniteRankAssociativeAlgebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra, basis
++ References:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
@@ 45694,6 +45640,9 @@ digraph pic {
FreeLieAlgebra examples
====================================================================
+The category of free Lie algebras. It is used by domains of
+noncommutative algebra such as LiePolynomial and XPBWPolynomial.
+
See Also:
o )show FreeLieAlgebra
@@ 45791,12 +45740,6 @@ These exports come from \refto{LieAlgebra}(CommutativeRing):
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Fix History: compilation v 2.1 le 13 dec 98
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of free Lie algebras.
++ It is used by domains of noncommutative algebra:
@@ 45954,6 +45897,13 @@ digraph pic {
IntegralDomain examples
====================================================================
+The category of commutative integral domains, i.e. commutative
+rings with no zero divisors.
+
+Conditional attributes:
+ canonicalUnitNormal  the canonical field is the same for all associates
+ canonicalsClosed  the product of two canonicals is itself canonical
+
See Also:
o )show IntegralDomain
@@ 46080,15 +46030,6 @@ These exports come from \refto{Algebra}(a:IntegralDomain):
\begin{chunk}{category INTDOM IntegralDomain}
)abbrev category INTDOM IntegralDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References: Davenport & Trager I
++ Description:
++ The category of commutative integral domains, i.e. commutative
++ rings with no zero divisors.
@@ 46243,6 +46184,18 @@ digraph pic {
MonogenicLinearOperator examples
====================================================================
+This is the category of linear operator rings with one generator.
+The generator is not named by the category but can always be
+constructed as monomial(1,1).
+
+For convenience, call the generator G.
+Then each value is equal to
+ sum(a(i)*G**i, i = 0..n)
+for some unique n and a(i) in R.
+
+Note that multiplication is not necessarily commutative.
+In fact, if a is in R, it is quite normal to have a*G ^= G*a.
+
See Also:
o )show MonogenicLinearOperator
@@ 46350,13 +46303,6 @@ These exports come from \refto{Algebra}(R:CommutativeRing):
++ Author: Stephen M. Watt
++ Date Created: 1986
++ Date Last Updated: May 30, 1991
++ Basic Operations:
++ Related Domains: NonCommutativeOperatorDivision
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ This is the category of linear operator rings with one generator.
++ The generator is not named by the category but can always be
@@ 46562,6 +46508,10 @@ digraph pic {
OctonionCategory examples
====================================================================
+OctonionCategory gives the categorial frame for the octonions, and
+eightdimensional nonassociative algebra, doubling the the quaternions
+in the same way as doubling the Complex numbers to get the quaternions.
+
See Also:
o )show OctonionCategory
@@ 46749,12 +46699,6 @@ These exports come from \refto{CharacteristicNonZero}():
++ Author: R. Wisbauer, J. Grabmeier
++ Date Created: 05 September 1990
++ Date Last Updated: 19 September 1990
++ Basic Operations: _+, _*, octon, real, imagi, imagj, imagk,
++ imagE, imagI, imagJ, imagK
++ Related Constructors: QuaternionCategory
++ Also See:
++ AMS Classifications:
++ Keywords: octonion, nonassociative algebra, CayleyDixon
++ References: e.g. I.L Kantor, A.S. Solodovnikov:
++ Hypercomplex Numbers, Springer Verlag Heidelberg, 1989,
++ ISBN 0387969802
@@ 47151,6 +47095,9 @@ digraph pic {
QuaternionCategory examples
====================================================================
+QuaternionCategory describes the category of quaternions and implements
+functions that are not representation specific.
+
See Also:
o )show QuaternionCategory
@@ 47367,15 +47314,7 @@ These exports come from \refto{CharacteristicNonZero}():
)abbrev category QUATCAT QuaternionCategory
++ Author: Robert S. Sutor
++ Date Created: 23 May 1990
++ Change History:
++ 10 September 1990
++ Basic Operations: (Algebra)
++ abs, conjugate, imagI, imagJ, imagK, norm, quatern, rational,
++ rational?, real
++ Related Constructors: Quaternion, QuaternionCategoryFunctions2
++ Also See: DivisionRing
++ AMS Classifications: 11R52
++ Keywords: quaternions, division ring, algebra
+++ Change History: 10 September 1990
++ Description:
++ \spadtype{QuaternionCategory} describes the category of quaternions
++ and implements functions that are not representation specific.
@@ 47735,6 +47674,11 @@ The SquareMatrix domain is for square matrices of fixed dimension.
SquareMatrixCategory examples
====================================================================
+SquareMatrixCategory is a general square matrix category which allows
+different representations and indexing schemes. Rows and columns may
+be extracted with rows returned as objects of type Row and colums
+returned as objects of type Col.
+
See Also:
o )show SquareMatrixCategory
@@ 47991,13 +47935,6 @@ These exports come from \refto{FullyLinearlyExplicitRingOver}(R:Ring):
++ Authors: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: SquareMatrix(ndim,R)
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ \spadtype{SquareMatrixCategory} is a general square matrix category which
++ allows different representations and indexing schemes. Rows and
@@ 48317,6 +48254,10 @@ digraph pic {
XPolynomialsCat examples
====================================================================
+The Category of polynomial rings with noncommutative variables.
+The coefficient ring may be noncommutative too.
+However coefficients commute with variables.
+
See Also:
o )show XPolynomialsCat
@@ 48463,16 +48404,10 @@ These exports come from \refto{SetCategory}():
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Fix History: compilation v 2.1 le 13 dec 98
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The Category of polynomial rings with noncommutative variables.
++ The coefficient ring may be noncommutative too.
++ However coefficients commute with vaiables.
+++ However coefficients commute with variables.
XPolynomialsCat(vl:OrderedSet,R:Ring):Category == Export where
WORD ==> OrderedFreeMonoid(vl)
@@ 48637,6 +48572,17 @@ digraph pic {
AbelianMonoidRing examples
====================================================================
+Abelian monoid ring elements (not necessarily of finite support)
+of this ring are of the form formal SUM (r_i * e_i)
+where the r_i are coefficents and the e_i, elements of the
+ordered abelian monoid, are thought of as exponents or monomials.
