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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declarations of the arithmetic operations, and the
## declarations of the categories for elements that allow those operations.
##
## This file contains the definitions of categories for elements in families
## that allow certain arithmetical operations,
## and the definition of properties, attributes, and operations for these
## elements.
##
## Note that the arithmetical operations are usually only partial functions.
## This means that a multiplicative element is simply an element whose
## family allows a multiplication of *some* of its elements. It does *not*
## mean that the product of *any* two elements in the family is defined,
##
#############################################################################
##
#C IsExtAElement( <obj> )
##
## <#GAPDoc Label="IsExtAElement">
## <ManSection>
## <Filt Name="IsExtAElement" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>external additive element</E> is an object that can be added via
## <C>+</C> with other elements
## (not necessarily in the same family, see <Ref Sect="Families"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsExtAElement", IsObject );
DeclareCategoryCollections( "IsExtAElement" );
DeclareCategoryCollections( "IsExtAElementCollection" );
DeclareSynonym( "IsExtAElementList",
IsExtAElementCollection and IsList );
DeclareSynonym( "IsExtAElementTable",
IsExtAElementCollColl and IsTable );
InstallTrueMethod( IsExtAElement,
IsExtAElementCollection );
#############################################################################
##
#C IsNearAdditiveElement( <obj> )
##
## <#GAPDoc Label="IsNearAdditiveElement">
## <ManSection>
## <Filt Name="IsNearAdditiveElement" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>near-additive element</E> is an object that can be added via
## <C>+</C> with elements in its family (see <Ref Sect="Families"/>);
## this addition is not necessarily commutative.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsNearAdditiveElement", IsExtAElement );
DeclareCategoryCollections( "IsNearAdditiveElement" );
DeclareCategoryCollections( "IsNearAdditiveElementCollection" );
DeclareCategoryCollections( "IsNearAdditiveElementCollColl" );
DeclareSynonym( "IsNearAdditiveElementList",
IsNearAdditiveElementCollection and IsList );
DeclareSynonym( "IsNearAdditiveElementTable",
IsNearAdditiveElementCollColl and IsTable );
InstallTrueMethod( IsNearAdditiveElement,
IsNearAdditiveElementList );
InstallTrueMethod( IsNearAdditiveElementList,
IsNearAdditiveElementTable );
#############################################################################
##
#C IsNearAdditiveElementWithZero( <obj> )
##
## <#GAPDoc Label="IsNearAdditiveElementWithZero">
## <ManSection>
## <Filt Name="IsNearAdditiveElementWithZero" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>near-additive element-with-zero</E> is an object that can be added
## via <C>+</C> with elements in its family
## (see <Ref Sect="Families"/>),
## and that is an admissible argument for the operation <Ref Attr="Zero"/>;
## this addition is not necessarily commutative.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsNearAdditiveElementWithZero", IsNearAdditiveElement );
DeclareCategoryCollections( "IsNearAdditiveElementWithZero" );
DeclareCategoryCollections( "IsNearAdditiveElementWithZeroCollection" );
DeclareCategoryCollections( "IsNearAdditiveElementWithZeroCollColl" );
DeclareSynonym( "IsNearAdditiveElementWithZeroList",
IsNearAdditiveElementWithZeroCollection and IsList );
DeclareSynonym( "IsNearAdditiveElementWithZeroTable",
IsNearAdditiveElementWithZeroCollColl and IsTable );
InstallTrueMethod(
IsNearAdditiveElementWithZero,
IsNearAdditiveElementWithZeroList );
InstallTrueMethod(
IsNearAdditiveElementWithZeroList,
IsNearAdditiveElementWithZeroTable );
#############################################################################
##
#C IsNearAdditiveElementWithInverse( <obj> )
##
## <#GAPDoc Label="IsNearAdditiveElementWithInverse">
## <ManSection>
## <Filt Name="IsNearAdditiveElementWithInverse" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>near-additive element-with-inverse</E> is an object that can be
## added via <C>+</C> with elements in its family
## (see <Ref Sect="Families"/>),
## and that is an admissible argument for the operations <Ref Attr="Zero"/>
## and <Ref Attr="AdditiveInverse"/>;
## this addition is not necessarily commutative.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsNearAdditiveElementWithInverse",
IsNearAdditiveElementWithZero );
DeclareCategoryCollections( "IsNearAdditiveElementWithInverse" );
DeclareCategoryCollections( "IsNearAdditiveElementWithInverseCollection" );
DeclareCategoryCollections( "IsNearAdditiveElementWithInverseCollColl" );
DeclareSynonym( "IsNearAdditiveElementWithInverseList",
IsNearAdditiveElementWithInverseCollection and IsList );
DeclareSynonym( "IsNearAdditiveElementWithInverseTable",
IsNearAdditiveElementWithInverseCollColl and IsTable );
InstallTrueMethod(
IsNearAdditiveElementWithInverse,
IsNearAdditiveElementWithInverseList );
InstallTrueMethod(
IsNearAdditiveElementWithInverseList,
IsNearAdditiveElementWithInverseTable );
#############################################################################
##
#C IsAdditiveElement( <obj> )
##
## <#GAPDoc Label="IsAdditiveElement">
## <ManSection>
## <Filt Name="IsAdditiveElement" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>additive element</E> is an object that can be added via <C>+</C>
## with elements in its family (see <Ref Sect="Families"/>);
## this addition is commutative.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsAdditiveElement", IsNearAdditiveElement );
DeclareCategoryCollections( "IsAdditiveElement" );
DeclareCategoryCollections( "IsAdditiveElementCollection" );
DeclareCategoryCollections( "IsAdditiveElementCollColl" );
DeclareSynonym( "IsAdditiveElementList",
IsAdditiveElementCollection and IsList );
DeclareSynonym( "IsAdditiveElementTable",
IsAdditiveElementCollColl and IsTable );
InstallTrueMethod( IsAdditiveElement,
IsAdditiveElementList );
InstallTrueMethod( IsAdditiveElementList,
IsAdditiveElementTable );
#############################################################################
##
#C IsAdditiveElementWithZero( <obj> )
##
## <#GAPDoc Label="IsAdditiveElementWithZero">
## <ManSection>
## <Filt Name="IsAdditiveElementWithZero" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>additive element-with-zero</E> is an object that can be added
## via <C>+</C> with elements in its family
## (see <Ref Sect="Families"/>),
## and that is an admissible argument for the operation <Ref Attr="Zero"/>;
## this addition is commutative.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsAdditiveElementWithZero",
IsNearAdditiveElementWithZero and IsAdditiveElement );
DeclareSynonym( "IsAdditiveElementWithZeroCollection",
IsNearAdditiveElementWithZeroCollection
and IsAdditiveElementCollection );
DeclareSynonym( "IsAdditiveElementWithZeroCollColl",
IsNearAdditiveElementWithZeroCollColl
and IsAdditiveElementCollColl );
DeclareSynonym( "IsAdditiveElementWithZeroCollCollColl",
IsNearAdditiveElementWithZeroCollCollColl
and IsAdditiveElementCollCollColl );
DeclareSynonym( "IsAdditiveElementWithZeroList",
IsAdditiveElementWithZeroCollection and IsList );
DeclareSynonym( "IsAdditiveElementWithZeroTable",
IsAdditiveElementWithZeroCollColl and IsTable );
InstallTrueMethod(
IsAdditiveElementWithZero,
IsAdditiveElementWithZeroList );
InstallTrueMethod(
IsAdditiveElementWithZeroList,
IsAdditiveElementWithZeroTable );
#############################################################################
##
#C IsAdditiveElementWithInverse( <obj> )
##
## <#GAPDoc Label="IsAdditiveElementWithInverse">
## <ManSection>
## <Filt Name="IsAdditiveElementWithInverse" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>additive element-with-inverse</E> is an object that can be
## added via <C>+</C> with elements in its family
## (see <Ref Sect="Families"/>),
## and that is an admissible argument for the operations <Ref Attr="Zero"/>
## and <Ref Attr="AdditiveInverse"/>;
## this addition is commutative.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsAdditiveElementWithInverse",
IsNearAdditiveElementWithInverse and IsAdditiveElement );
DeclareSynonym( "IsAdditiveElementWithInverseCollection",
IsNearAdditiveElementWithInverseCollection
and IsAdditiveElementCollection );
DeclareSynonym( "IsAdditiveElementWithInverseCollColl",
IsNearAdditiveElementWithInverseCollColl
and IsAdditiveElementCollColl );
DeclareSynonym( "IsAdditiveElementWithInverseCollCollColl",
IsNearAdditiveElementWithInverseCollCollColl
and IsAdditiveElementCollCollColl );
DeclareSynonym( "IsAdditiveElementWithInverseList",
IsAdditiveElementWithInverseCollection and IsList );
DeclareSynonym( "IsAdditiveElementWithInverseTable",
IsAdditiveElementWithInverseCollColl and IsTable );
InstallTrueMethod(
IsAdditiveElementWithInverse,
IsAdditiveElementWithInverseList );
InstallTrueMethod(
IsAdditiveElementWithInverseList,
IsAdditiveElementWithInverseTable );
#############################################################################
##
#C IsExtLElement( <obj> )
##
## <#GAPDoc Label="IsExtLElement">
## <ManSection>
## <Filt Name="IsExtLElement" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>external left element</E> is an object that can be multiplied
## from the left, via <C>*</C>, with other elements
## (not necessarily in the same family, see <Ref Sect="Families"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsExtLElement", IsObject );
DeclareCategoryCollections( "IsExtLElement" );
DeclareCategoryCollections( "IsExtLElementCollection" );
DeclareSynonym( "IsExtLElementList",
IsExtLElementCollection and IsList );
DeclareSynonym( "IsExtLElementTable",
IsExtLElementCollColl and IsTable );
InstallTrueMethod(
IsExtLElement,
IsExtLElementCollection );
#############################################################################
##
#C IsExtRElement( <obj> )