+The monomials commute with each other, and with
+the coefficients (which themselves may or may not be commutative).
+
+See FiniteAbelianMonoidRing for the case of finite support
+a useful common model for polynomials and power series.
+Conceptually at least, only the nonzero terms are ever operated on.
+
See Also:
o )show AbelianMonoidRing
@@ 48787,15 +48733,6 @@ These exports come from \refto{Algebra}(Fraction(Integer)):
\begin{chunk}{category AMR AbelianMonoidRing}
)abbrev category AMR AbelianMonoidRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Abelian monoid ring elements (not necessarily of finite support)
++ of this ring are of the form formal SUM (r_i * e_i)
@@ 48986,6 +48923,9 @@ digraph pic {
FortranMachineTypeCategory examples
====================================================================
+A category of domains which model machine arithmetic used by machines
+in the AXIOMNAG link.
+
See Also:
o )show FortranMachineTypeCategory
@@ 49106,14 +49046,6 @@ These exports come from \refto{RetractableTo}(Integer):
)abbrev category FMTC FortranMachineTypeCategory
++ Author: Mike Dewar
++ Date Created: December 1993
++ Date Last Updated:
++ Basic Operations:
++ Related Domains:
++ Also See: FortranExpression, MachineInteger, MachineFloat, MachineComplex
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ A category of domains which model machine arithmetic
++ used by machines in the AXIOMNAG link.
@@ 49300,6 +49232,10 @@ digraph pic {
FramedNonAssociativeAlgebra examples
====================================================================
+FramedNonAssociativeAlgebra(R) is a FiniteRankNonAssociativeAlgebra
+(i.e. a non associative algebra over R which is a free Rmodule of
+finite rank) over a commutative ring R together with a fixed Rmodule basis.
+
See Also:
o )show FramedNonAssociativeAlgebra
@@ 49517,12 +49453,6 @@ where R:CommutativeRing:
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +,,*,**,basis
++ Related Constructors: FiniteRankNonAssociativeAlgebra, FramedAlgebra,
++ FiniteRankAssociativeAlgebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra, basis
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
@@ 49950,6 +49880,11 @@ digraph pic {
GcdDomain examples
====================================================================
+This category describes domains where gcd can be computed but where
+there is no guarantee of the existence of factor operation for factorisation
+into irreducibles. However, if such a factor operation exist, factorization
+will be unique up to order and units.
+
See Also:
o )show GcdDomain
@@ 50064,15 +49999,6 @@ These exports come from \refto{IntegralDomain}():
\begin{chunk}{category GCDDOM GcdDomain}
)abbrev category GCDDOM GcdDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References: Davenport & Trager 1
++ Description:
++ This category describes domains where
++ \spadfun{gcd} can be computed but where there is no guarantee
@@ 50234,6 +50160,9 @@ digraph pic {
OrderedIntegralDomain examples
====================================================================
+The category of ordered commutative integral domains, where ordering
+and the arithmetic operations are compatible
+
See Also:
o )show OrderedIntegralDomain
@@ 50354,12 +50283,6 @@ These exports come from \refto{OrderedRing}():
)abbrev category OINTDOM OrderedIntegralDomain
++ Author: JH Davenport (after L GonzalezVega)
++ Date Created: 30.1.96
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Description:
++ The category of ordered commutative integral domains, where ordering
++ and the arithmetic operations are compatible
@@ 50507,6 +50430,10 @@ digraph pic {
FiniteAbelianMonoidRing examples
====================================================================
+This category is similar to AbelianMonoidRing, except that the sum is
+assumed to be finite. It is a useful model for polynomials, but is
+somewhat more general.
+
See Also:
o )show FiniteAbelianMonoidRing
@@ 50678,15 +50605,7 @@ These exports come from \refto{FullyRetractableTo}(R:Ring):
\begin{chunk}{category FAMR FiniteAbelianMonoidRing}
)abbrev category FAMR FiniteAbelianMonoidRing
++ Author:
++ Date Created:
++ Date Last Updated: 14.08.2000 Exported pomopo! and binomThmExpt [MMM]
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category is similar to AbelianMonoidRing, except that the sum is
++ assumed to be finite. It is a useful model for polynomials,
@@ 50963,6 +50882,9 @@ digraph pic {
IntervalCategory examples
====================================================================
+This category implements of interval arithmetic and transcendental
+functions over intervals.
+
See Also:
o )show IntervalCategory
@@ 51191,13 +51113,6 @@ These exports come from \refto{RetractableTo}(Integer):
)abbrev category INTCAT IntervalCategory
++ Author: Mike Dewar
++ Date Created: November 1996
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category implements of interval arithmetic and transcendental
++ functions over intervals.
@@ 51381,6 +51296,9 @@ digraph pic {
PowerSeriesCategory examples
====================================================================
+PowerSeriesCategory is the most general power series category with
+exponents in an ordered abelian monoid.
+
See Also:
o )show PowerSeriesCategory
@@ 51523,13 +51441,6 @@ where Coef:Ring and Expon:OrderedAbelianMonoid:
++ Author: Clifton J. Williamson
++ Date Created: 21 December 1989
++ Date Last Updated: 25 February 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: power series
++ Examples:
++ References:
++ Description:
++ \spadtype{PowerSeriesCategory} is the most general power series
++ category with exponents in an ordered abelian monoid.
@@ 51717,6 +51628,12 @@ digraph pic {
PrincipalIdealDomain examples
====================================================================
+The category of constructive principal ideal domains, i.e. where a
+single generator can be constructively found for any ideal given by
+a finite set of generators. Note that this constructive definition
+only implies that finitely generated ideals are principal. It is not
+clear what we would mean by an infinitely generated ideal.
+
See Also:
o )show PrincipalIdealDomain
@@ 51832,15 +51749,6 @@ These exports come from \refto{GcdDomain}():
\begin{chunk}{category PID PrincipalIdealDomain}
)abbrev category PID PrincipalIdealDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of constructive principal ideal domains, i.e.