##
## <#GAPDoc Label="IsExtRElement">
## <ManSection>
## <Filt Name="IsExtRElement" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>external right element</E> is an object that can be multiplied
## from the right, via <C>*</C>, with other elements
## (not necessarily in the same family, see <Ref Sect="Families"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsExtRElement", IsObject );
DeclareCategoryCollections( "IsExtRElement" );
DeclareCategoryCollections( "IsExtRElementCollection" );
DeclareSynonym( "IsExtRElementList",
IsExtRElementCollection and IsList );
DeclareSynonym( "IsExtRElementTable",
IsExtRElementCollColl and IsTable );
InstallTrueMethod(
IsExtRElement,
IsExtRElementCollection );
#############################################################################
##
#C IsMultiplicativeElement( <obj> )
##
## <#GAPDoc Label="IsMultiplicativeElement">
## <ManSection>
## <Filt Name="IsMultiplicativeElement" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>multiplicative element</E> is an object that can be multiplied via
## <C>*</C> with elements in its family (see <Ref Sect="Families"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsMultiplicativeElement",
IsExtLElement and IsExtRElement );
DeclareCategoryCollections( "IsMultiplicativeElement" );
DeclareCategoryCollections( "IsMultiplicativeElementCollection" );
DeclareCategoryCollections( "IsMultiplicativeElementCollColl" );
DeclareSynonym( "IsMultiplicativeElementList",
IsMultiplicativeElementCollection and IsList );
DeclareSynonym( "IsMultiplicativeElementTable",
IsMultiplicativeElementCollColl and IsTable );
#############################################################################
##
#C IsMultiplicativeElementWithOne( <obj> )
##
## <#GAPDoc Label="IsMultiplicativeElementWithOne">
## <ManSection>
## <Filt Name="IsMultiplicativeElementWithOne" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>multiplicative element-with-one</E> is an object that can be
## multiplied via <C>*</C> with elements in its family
## (see <Ref Sect="Families"/>),
## and that is an admissible argument for the operation <Ref Attr="One"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsMultiplicativeElementWithOne",
IsMultiplicativeElement );
DeclareCategoryCollections( "IsMultiplicativeElementWithOne" );
DeclareCategoryCollections( "IsMultiplicativeElementWithOneCollection" );
DeclareCategoryCollections( "IsMultiplicativeElementWithOneCollColl" );
DeclareSynonym( "IsMultiplicativeElementWithOneList",
IsMultiplicativeElementWithOneCollection and IsList );
DeclareSynonym( "IsMultiplicativeElementWithOneTable",
IsMultiplicativeElementWithOneCollColl and IsTable );
## <#GAPDoc Label="IsMultiplicativeElementWithZero">
## <ManSection>
## <Filt Name="IsMultiplicativeElementWithZero" Arg='elt' Type='Category'/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
## <Description>
## This is the category of elements in a family which can be the operands of
## <C>*</C> (multiplication) and the operation
## <Ref Oper="MultiplicativeZeroOp"/>.
## <Example><![CDATA[
## gap> S:=Semigroup(Transformation( [ 1, 1, 1 ] ));;
## gap> M:=MagmaWithZeroAdjoined(S);
## <<commutative transformation semigroup of degree 3 with 1 generator>
## with 0 adjoined>
## gap> x:=Representative(M);
## <semigroup with 0 adjoined elt: Transformation( [ 1, 1, 1 ] )>
## gap> IsMultiplicativeElementWithZero(x);
## true
## gap> MultiplicativeZeroOp(x);
## <semigroup with 0 adjoined elt: 0>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareCategory("IsMultiplicativeElementWithZero",IsMultiplicativeElement);
DeclareCategoryCollections("IsMultiplicativeElementWithZero");
#############################################################################
##
#C IsMultiplicativeElementWithInverse( <obj> )
##
## <#GAPDoc Label="IsMultiplicativeElementWithInverse">
## <ManSection>
## <Filt Name="IsMultiplicativeElementWithInverse" Arg='obj'
## Type='Category'/>
##
## <Description>
## A <E>multiplicative element-with-inverse</E> is an object that can be
## multiplied via <C>*</C> with elements in its family
## (see <Ref Sect="Families"/>),
## and that is an admissible argument for the operations <Ref Attr="One"/>
## and <Ref Attr="Inverse"/>. (Note the word <Q>admissible</Q>: an
## object in this category does not necessarily have an inverse,
## <Ref Attr="Inverse"/> may return <K>fail</K>.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsMultiplicativeElementWithInverse",
IsMultiplicativeElementWithOne );
DeclareCategoryCollections( "IsMultiplicativeElementWithInverse" );
DeclareCategoryCollections( "IsMultiplicativeElementWithInverseCollection" );
DeclareCategoryCollections( "IsMultiplicativeElementWithInverseCollColl" );
DeclareSynonym( "IsMultiplicativeElementWithInverseList",
IsMultiplicativeElementWithInverseCollection and IsList );
DeclareSynonym( "IsMultiplicativeElementWithInverseTable",
IsMultiplicativeElementWithInverseCollColl and IsTable );
#############################################################################
##
#C IsVector( <obj> )
##
## <#GAPDoc Label="IsVector">
## <ManSection>
## <Filt Name="IsVector" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>vector</E> is an additive-element-with-inverse that can be