++ where a single generator can be constructively found for
@@ 51970,6 +51878,10 @@ digraph pic {
UniqueFactorizationDomain examples
====================================================================
+A constructive unique factorization domain, i.e. where we can
+constructively factor members into a product of a finite number
+of irreducible elements.
+
See Also:
o )show UniqueFactorizationDomain
@@ 52095,15 +52007,6 @@ These exports come from \refto{GcdDomain}():
\begin{chunk}{category UFD UniqueFactorizationDomain}
)abbrev category UFD UniqueFactorizationDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A constructive unique factorization domain, i.e. where
++ we can constructively factor members into a product of
@@ 52240,6 +52143,8 @@ digraph pic {
DivisorCategory examples
====================================================================
+This category exports the function for domains.
+
See Also:
o )show DivisorCategory
@@ 52503,13 +52408,22 @@ digraph pic {
EuclideanDomain examples
====================================================================
+A constructive euclidean domain, i.e. one can divide producing
+a quotient and a remainder where the remainder is either zero
+or is smaller (euclideanSize) than the divisor.
+
+Conditional attributes:
+ multiplicativeValuation  Size(a*b)=Size(a)*Size(b)
+ additiveValuation  Size(a*b)=Size(a)+Size(b)
+
+Principal Ideal Domains are a subset of Euclidean Domains.
+Euclidean Domains are a subset of Fields.
+
See Also:
o )show EuclideanDomain
\end{chunk}
Principal Ideal Domains are a subset of Euclidean Domains.
\pagefrom{PrincipalIdealDomain}{PID}.
Euclidean Domains are a subset of Fields.
\pageto{Field}{FIELD}
{\bf See:}
@@ 52640,15 +52554,6 @@ These exports come from \refto{PrincipalIdealDomain}():
\begin{chunk}{category EUCDOM EuclideanDomain}
)abbrev category EUCDOM EuclideanDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A constructive euclidean domain, i.e. one can divide producing
++ a quotient and a remainder where the remainder is either zero
@@ 52961,6 +52866,9 @@ digraph pic {
MultivariateTaylorSeriesCategory examples
====================================================================
+MultivariateTaylorSeriesCategory is the most general multivariate
+Taylor series category.
+
See Also:
o )show MultivariateTaylorSeriesCategory
@@ 53211,13 +53119,6 @@ These exports come from \refto{TranscendentalFunctionCategory}():
++ Author: Clifton J. Williamson
++ Date Created: 6 March 1990
++ Date Last Updated: 6 March 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: multivariate, Taylor, series
++ Examples:
++ References:
++ Description:
++ \spadtype{MultivariateTaylorSeriesCategory} is the most general
++ multivariate Taylor series category.
@@ 53377,6 +53278,12 @@ digraph pic {
PolynomialFactorizationExplicit examples
====================================================================
+This is the category of domains that know "enough" about themselves
+in order to factor univariate polynomials over themselves. This will
+be used in future releases for supporting factorization over finitely
+generated coefficient fields, it is not yet available in the current
+release of Axiom.
+
See Also:
o )show PolynomialFactorizationExplicit
@@ 53520,14 +53427,6 @@ These exports come from \refto{UniqueFactorizationDomain}():
\begin{chunk}{category PFECAT PolynomialFactorizationExplicit}
)abbrev category PFECAT PolynomialFactorizationExplicit
++ Author: James Davenport
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is the category of domains that know "enough" about
++ themselves in order to factor univariate polynomials over themselves.
@@ 53741,6 +53640,12 @@ digraph pic {
UnivariatePowerSeriesCategory examples
====================================================================
+UnivariatePowerSeriesCategory is the most general univariate power
+series category with exponents in an ordered abelian monoid. Note that
+this category exports a substitution function if it is possible to
+multiply exponents. Also note that this category exports a derivative
+operation if it is possible to multiply coefficients by exponents.
+
See Also:
o )show UnivariatePowerSeriesCategory
@@ 53956,13 +53861,6 @@ These exports come from \refto{PartialDifferentialRing}(Symbol):
++ Author: Clifton J. Williamson
++ Date Created: 21 December 1989
++ Date Last Updated: 20 September 1993
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ \spadtype{UnivariatePowerSeriesCategory} is the most general
++ univariate power series category with exponents in an ordered
@@ 54218,6 +54116,14 @@ digraph pic {
Field examples
====================================================================
+The category of commutative fields, i.e. commutative rings where all
+nonzero elements have multiplicative inverses. The factor operation
+while trivial is useful to have defined.
+
+Axioms:
+ a*(b/a) = b
+ inv(a) = 1/a
+
See Also:
o )show Field
@@ 54391,15 +54297,6 @@ These exports come from \refto{DivisionRing}():
\begin{chunk}{category FIELD Field}
)abbrev category FIELD Field
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of commutative fields, i.e. commutative rings
++ where all nonzero elements have multiplicative inverses.
@@ 54571,6 +54468,8 @@ digraph pic {
IntegerNumberSystem examples
====================================================================
+An IntegerNumberSystem is a model for the integers.
+
See Also:
o )show IntegerNumberSystem
@@ 54847,7 +54746,6 @@ These exports come from \refto{LinearlyExplicitRingOver}(Integer):
)abbrev category INS IntegerNumberSystem
++ Author: Stephen M. Watt
++ Date Created: January 1988
++ Change History:
++ Description:
++ An \spad{IntegerNumberSystem} is a model for the integers.
@@ 55626,6 +55524,8 @@ digraph pic {
PAdicIntegerCategory examples
====================================================================
+This is the category of streambased representations of the padic integers.
+
See Also:
o )show PAdicIntegerCategory
@@ 55775,13 +55675,6 @@ These exports come from \refto{EuclideanDomain}():
++ Author: Clifton J. Williamson
++ Date Created: 15 May 1990
++ Date Last Updated: 15 May 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: padic, completion
++ Examples:
++ References:
++ Description:
++ This is the category of streambased representations of
++ the padic integers.
@@ 56028,6 +55921,9 @@ digraph pic {
PolynomialCategory examples
====================================================================
+The category for general multivariate polynomials over a ring R,
+in variables from VarSet, with exponents from the OrderedAbelianMonoidSup.
+
See Also:
o )show PolynomialCategory
@@ 56404,15 +56300,6 @@ These exports come from \refto{PolynomialFactorizationExplicit}():
\begin{chunk}{category POLYCAT PolynomialCategory}
)abbrev category POLYCAT PolynomialCategory
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions: Ring, monomial, coefficient, differentiate, eval
++ Related Constructors: Polynomial, DistributedMultivariatePolynomial
++ Also See: UnivariatePolynomialCategory
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category for general multivariate polynomials over a ring
++ R, in variables from VarSet, with exponents from the
@@ 57078,6 +56965,9 @@ digraph pic {
UnivariateTaylorSeriesCategory examples
====================================================================
+UnivariateTaylorSeriesCategory is the category of Taylor series
+in one variable.
+
See Also:
o )show UnivariateTaylorSeriesCategory
@@ 57370,13 +57260,6 @@ These exports come from \refto{RadicalCategory}():
++ Author: Clifton J. Williamson
++ Date Created: 21 December 1989
++ Date Last Updated: 26 May 1994
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Taylor, linebacker
++ Examples:
++ References:
++ Description:
++ \spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor
++ series in one variable.
@@ 57879,6 +57762,8 @@ zerosOf(sup,x)
AlgebraicallyClosedField examples
====================================================================
+This category is a model for algebraically closed fields.
+
Given the polynomial:
pi:Polynomial(Integer):=3*x^3+2*x+13
@@ 58139,7 +58024,6 @@ These exports come from \refto{RadicalCategory}():
++ Author: Manuel Bronstein
++ Date Created: 22 Mar 1988
++ Date Last Updated: 27 November 1991
++ Keywords: algebraic, closure, field.
++ Description:
++ Model for algebraically closed fields.
@@ 58545,6 +58429,28 @@ digraph pic {
DifferentialPolynomialCategory examples
====================================================================
+DifferentialPolynomialCategory is a category constructor specifying
+basic functions in an ordinary differential polynomial ring with a
+given ordered set of differential indeterminates. In addition, it
+implements defaults for the basic functions.
+
+The functions order and weight are extended from the set of
+derivatives of differential indeterminates to the set of differential
+polynomials. Other operations provided on differential polynomials are
+leader, initial, separant, differentialVariables, and isobaric?.
+Furthermore, if the ground ring is a differential ring, then evaluation
+(substitution of differential indeterminates by elements of the ground ring
+or by differential polynomials) is provided by eval.
+
+A convenient way of referencing derivatives is provided by the functions
+makeVariable.
+
+To construct a domain using this constructor, one needs to provide a
+ground ring R, an ordered set S of differential indeterminates, a ranking
+V on the set of derivatives of the differential indeterminates, and a set
+E of exponents in bijection with the set of differential monomials
+in the given differential indeterminates.
+
See Also:
o )show DifferentialPolynomialCategory
@@ 58941,12 +58847,6 @@ These exports come from \refto{Evalable}(%:DPOLCAT):
++ Author: William Sit
++ Date Created: 19 July 1990
++ Date Last Updated: 13 September 1991
++ Basic Operations:PolynomialCategory
++ Related Constructors:DifferentialVariableCategory
++ See Also:
++ AMS Classifications:12H05
++ Keywords: differential indeterminates, ranking, differential polynomials,
++ order, weight, leader, separant, initial, isobaric
++ References:Kolchin, E.R. "Differential Algebra and Algebraic Groups"
++ (Academic Press, 1973).
++ Description:
@@ 59330,6 +59230,11 @@ digraph pic {
FieldOfPrimeCharacteristic examples
====================================================================
+FieldOfPrimeCharacteristic is the category of fields of prime
+characteristic, e.g. finite fields, algebraic closures of
+fields of prime characteristic, transcendental extensions of
+of fields of prime characteristic.
+
See Also:
o )show FieldOfPrimeCharacteristic
@@ 59500,11 +59405,6 @@ These exports come from \refto{CharacteristicNonZero}():
++ Author: J. Grabmeier, A. Scheerhorn
++ Date Created: 10 March 1991
++ Date Last Updated: 31 March 1991
++ Basic Operations: _+, _*
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: field, finite field, prime characteristic
++ References:
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
@@ 59634,6 +59534,9 @@ digraph pic {
FiniteRankAlgebra examples
====================================================================
+A FiniteRankAlgebra is an algebra over a commutative ring R which
+is a free Rmodule of finite rank.
+
See Also:
o )show FiniteRankAlgebra
@@ 59760,14 +59663,6 @@ These exports come from \refto{CharacteristicZero}():
\begin{chunk}{category FINRALG FiniteRankAlgebra}
)abbrev category FINRALG FiniteRankAlgebra
++ Author: Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A FiniteRankAlgebra is an algebra over a commutative ring R which
++ is a free Rmodule of finite rank.
@@ 60091,6 +59986,9 @@ digraph pic {
FunctionSpace examples
====================================================================
+This is the category for formal functions.
+A space of formal functions with arguments in an arbitrary ordered set.
+
See Also:
o )show FunctionSpace
@@ 60566,12 +60464,11 @@ These exports come from \refto{RetractableTo}(Fraction(Integer)):
\begin{chunk}{category FS FunctionSpace}
)abbrev category FS FunctionSpace
++ Category for formal functions
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Date Last Updated: 14 February 1994
++ Keywords: operator, kernel, function.
++ Description:
+++ Category for formal functions
++ A space of formal functions with arguments in an arbitrary ordered set.
FunctionSpace(R:OrderedSet): Category == Definition where
@@ 61422,6 +61319,8 @@ digraph pic {
InfinitlyClosePointCategory examples
====================================================================
+This category is part of the PAFF package
+
See Also:
o )show InfinitlyClosePointCategory
@@ 61689,6 +61588,27 @@ digraph pic {
PseudoAlgebraicClosureOfPerfectFieldCategory examples
====================================================================
+This category exports the function for domains which implement dynamic
+extension using the simple notion of tower extensions. A tower extension
+T of the ground field K is any sequence of field extensions
+ (T : K_0, K_1, ..., K_i...,K_n) where K_0 = K
+and for
+ i =1,2,...,n, K_i is an extension of K_{i1} of degree > 1
+and defined by an irreducible polynomial p(Z) in K_{i1}.