## multiplied from the left and right with other objects
## (not necessarily of the same type).
## Examples are cyclotomics, finite field elements,
## and of course row vectors (see below).
## <P/>
## Note that not all lists of ring elements are regarded as vectors,
## for example lists of matrices are not vectors.
## This is because although the category
## <Ref Filt="IsAdditiveElementWithInverse"/> is
## implied by the meet of its collections category and <Ref Filt="IsList"/>,
## the family of a list entry may not imply
## <Ref Filt="IsAdditiveElementWithInverse"/> for all its elements.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsVector",
IsAdditiveElementWithInverse
and IsExtLElement
and IsExtRElement );
DeclareSynonym( "IsVectorCollection",
IsAdditiveElementWithInverseCollection
and IsExtLElementCollection
and IsExtRElementCollection );
DeclareSynonym( "IsVectorCollColl",
IsAdditiveElementWithInverseCollColl
and IsExtLElementCollColl
and IsExtRElementCollColl );
DeclareSynonym( "IsVectorList",
IsAdditiveElementWithInverseList
and IsExtLElementList
and IsExtRElementList );
DeclareSynonym( "IsVectorTable",
IsAdditiveElementWithInverseTable
and IsExtLElementTable
and IsExtRElementTable );
#############################################################################
##
#F IsOddAdditiveNestingDepthFamily( <Fam> )
#F IsOddAdditiveNestingDepthObject( <Fam> )
##
## <ManSection>
## <Filt Name="IsOddAdditiveNestingDepthFamily" Arg='Fam'/>
## <Filt Name="IsOddAdditiveNestingDepthObject" Arg='Fam'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareFilter( "IsOddAdditiveNestingDepthFamily" );
DeclareFilter( "IsOddAdditiveNestingDepthObject" );
#############################################################################
##
#C IsRowVector( <obj> )
##
## <#GAPDoc Label="IsRowVector">
## <ManSection>
## <Filt Name="IsRowVector" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>row vector</E> is a vector (see <Ref Filt="IsVector"/>)
## that is also a homogeneous list of odd additive nesting depth
## (see <Ref Sect="Filters Controlling the Arithmetic Behaviour of Lists"/>).
## Typical examples are lists of integers and rationals,
## lists of finite field elements of the same characteristic,
## and lists of polynomials from a common polynomial ring.
## Note that matrices are <E>not</E> regarded as row vectors, because they have
## even additive nesting depth.
## <P/>
## The additive operations of the vector must thus be compatible with
## that for lists, implying that the list entries are the
## coefficients of the vector with respect to some basis.
## <P/>
## Note that not all row vectors admit a multiplication via <C>*</C>
## (which is to be understood as a scalar product);
## for example, class functions are row vectors but the product of two
## class functions is defined in a different way.
## For the installation of a scalar product of row vectors, the entries of
## the vector must be ring elements; note that the default method expects
## the row vectors to lie in <C>IsRingElementList</C>,
## and this category may not be implied by <Ref Filt="IsRingElement"/>
## for all entries of the row vector
## (see the comment in <Ref Filt="IsVector"/>).
## <P/>
## Note that methods for special types of row vectors really must be
## installed with the requirement <Ref Filt="IsRowVector"/>,
## since <Ref Filt="IsVector"/> may lead to a rank of the method below
## that of the default method for row vectors (see file <F>lib/vecmat.gi</F>).
## <P/>
## <Example><![CDATA[
## gap> IsRowVector([1,2,3]);
## true
## ]]></Example>
## <P/>
## Because row vectors are just a special case of lists, all operations
## and functions for lists are applicable to row vectors as well (see
## Chapter <Ref Chap="Lists"/>).
## This especially includes accessing elements of a row vector
## (see <Ref Sect="List Elements"/>), changing elements of a mutable row
## vector (see <Ref Sect="List Assignment"/>),
## and comparing row vectors (see <Ref Sect="Comparisons of Lists"/>).
## <P/>
## Note that, unless your algorithms specifically require you to be able
## to change entries of your vectors, it is generally better and faster
## to work with immutable row vectors.
## See Section <Ref Sect="Mutability and Copyability"/> for more
## details.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsRowVector",
IsVector and IsHomogeneousList and IsOddAdditiveNestingDepthObject );
#############################################################################
##
## Filters Controlling the Arithmetic Behaviour of Lists
## <#GAPDoc Label="[1]{arith}">
## The arithmetic behaviour of lists is controlled by their types.
## The following categories and attributes are used for that.
## <P/>
## Note that we distinguish additive and multiplicative behaviour.
## For example, Lie matrices have the usual additive behaviour but not the
## usual multiplicative behaviour.
## <#/GAPDoc>
##
#############################################################################
##
#C IsGeneralizedRowVector( <list> ) . . . objects that comply with new list
## addition rules
##
## <#GAPDoc Label="IsGeneralizedRowVector">
## <ManSection>
## <Filt Name="IsGeneralizedRowVector" Arg='list' Type='Category'/>
##
## <Description>
## For a list <A>list</A>, the value <K>true</K> for
## <Ref Filt="IsGeneralizedRowVector"/>
## indicates that the additive arithmetic behaviour of <A>list</A> is
## as defined in <Ref Sect="Additive Arithmetic for Lists"/>,
## and that the attribute <Ref Attr="NestingDepthA"/>
## will return a nonzero value when called with <A>list</A>.
## <P/>
## <Example><![CDATA[
## gap> IsList( "abc" ); IsGeneralizedRowVector( "abc" );
## true
## false
## gap> liemat:= LieObject( [ [ 1, 2 ], [ 3, 4 ] ] );
## LieObject( [ [ 1, 2 ], [ 3, 4 ] ] )
## gap> IsGeneralizedRowVector( liemat );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsGeneralizedRowVector",
IsList and IsAdditiveElementWithInverse );
#############################################################################
##
#C IsMultiplicativeGeneralizedRowVector( <list> ) . . . .