+
+Two towers
+ (T_1: K_01, K_11,...,K_i1,...,K_n1)
+and
+ (T_2: K_02, K_12,...,K_i2,...,K_n2)
+are said to be related if
+ T_1 <= T_2 (or T_1 >= T_2),
+that is if
+ K_i1 = K_i2 for i=1,2,...,n1 (or i=1,2,...,n2).
+
+Any algebraic operations defined for several elements are only defined
+if all of the concerned elements are coming from a set of related tower
+extensions.
+
See Also:
o )show PseudoAlgebraicClosureOfPerfectFieldCategory
@@ 61880,7 +61800,8 @@ These exports come from \refto{DivisionRing}():
++ Authors: Gaetan Hache
++ Date Created: may 1997
++ Date Last Updated: April 2010, by Tim Daly
++ Description: This category exports the function for domains
+++ Description:
+++ This category exports the function for domains
++ which implement dynamic extension using the simple notion of tower
++ extensions. ++ A tower extension T of the ground
++ field K is any sequence of field extension
@@ 61892,8 +61813,8 @@ These exports come from \refto{DivisionRing}():
++ are said to be related if T_1 <= T_2 (or T_1 >= T_2),
++ that is if K_i1 = K_i2 for i=1,2,...,n1 (or i=1,2,...,n2).
++ Any algebraic operations defined for several elements
++ are only defined if all of the concerned elements are comming from
++ a set of related tour extensions.
+++ are only defined if all of the concerned elements are coming from
+++ a set of related tower extensions.
PseudoAlgebraicClosureOfPerfectFieldCategory() : Category == PUB where
INT ==> Integer
@@ 62099,6 +62020,8 @@ digraph pic {
QuotientFieldCategory examples
====================================================================
+QuotientField(S) is the category of fractions of an Integral Domain S.
+
See Also:
o )show QuotientFieldCategory
@@ 62454,15 +62377,7 @@ These exports come from \refto{PolynomialFactorizationExplicit}():
\begin{chunk}{category QFCAT QuotientFieldCategory}
)abbrev category QFCAT QuotientFieldCategory
++ Author:
++ Date Created:
++ Date Last Updated: 5th March 1996
++ Basic Functions: +  * / numer denom
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ QuotientField(S) is the category of fractions of an Integral Domain S.
@@ 62769,6 +62684,9 @@ digraph pic {
RealClosedField examples
====================================================================
+RealClosedField provides common access functions for all real closed fields.
+It provides computations with generic real roots of polynomials.
+
See Also:
o )show RealClosedField
@@ 63020,16 +62938,10 @@ These exports come from \refto{Algebra}(Integer):
++ Author: Renaud Rioboo
++ Date Created: may 1993
++ Date Last Updated: January 2004
++ Basic Functions: provides computations with generic real roots of
++ polynomials
++ Related Constructors: SimpleOrderedAlgebraicExtension, RealClosure
++ Also See:
++ AMS Classifications:
++ Keywords: Real Algebraic Numbers
++ References:
++ Description:
++ \axiomType{RealClosedField} provides common acces
+++ \axiomType{RealClosedField} provides common access
++ functions for all real closed fields.
+++ provides computations with generic real roots of polynomials
RealClosedField : Category == PUB where
@@ 63308,8 +63220,14 @@ digraph pic {
RealNumberSystem examples
====================================================================
+The real number system category is intended as a model for the real
+numbers. The real numbers form an ordered normed field. Note that
+we have purposely not included DifferentialRing or the elementary
+functions (see TranscendentalFunctionCategory) in the definition.
+
See Also:
o )show RealNumberSystem
+o )show TranscendentalFunctionCategory
\end{chunk}
{\bf See:}
@@ 63532,11 +63450,7 @@ These exports come from \refto{CharacteristicZero}():
\begin{chunk}{category RNS RealNumberSystem}
)abbrev category RNS RealNumberSystem
++ Author: Michael Monagan and Stephen M. Watt
++ Date Created:
++ January 1988
++ Change History:
++ Related Constructors:
++ Keywords: real numbers
+++ Date Created: January 1988
++ Description:
++ The real number system category is intended as a model for the real
++ numbers. The real numbers form an ordered normed field. Note that
@@ 63861,6 +63775,15 @@ digraph pic {
RecursivePolynomialCategory examples
====================================================================
+A category for general multivariate polynomials with coefficients
+in a ring, variables in an ordered set, and exponents from an
+ordered abelian monoid, with a sup operation.
+
+When not constant, such a polynomial is viewed as a univariate polynomial
+in its main variable w. r. t. to the total ordering on the elements in
+the ordered set, so that some operations usually defined for univariate
+polynomials make sense here.
+
See Also:
o )show RecursivePolynomialCategory
@@ 64355,19 +64278,7 @@ where R:Ring, E:OrderedAbelianMonoidSup, V:OrderedSet:
++ Author: Marc Moreno Maza
++ Date Created: 04/22/1994
++ Date Last Updated: 14/12/1998
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References:
++ Description:
++ A category for general multivariate polynomials with coefficients
++ in a ring, variables in an ordered set, and exponents from an
++ ordered abelian monoid, with a \axiomOp{sup} operation.
++ When not constant, such a polynomial is viewed as a univariate polynomial
++ in its main variable w. r. t. to the total ordering on the elements in
++ the ordered set, so that some operations usually defined for univariate
++ polynomials make sense here.
RecursivePolynomialCategory(R:Ring, E:OrderedAbelianMonoidSup, V:OrderedSet):_
Category ==
@@ 65790,6 +65701,9 @@ digraph pic {
UnivariateLaurentSeriesCategory examples
====================================================================
+UnivariateLaurentSeriesCategory is the category of Laurent series
+in one variable.
+
See Also:
o )show UnivariateLaurentSeriesCategory
@@ 66142,13 +66056,6 @@ These exports come from \refto{Field}():
++ Author: Clifton J. Williamson
++ Date Created: 21 December 1989
++ Date Last Updated: 20 September 1993
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Laurent
++ Examples:
++ References:
++ Description:
++ \spadtype{UnivariateLaurentSeriesCategory} is the category of
++ Laurent series in one variable.
@@ 66423,6 +66330,9 @@ digraph pic {
UnivariatePuiseuxSeriesCategory examples
====================================================================
+UnivariatePuiseuxSeriesCategory is the category of Puiseux series
+in one variable.
+
See Also:
o )show UnivariatePuiseuxSeriesCategory
@@ 66768,13 +66678,6 @@ These exports come from \refto{RadicalCategory}():
++ Author: Clifton J. Williamson
++ Date Created: 21 December 1989
++ Date Last Updated: 20 September 1993
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Puiseux
++ Examples:
++ References:
++ Description:
++ \spadtype{UnivariatePuiseuxSeriesCategory} is the category of Puiseux
++ series in one variable.
@@ 67091,6 +66994,9 @@ digraph pic {
UnivariatePolynomialCategory examples
====================================================================
+The category of univariate polynomials over a ring R. No particular
+model is assumed  implementations can be either sparse or dense.
+
See Also:
o )show UnivariatePolynomialCategory
@@ 67601,16 +67507,6 @@ These exports come from \refto{PolynomialFactorizationExplicit}()
\begin{chunk}{category UPOLYC UnivariatePolynomialCategory}
)abbrev category UPOLYC UnivariatePolynomialCategory
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions: Ring, monomial, coefficient, reductum, differentiate,
++ elt, map, resultant, discriminant
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of univariate polynomials over a ring R.
++ No particular model is assumed  implementations can be either
@@ 68273,6 +68169,8 @@ digraph pic {
AlgebraicallyClosedFunctionSpace examples
====================================================================
+Model for algebraically closed function spaces.
+
See Also:
o )show AlgebraicallyClosedFunctionSpace
@@ 68683,7 +68581,6 @@ where R:Join(OrderedSet, IntegralDomain)):
++ Author: Manuel Bronstein
++ Date Created: 31 October 1988
++ Date Last Updated: 7 October 1991
++ Keywords: algebraic, closure, field.
++ Description:
++ Model for algebraically closed function spaces.
@@ 68935,6 +68832,8 @@ digraph pic {
ExtensionField examples
====================================================================
+ExtensionField F is the category of fields which extend the field F
+
See Also:
o )show ExtensionField
@@ 69147,11 +69046,6 @@ These exports come from \refto{FieldOfPrimeCharacteristic}():
++ Author: J. Grabmeier, A. Scheerhorn
++ Date Created: 10 March 1991
++ Date Last Updated: 31 March 1991
++ Basic Operations: _+, _*, extensionDegree, algebraic?, transcendent?
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: field, extension field
++ References:
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
@@ 69348,6 +69242,8 @@ digraph pic {
FiniteFieldCategory examples
====================================================================
+FiniteFieldCategory is the category of finite fields
+
See Also:
o )show FiniteFieldCategory
@@ 69564,12 +69460,6 @@ These exports come from \refto{DifferentialRing}():
++ Author: J. Grabmeier, A. Scheerhorn
++ Date Created: 11 March 1991
++ Date Last Updated: 31 March 1991
++ Basic Operations: _+, _*, extensionDegree, order, primitiveElement
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: field, extension field, algebraic extension, finite field
++ Galois field
++ References:
++ D.Lipson, Elements of Algebra and Algebraic Computing, The
++ Benjamin/Cummings Publishing Company, Inc.Menlo Park, California, 1981.
@@ 69947,6 +69837,25 @@ digraph pic {
FloatingPointSystem examples
====================================================================
+This category is intended as a model for floating point systems.
+A floating point system is a model for the real numbers. In fact,
+it is an approximation in the sense that not all real numbers are
+exactly representable by floating point numbers.
+
+A floating point system is characterized by the following:
+
+ 1: base of the exponent where the actual implemenations are
+ usually binary or decimal)
+ 2: precision of the mantissa (arbitrary or fixed)
+ 3: rounding error for operations
+ 4: when, and what happens if exponent overflow/underflow occurs
+
+Because a Float is an approximation to the real numbers, even though
+it is defined to be a join of a Field and OrderedRing, some of
+the attributes do not hold. In particular associative("+")
+does not hold. Algorithms defined over a field need special
+considerations when the field is a floating point system.
+
See Also:
o )show FloatingPointSystem
@@ 70192,13 +70101,6 @@ These exports come from \refto{RealNumberSystem}():
\begin{chunk}{category FPS FloatingPointSystem}
)abbrev category FPS FloatingPointSystem
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations: approximate, base, bits, digits, exponent, float,
++ mantissa, order, precision, round?
++ Related Constructors:
++ Keywords: float, floating point
++ Description:
++ This category is intended as a model for floating point systems.
++ A floating point system is a model for the real numbers. In fact,
@@ 70392,6 +70294,8 @@ digraph pic {
FramedAlgebra examples
====================================================================
+A FramedAlgebra is a FiniteRankAlgebra together with a fixed Rmodule basis.
+
See Also:
o )show FramedAlgebra
@@ 70515,14 +70419,6 @@ where R:CommutativeRing and UP:UnivariatePolynomialCategory R):
\begin{chunk}{category FRAMALG FramedAlgebra}
)abbrev category FRAMALG FramedAlgebra
++ Author: Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A \spadtype{FramedAlgebra} is a \spadtype{FiniteRankAlgebra} together
++ with a fixed Rmodule basis.
@@ 70757,6 +70653,28 @@ digraph pic {
PseudoAlgebraicClosureOfFiniteFieldCategory examples
====================================================================
+This category exports the function for the domain
+PseudoAlgebraicClosureOfFiniteField which implement dynamic extension
+using the simple notion of tower extensions.
+
+A tower extension T of the ground field K is any sequence of field extension
+ (T : K_0, K_1, ..., K_i...,K_n)
+where K_0 = K and for i =1,2,...,n, K_i is an extension
+of K_{i1} of degree > 1 and defined by an irreducible polynomial
+p(Z) in K_{i1}.