## objects that comply with new list multiplication rules
##
## <#GAPDoc Label="IsMultiplicativeGeneralizedRowVector">
## <ManSection>
## <Filt Name="IsMultiplicativeGeneralizedRowVector" Arg='list'
## Type='Category'/>
##
## <Description>
## For a list <A>list</A>, the value <K>true</K> for
## <Ref Filt="IsMultiplicativeGeneralizedRowVector"/> indicates that the
## multiplicative arithmetic behaviour of <A>list</A> is as defined
## in <Ref Sect="Multiplicative Arithmetic for Lists"/>,
## and that the attribute <Ref Attr="NestingDepthM"/>
## will return a nonzero value when called with <A>list</A>.
## <P/>
## <Example><![CDATA[
## gap> IsMultiplicativeGeneralizedRowVector( liemat );
## false
## gap> bas:= CanonicalBasis( FullRowSpace( Rationals, 3 ) );
## CanonicalBasis( ( Rationals^3 ) )
## gap> IsMultiplicativeGeneralizedRowVector( bas );
## true
## ]]></Example>
## <P/>
## Note that the filters <Ref Filt="IsGeneralizedRowVector"/>,
## <Ref Filt="IsMultiplicativeGeneralizedRowVector"/>
## do <E>not</E> enable default methods for addition or multiplication
## (cf. <Ref Filt="IsListDefault"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsMultiplicativeGeneralizedRowVector",
IsGeneralizedRowVector );
#############################################################################
##
#A NestingDepthA( <obj> )
##
## <#GAPDoc Label="NestingDepthA">
## <ManSection>
## <Attr Name="NestingDepthA" Arg='obj'/>
##
## <Description>
## For a &GAP; object <A>obj</A>,
## <Ref Attr="NestingDepthA"/> returns the <E>additive nesting depth</E>
## of <A>obj</A>.
## This is defined recursively
## as the integer <M>0</M> if <A>obj</A> is not in
## <Ref Filt="IsGeneralizedRowVector"/>,
## as the integer <M>1</M> if <A>obj</A> is an empty list in
## <Ref Filt="IsGeneralizedRowVector"/>,
## and as <M>1</M> plus the additive nesting depth of the first bound entry
## in <A>obj</A> otherwise.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "NestingDepthA", IsObject );
#############################################################################
##
#A NestingDepthM( <obj> )
##
## <#GAPDoc Label="NestingDepthM">
## <ManSection>
## <Attr Name="NestingDepthM" Arg='obj'/>
##
## <Description>
## For a &GAP; object <A>obj</A>,
## <Ref Attr="NestingDepthM"/> returns the
## <E>multiplicative nesting depth</E> of <A>obj</A>.
## This is defined recursively as the
## integer <M>0</M> if <A>obj</A> is not in
## <Ref Filt="IsMultiplicativeGeneralizedRowVector"/>,
## as the integer <M>1</M> if <A>obj</A> is an empty list in
## <Ref Filt="IsMultiplicativeGeneralizedRowVector"/>,
## and as <M>1</M> plus the multiplicative nesting depth of the first bound
## entry in <A>obj</A> otherwise.
## <Example><![CDATA[
## gap> NestingDepthA( v ); NestingDepthM( v );
## 1
## 1
## gap> NestingDepthA( m ); NestingDepthM( m );
## 2
## 2
## gap> NestingDepthA( liemat ); NestingDepthM( liemat );
## 2
## 0
## gap> l1:= [ [ 1, 2 ], 3 ];; l2:= [ 1, [ 2, 3 ] ];;
## gap> NestingDepthA( l1 ); NestingDepthM( l1 );
## 2
## 2
## gap> NestingDepthA( l2 ); NestingDepthM( l2 );
## 1
## 1
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "NestingDepthM", IsObject );
#############################################################################
##
#C IsNearRingElement( <obj> )
##
## <#GAPDoc Label="IsNearRingElement">
## <ManSection>
## <Filt Name="IsNearRingElement" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsNearRingElement"/> is just a synonym for the meet of
## <Ref Filt="IsNearAdditiveElementWithInverse"/> and
## <Ref Filt="IsMultiplicativeElement"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsNearRingElement",
IsNearAdditiveElementWithInverse
and IsMultiplicativeElement );
DeclareSynonym( "IsNearRingElementCollection",
IsNearAdditiveElementWithInverseCollection
and IsMultiplicativeElementCollection );
DeclareSynonym( "IsNearRingElementCollColl",
IsNearAdditiveElementWithInverseCollColl
and IsMultiplicativeElementCollColl );
DeclareSynonym( "IsNearRingElementCollCollColl",
IsNearAdditiveElementWithInverseCollCollColl
and IsMultiplicativeElementCollCollColl );
DeclareSynonym( "IsNearRingElementList",
IsNearAdditiveElementWithInverseList
and IsMultiplicativeElementList );
DeclareSynonym( "IsNearRingElementTable",
IsNearAdditiveElementWithInverseTable
and IsMultiplicativeElementTable );
InstallTrueMethod(
IsNearRingElement,
IsNearRingElementTable );
DeclareCategoryFamily( "IsNearRingElement" );
#############################################################################
##
#C IsNearRingElementWithOne( <obj> )
##
## <#GAPDoc Label="IsNearRingElementWithOne">
## <ManSection>
## <Filt Name="IsNearRingElementWithOne" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsNearRingElementWithOne"/> is just a synonym for the meet of
## <Ref Filt="IsNearAdditiveElementWithInverse"/> and
## <Ref Filt="IsMultiplicativeElementWithOne"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsNearRingElementWithOne",
IsNearAdditiveElementWithInverse
and IsMultiplicativeElementWithOne );
DeclareSynonym( "IsNearRingElementWithOneCollection",
IsNearAdditiveElementWithInverseCollection
and IsMultiplicativeElementWithOneCollection );