+
+Two towers
+ (T_1: K_01, K_11,...,K_i1,...,K_n1)
+and
+ (T_2: K_02, K_12,...,K_i2,...,K_n2)
+are said to be related if
+ T_1 <= T_2 (or T_1 >= T_2),
+that is if
+ K_i1 = K_i2 for i=1,2,...,n1
+(or i=1,2,...,n2). Any algebraic operations defined for several elements
+are only defined if all of the concerned elements are comming from
+a set of related tour extensions.
+
See Also:
o )show PseudoAlgebraicClosureOfFiniteFieldCategory
@@ 70980,9 +70898,8 @@ These exports come from \refto{FiniteFieldCategory}():
 PseudoAlgebraicClosureOfFiniteFieldCategory
++ Authors: Gaetan Hache
++ Date Created: june 1996
++ Date Last Updated:
++ References:
++ Description: This category exports the function for the domain
+++ Description:
+++ This category exports the function for the domain
++ PseudoAlgebraicClosureOfFiniteField which implement dynamic extension
++ using the simple notion of tower extensions.
++ A tower extension T of the ground
@@ 71245,6 +71162,11 @@ digraph pic {
UnivariateLaurentSeriesConstructorCategory examples
====================================================================
+This is a category of univariate Laurent series constructed from
+univariate Taylor series. A Laurent series is represented by a pair
+[n,f(x)], where n is an arbitrary integer and f(x) is a Taylor series.
+This pair represents the Laurent series x**n * f(x).
+
See Also:
o )show UnivariateLaurentSeriesConstructorCategory
@@ 71767,13 +71689,6 @@ where UTS:UnivariateLaurentSeriesCategory(Coef:Ring)
++ Author: Clifton J. Williamson
++ Date Created: 6 February 1990
++ Date Last Updated: 10 May 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Laurent, Taylor
++ Examples:
++ References:
++ Description:
++ This is a category of univariate Laurent series constructed from
++ univariate Taylor series. A Laurent series is represented by a pair
@@ 72056,6 +71971,11 @@ digraph pic {
UnivariatePuiseuxSeriesConstructorCategory examples
====================================================================
+This is a category of univariate Puiseux series constructed from
+univariate Laurent series. A Puiseux series is represented by a pair
+[r,f(x)], where r is a positive rational number and f(x) is a Laurent
+series. This pair represents the Puiseux series f(x^r).
+
See Also:
o )show UnivariatePuiseuxSeriesConstructorCategory
@@ 72400,13 +72320,6 @@ These exports come from \refto{UnivariatePuiseuxSeriesCategory}(Coef:Ring):
++ Author: Clifton J. Williamson
++ Date Created: 6 February 1990
++ Date Last Updated: 22 March 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Puiseux, Laurent
++ Examples:
++ References:
++ Description:
++ This is a category of univariate Puiseux series constructed
++ from univariate Laurent series. A Puiseux series is represented
@@ 72643,6 +72556,31 @@ digraph pic {
FiniteAlgebraicExtensionField examples
====================================================================
+FiniteAlgebraicExtensionField F is the category of fields
+which are finite algebraic extensions of the field F.
+
+If F is finite then any finite algebraic extension of F is finite, too.
+Let K be a finite algebraic extension of the finite field F. The
+exponentiation of elements of K defines a Zmodule structure on the
+multiplicative group of K.
+
+The additive group of K becomes a module over the ring of polynomials
+over F via the operation
+ linearAssociatedExp(a:K,f:SparseUnivariatePolynomial F)
+which is linear over F, i.e. for elements a from K, c,d from F and
+f,g univariate polynomials over F we have linearAssociatedExp}(a,cf+dg)
+equals c times linearAssociatedExp}(a,f) plus d times
+linearAssociatedExp}(a,g).
+
+Therefore linearAssociatedExp is defined completely by its action on
+monomials from F[X]: linearAssociatedExp(a,monomial(1,k)\$SUP(F)) is
+defined to be Frobenius(a,k) which is a**(q**k) where q=size()\$F.
+
+The operations order and discreteLog associated with the multiplicative
+exponentiation have additive analogues associated to the operation
+linearAssociatedExp. These are the functions linearAssociatedOrder
+and linearAssociatedLog, respectively.
+
See Also:
o )show FiniteAlgebraicExtensionField
@@ 72934,11 +72872,6 @@ These exports come from \refto{FiniteFieldCategory}():
++ Author: J. Grabmeier, A. Scheerhorn
++ Date Created: 11 March 1991
++ Date Last Updated: 31 March 1991
++ Basic Operations: _+, _*, extensionDegree,
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: field, extension field, algebraic extension, finite extension
++ References:
++ R.Lidl, H.Niederreiter: Finite Field, Encycoldia of Mathematics and
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983,
@@ 73437,6 +73370,9 @@ digraph pic {
MonogenicAlgebra examples
====================================================================
+A MonogenicAlgebra is an algebra of finite rank which can be
+generated by a single element.
+
See Also:
o )show MonogenicAlgebra
@@ 73786,14 +73722,6 @@ These exports come from \refto{FiniteFieldCategory}():
\begin{chunk}{category MONOGEN MonogenicAlgebra}
)abbrev category MONOGEN MonogenicAlgebra
++ Author: Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A \spadtype{MonogenicAlgebra} is an algebra of finite rank which
++ can be generated by a single element.
@@ 74033,6 +73961,27 @@ digraph pic {
PseudoAlgebraicClosureOfRationalNumberCategory examples
====================================================================
+This category exports the function for the domain
+PseudoAlgebraicClosureOfRationalNumber which implement dynamic extension
+using the simple notion of tower extensions. A tower extension T of the
+ground field K is any sequence of field extension
+ (T : K_0, K_1, ..., K_i...,K_n)
+where K_0 = K and for i =1,2,...,n, K_i is an extension
+of K_{i1} of degree > 1 and defined by an irreducible polynomial
+p(Z) in K_{i1}.
+
+Two towers
+ (T_1: K_01, K_11,...,K_i1,...,K_n1)
+and
+ (T_2: K_02, K_12,...,K_i2,...,K_n2)
+are said to be related if
+ T_1 <= T_2 (or T_1 >= T_2),
+that is if
+ K_i1 = K_i2 for i=1,2,...,n1
+(or i=1,2,...,n2). Any algebraic operations defined for several elements
+are only defined if all of the concerned elements are comming from
+a set of related tour extensions.