DeclareSynonym( "IsNearRingElementWithOneCollColl",
IsNearAdditiveElementWithInverseCollColl
and IsMultiplicativeElementWithOneCollColl );
DeclareSynonym( "IsNearRingElementWithOneCollCollColl",
IsNearAdditiveElementWithInverseCollCollColl
and IsMultiplicativeElementWithOneCollCollColl );
DeclareSynonym( "IsNearRingElementWithOneList",
IsNearAdditiveElementWithInverseList
and IsMultiplicativeElementWithOneList );
DeclareSynonym( "IsNearRingElementWithOneTable",
IsNearAdditiveElementWithInverseTable
and IsMultiplicativeElementWithOneTable );
InstallTrueMethod(
IsNearRingElementWithOne,
IsNearRingElementWithOneTable );
#############################################################################
##
#C IsNearRingElementWithInverse( <obj> )
##
## <#GAPDoc Label="IsNearRingElementWithInverse">
## <ManSection>
## <Filt Name="IsNearRingElementWithInverse" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsNearRingElementWithInverse"/> is just a synonym for the meet of
## <Ref Filt="IsNearAdditiveElementWithInverse"/> and
## <Ref Filt="IsMultiplicativeElementWithInverse"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsNearRingElementWithInverse",
IsNearAdditiveElementWithInverse
and IsMultiplicativeElementWithInverse );
DeclareSynonym( "IsNearRingElementWithInverseCollection",
IsNearAdditiveElementWithInverseCollection
and IsMultiplicativeElementWithInverseCollection );
DeclareSynonym( "IsNearRingElementWithInverseCollColl",
IsNearAdditiveElementWithInverseCollColl
and IsMultiplicativeElementWithInverseCollColl );
DeclareSynonym( "IsNearRingElementWithInverseCollCollColl",
IsNearAdditiveElementWithInverseCollCollColl
and IsMultiplicativeElementWithInverseCollCollColl );
DeclareSynonym( "IsNearRingElementWithInverseList",
IsNearAdditiveElementWithInverseList
and IsMultiplicativeElementWithInverseList );
DeclareSynonym( "IsNearRingElementWithInverseTable",
IsNearAdditiveElementWithInverseTable
and IsMultiplicativeElementWithInverseTable );
InstallTrueMethod(
IsNearRingElementWithInverse,
IsNearRingElementWithInverseTable );
#############################################################################
##
#C IsRingElement( <obj> )
##
## <#GAPDoc Label="IsRingElement">
## <ManSection>
## <Filt Name="IsRingElement" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsRingElement"/> is just a synonym for the meet of
## <Ref Filt="IsAdditiveElementWithInverse"/> and
## <Ref Filt="IsMultiplicativeElement"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsRingElement",
IsAdditiveElementWithInverse
and IsMultiplicativeElement );
DeclareSynonym( "IsRingElementCollection",
IsAdditiveElementWithInverseCollection
and IsMultiplicativeElementCollection );
DeclareSynonym( "IsRingElementCollColl",
IsAdditiveElementWithInverseCollColl
and IsMultiplicativeElementCollColl );
DeclareSynonym( "IsRingElementCollCollColl",
IsAdditiveElementWithInverseCollCollColl
and IsMultiplicativeElementCollCollColl );
DeclareSynonym( "IsRingElementList",
IsAdditiveElementWithInverseList
and IsMultiplicativeElementList );
DeclareSynonym( "IsRingElementTable",
IsAdditiveElementWithInverseTable
and IsMultiplicativeElementTable );
InstallTrueMethod(
IsRingElement,
IsRingElementTable );
DeclareCategoryFamily( "IsRingElement" );
#############################################################################
##
#C IsRingElementWithOne( <obj> )
##
## <#GAPDoc Label="IsRingElementWithOne">
## <ManSection>
## <Filt Name="IsRingElementWithOne" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsRingElementWithOne"/> is just a synonym for the meet of
## <Ref Filt="IsAdditiveElementWithInverse"/> and
## <Ref Filt="IsMultiplicativeElementWithOne"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsRingElementWithOne",
IsAdditiveElementWithInverse
and IsMultiplicativeElementWithOne );
DeclareSynonym( "IsRingElementWithOneCollection",
IsAdditiveElementWithInverseCollection
and IsMultiplicativeElementWithOneCollection );
DeclareSynonym( "IsRingElementWithOneCollColl",
IsAdditiveElementWithInverseCollColl
and IsMultiplicativeElementWithOneCollColl );
DeclareSynonym( "IsRingElementWithOneCollCollColl",
IsAdditiveElementWithInverseCollCollColl
and IsMultiplicativeElementWithOneCollCollColl );
DeclareSynonym( "IsRingElementWithOneList",
IsAdditiveElementWithInverseList
and IsMultiplicativeElementWithOneList );
DeclareSynonym( "IsRingElementWithOneTable",
IsAdditiveElementWithInverseTable
and IsMultiplicativeElementWithOneTable );
InstallTrueMethod(
IsRingElementWithOne,
IsRingElementWithOneTable );
#############################################################################
##
#C IsRingElementWithInverse( <obj> )
#C IsScalar( <obj> )
##
## <#GAPDoc Label="IsRingElementWithInverse">
## <ManSection>
## <Filt Name="IsRingElementWithInverse" Arg='obj' Type='Category'/>
## <Filt Name="IsScalar" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsRingElementWithInverse"/> and <Ref Filt="IsScalar"/>
## are just synonyms for the meet of
## <Ref Filt="IsAdditiveElementWithInverse"/> and
## <Ref Filt="IsMultiplicativeElementWithInverse"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsRingElementWithInverse",
IsAdditiveElementWithInverse
and IsMultiplicativeElementWithInverse );