+
See Also:
o )show PseudoAlgebraicClosureOfRationalNumberCategory
@@ 74264,8 +74213,8 @@ These exports come from \refto{ExtensionField}(Fraction(Integer)):
)abbrev category PACRATC PseudoAlgebraicClosureOfRationalNumberCategory
++ Authors: Gaetan Hache
++ Date Created: feb 1997
++ Date Last Updated:
++ Description: This category exports the function for the domain
+++ Description:
+++ This category exports the function for the domain
++ PseudoAlgebraicClosureOfRationalNumber
++ which implement dynamic extension using the simple notion of tower
++ extensions. A tower extension T of the ground
@@ 74524,6 +74473,8 @@ digraph pic {
ComplexCategory examples
====================================================================
+This category represents the extension of a ring by a square root of 1.
+
See Also:
o )show ComplexCategory
@@ 74960,15 +74911,7 @@ These exports come from \refto{PolynomialFactorizationExplicit}():
\begin{chunk}{category COMPCAT ComplexCategory}
)abbrev category COMPCAT ComplexCategory
++ Author:
++ Date Created:
++ Date Last Updated: 18 March 1994
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: complex, gaussian
++ References:
++ Description:
++ This category represents the extension of a ring by a square root of 1.
@@ 75642,6 +75585,8 @@ digraph pic {
FunctionFieldCategory examples
====================================================================
+This category is a model for the function field of a plane algebraic curve.
+
See Also:
o )show FunctionFieldCategory
@@ 76162,12 +76107,11 @@ UPUP:UnivariatePolynomialCategory Fraction UP
\begin{chunk}{category FFCAT FunctionFieldCategory}
)abbrev category FFCAT FunctionFieldCategory
++ Function field of a curve
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Date Last Updated: 19 Mai 1993
++ Keywords: algebraic, curve, function, field.
++ Description:
+++ Function field of a curve
++ This category is a model for the function field of a
++ plane algebraic curve.
@@ 76736,6 +76680,27 @@ digraph pic {
PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory examples
====================================================================
+This category exports the function for the domain
+PseudoAlgebraicClosureOfAlgExtOfRationalNumber which implement dynamic
+extension using the simple notion of tower extensions. A tower extension
+T of the ground field K is any sequence of field extension
+ (T : K_0, K_1, ..., K_i...,K_n)
+where K_0 = K and for i =1,2,...,n,
+ K_i is an extension of K_{i1} of degree > 1
+and defined by an irreducible polynomial p(Z) in K_{i1}.
+
+Two towers
+ (T_1: K_01, K_11,...,K_i1,...,K_n1)
+and
+ (T_2: K_02, K_12,...,K_i2,...,K_n2)
+are said to be related if
+ T_1 <= T_2 (or T_1 >= T_2),
+that is if
+ K_i1 = K_i2 for i=1,2,...,n1 (or i=1,2,...,n2).
+Any algebraic operations defined for several elements
+are only defined if all of the concerned elements are comming from
+a set of related tour extensions.
+
See Also:
o )show PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory
@@ 92092,33 +92057,6 @@ digraph dotfull {
\end{chunk}
\eject
\begin{thebibliography}{99}
\bibitem{1} N. Jacobson: Structure and Representations of Jordan Algebras
AMS, Providence, 1968
\bibitem{2} MacLane and Birkhoff, Algebra 2d Edition, MacMillan 1979
\bibitem{3} Encyclopedic Dictionary of Mathematics, MIT Press, 1977
\bibitem{4} R.D. Schafer: An Introduction to Nonassociative Algebras
Academic Press, New York, 1966
\bibitem{5} R. Wisbauer: Bimodule Structure of Algebra
Lecture Notes Univ. Duesseldorf 1991
\bibitem{6} J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
AXIOM Technical Report Series, ATR/5 NP2522.
\bibitem{7} R. Rioboo,
{\sl Real Algebraic Closure of an ordered Field : Implementation in Axiom.},
In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206215.
\bibitem{8} Z. Ligatsikas, R. Rioboo, M. F. Roy
{\sl Generic computation of the real closure of an ordered field.},
In Mathematics and Computers in Simulation Volume 42, Issue 46,
November 1996.
\bibitem{9} D. LAZARD ``A new method for solving algebraic systems of
positive dimension'' Discr. App. Math. 33:147160,1991
\bibitem{10} P. AUBRY, D. LAZARD and M. MORENO MAZA ``On the Theories
of Triangular Sets'' Journal of Symbol. Comp. (to appear)
\bibitem{11} M. MORENO MAZA and R. RIOBOO ``Computations of gcd over
algebraic towers of simple extensions'' In proceedings of AAECC11
Paris, 1995.
\bibitem{12} M. MORENO MAZA ``Calculs de pgcd audessus des tours
d'extensions simples et resolution des systemes d'equations
algebriques'' These, Universite P.etM. Curie, Paris, 1997.
\end{thebibliography}
\printindex
\end{document}
diff git a/changelog b/changelog
index b8bfaaa..328dc18 100644
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+++ b/changelog
@@ 1,3 +1,5 @@
+20130228 tpd src/axiomwebsite/patches.html 20130228.02.tpd.patch
+20130228 tpd books/bookvol10.2 write help documentation for all categories
20130228 tpd src/axiomwebsite/patches.html 20130228.01.tpd.patch
20130228 tpd books/bookvolbib add references
20130227 tpd src/axiomwebsite/patches.html 20130227.02.tpd.patch
diff git a/src/axiomwebsite/patches.html b/src/axiomwebsite/patches.html
index 96222a4..e3fb43b 100644
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+++ b/src/axiomwebsite/patches.html
@@ 3997,5 +3997,7 @@ books/bookvol10.3 add U8Matrix
books/bookvol10.4 add U32VectorPolynomialOperations
20130228.01.tpd.patch
books/bookvolbib add references
+20130228.02.tpd.patch
+books/bookvol10.2 write help documentation for all categories