DeclareSynonym( "IsRingElementWithInverseCollection",
IsAdditiveElementWithInverseCollection
and IsMultiplicativeElementWithInverseCollection );
DeclareSynonym( "IsRingElementWithInverseCollColl",
IsAdditiveElementWithInverseCollColl
and IsMultiplicativeElementWithInverseCollColl );
DeclareSynonym( "IsRingElementWithInverseCollCollColl",
IsAdditiveElementWithInverseCollCollColl
and IsMultiplicativeElementWithInverseCollCollColl );
DeclareSynonym( "IsRingElementWithInverseList",
IsAdditiveElementWithInverseList
and IsMultiplicativeElementWithInverseList );
DeclareSynonym( "IsRingElementWithInverseTable",
IsAdditiveElementWithInverseTable
and IsMultiplicativeElementWithInverseTable );
InstallTrueMethod(
IsRingElementWithInverse,
IsRingElementWithInverseTable );
DeclareSynonym( "IsScalar", IsRingElementWithInverse );
DeclareSynonym( "IsScalarCollection", IsRingElementWithInverseCollection );
DeclareSynonym( "IsScalarCollColl", IsRingElementWithInverseCollColl );
DeclareSynonym( "IsScalarList", IsRingElementWithInverseList );
DeclareSynonym( "IsScalarTable", IsRingElementWithInverseTable );
#############################################################################
##
#C IsZDFRE( <obj> )
##
## <ManSection>
## <Filt Name="IsZDFRE" Arg='obj' Type='Category'/>
##
## <Description>
## This category (<Q>is zero divisor free ring element</Q>) indicates elements
## from a ring which contains no zero divisors. For matrix operations over
## this ring, a standard Gauss algorithm can be used.
## </Description>
## </ManSection>
##
DeclareCategory("IsZDFRE",IsRingElementWithInverse);
DeclareCategoryCollections("IsZDFRE");
DeclareCategoryCollections("IsZDFRECollection");
#############################################################################
##
#C IsMatrix( <obj> )
##
## <#GAPDoc Label="IsMatrix">
## <ManSection>
## <Filt Name="IsMatrix" Arg='obj' Type='Category'/>
##
## <Description>
## By convention <E>matrix</E> is a list of lists of equal length whose
## entries lie in a common ring.
## <P/>
## For technical reasons laid out at the top of Chapter <Ref Chap="Matrices"/>,
## the filter <Ref Filt="IsMatrix"/> is a synonym for a table of ring elements,
## (see <Ref Filt="IsTable"/> and <Ref Filt="IsRingElement"/>). This means that
## <Ref Filt="IsMatrix"/> returns <K>true</K> for tables such as
## <C>[[1,2],[3]]</C>.
## If necessary, <Ref Prop="IsRectangularTable"/> can be used to test whether
## an object is a list of homogeneous lists of equal lengths manually.
## <P/>
## Note that matrices may have different multiplications,
## besides the usual matrix product there is for example the Lie product.
## So there are categories such as
## <Ref Filt="IsOrdinaryMatrix"/> and <Ref Filt="IsLieMatrix"/>
## that describe the matrix multiplication.
## One can form the product of two matrices only if they support the same
## multiplication.
## <P/>
## <Example><![CDATA[
## gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];
## [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
## gap> IsMatrix(mat);
## true
## gap> mat:=[[1,2],[3]];
## [ [ 1, 2 ], [ 3 ] ]
## gap> IsMatrix(mat);
## true
## gap> IsRectangularTable(mat);
## false
## ]]></Example>
## <P/>
## Note that the empty list <C>[]</C> and more complex
## <Q>empty</Q> structures such as <C>[[]]</C> are <E>not</E> matrices,
## although special methods allow them be used in place of matrices in some
## situations. See <Ref Func="EmptyMatrix"/> below.
## <P/>
## <Example><![CDATA[
## gap> [[0]]*[[]];
## [ [ ] ]
## gap> IsMatrix([[]]);
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
#T
#T In order to avoid that a matrix supports more than one multiplication,
#T appropriate immediate methods are installed (see~arith.gi).
##
DeclareSynonym( "IsMatrix", IsRingElementTable );
DeclareCategoryCollections( "IsMatrix" );
#############################################################################
##
#C IsOrdinaryMatrix( <obj> )
##
## <#GAPDoc Label="IsOrdinaryMatrix">
## <ManSection>
## <Filt Name="IsOrdinaryMatrix" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>ordinary matrix</E> is a matrix whose multiplication is the ordinary
## matrix multiplication.
## <P/>
## Each matrix in internal representation is in the category
## <Ref Filt="IsOrdinaryMatrix"/>,
## and arithmetic operations with objects in <Ref Filt="IsOrdinaryMatrix"/>
## produce again matrices in <Ref Filt="IsOrdinaryMatrix"/>.
## <P/>
## Note that we want that Lie matrices shall be matrices that behave in the
## same way as ordinary matrices, except that they have a different
## multiplication.
## So we must distinguish the different matrix multiplications,
## in order to be able to describe the applicability of multiplication,
## and also in order to form a matrix of the appropriate type as the
## sum, difference etc. of two matrices
## which have the same multiplication.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsOrdinaryMatrix", IsMatrix );
DeclareCategoryCollections( "IsOrdinaryMatrix" );
#T get rid of this filter!!
InstallTrueMethod( IsOrdinaryMatrix, IsMatrix and IsInternalRep );
InstallTrueMethod( IsGeneralizedRowVector, IsMatrix );
#T get rid of that hack!
InstallTrueMethod( IsMultiplicativeGeneralizedRowVector,
IsOrdinaryMatrix );
#############################################################################
##
#C IsLieMatrix( <mat> )
##
## <#GAPDoc Label="IsLieMatrix">
## <ManSection>
## <Filt Name="IsLieMatrix" Arg='mat' Type='Category'/>
##
## <Description>
## A <E>Lie matrix</E> is a matrix whose multiplication is given by the
## Lie bracket.
## (Note that a matrix with ordinary matrix multiplication is in the
## category <Ref Filt="IsOrdinaryMatrix"/>.)
## <P/>
## Each matrix created by <Ref Attr="LieObject"/> is in the category
## <Ref Filt="IsLieMatrix"/>,
## and arithmetic operations with objects in <Ref Filt="IsLieMatrix"/>
## produce again matrices in <Ref Filt="IsLieMatrix"/>.
## <!-- We do not claim that every object in <Ref Func="IsLieMatrix"/>
## is also contained in <Ref Func="IsLieObject"/>,
## since the former describes the containment in a certain
## family and the latter describes a certain matrix multiplication;
## probably this distinction is unnecessary. -->
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsLieMatrix", IsGeneralizedRowVector and IsMatrix );
#############################################################################
##
#C IsAssociativeElement( <obj> ) . . . elements belonging to assoc. families
#C IsAssociativeElementCollection( <obj> )
#C IsAssociativeElementCollColl( <obj> )
##
## <#GAPDoc Label="IsAssociativeElement">
## <ManSection>
## <Filt Name="IsAssociativeElement" Arg='obj' Type='Category'/>
## <Filt Name="IsAssociativeElementCollection" Arg='obj' Type='Category'/>
## <Filt Name="IsAssociativeElementCollColl" Arg='obj' Type='Category'/>
##
## <Description>
## An element <A>obj</A> in the category <Ref Filt="IsAssociativeElement"/>
## knows that the multiplication of any elements in the family of <A>obj</A>
## is associative.
## For example, all permutations lie in this category, as well as those
## ordinary matrices (see <Ref Filt="IsOrdinaryMatrix"/>)
## whose entries are also in <Ref Filt="IsAssociativeElement"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsAssociativeElement", IsMultiplicativeElement );
DeclareCategoryCollections( "IsAssociativeElement" );
DeclareCategoryCollections( "IsAssociativeElementCollection" );
#############################################################################
##
#M IsAssociativeElement( <mat> ) . . . . . . . for certain ordinary matrices
##
## Matrices with associative multiplication
## and with entries in an associative family
## are themselves associative elements.
##
InstallTrueMethod( IsAssociativeElement,
IsOrdinaryMatrix and IsAssociativeElementCollColl );
#############################################################################
##
#C IsAdditivelyCommutativeElement( <obj> )
#C IsAdditivelyCommutativeElementCollection( <obj> )
#C IsAdditivelyCommutativeElementCollColl( <obj> )
#C IsAdditivelyCommutativeElementFamily( <obj> )
##
## <#GAPDoc Label="IsAdditivelyCommutativeElement">
## <ManSection>
## <Filt Name="IsAdditivelyCommutativeElement" Arg='obj' Type='Category'/>
## <Filt Name="IsAdditivelyCommutativeElementCollection" Arg='obj'
## Type='Category'/>
## <Filt Name="IsAdditivelyCommutativeElementCollColl" Arg='obj'
## Type='Category'/>
## <Filt Name="IsAdditivelyCommutativeElementFamily" Arg='obj'
## Type='Category'/>
##
## <Description>
## An element <A>obj</A> in the category
## <Ref Filt="IsAdditivelyCommutativeElement"/> knows
## that the addition of any elements in the family of <A>obj</A>
## is commutative.
## For example, each finite field element and each rational number lies in
## this category.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsAdditivelyCommutativeElement", IsAdditiveElement );
DeclareCategoryCollections( "IsAdditivelyCommutativeElement" );
DeclareCategoryCollections( "IsAdditivelyCommutativeElementCollection" );
DeclareCategoryFamily( "IsAdditivelyCommutativeElement" );
#############################################################################
##
#M IsAdditivelyCommutativeElement( <mat> ) . . . . . . for certain matrices
##
## Matrices with entries in an additively commutative family
## are themselves additively commutative elements.
##
InstallTrueMethod( IsAdditivelyCommutativeElement,
IsMatrix and IsAdditivelyCommutativeElementCollColl );
#############################################################################
##
#M IsAdditivelyCommutativeElement( <mat> ) . . . . . for certain row vectors
##
## Row vectors with entries in an additively commutative family
## are themselves additively commutative elements.
##
InstallTrueMethod( IsAdditivelyCommutativeElement,
IsRowVector and IsAdditivelyCommutativeElementCollection );
#############################################################################
##
#C IsCommutativeElement( <obj> ) . . . elements belonging to comm. families
#C IsCommutativeElementCollection( <obj> )
#C IsCommutativeElementCollColl( <obj> )
##
## <#GAPDoc Label="IsCommutativeElement">
## <ManSection>
## <Filt Name="IsCommutativeElement" Arg='obj' Type='Category'/>
## <Filt Name="IsCommutativeElementCollection" Arg='obj' Type='Category'/>
## <Filt Name="IsCommutativeElementCollColl" Arg='obj' Type='Category'/>
##
## <Description>
## An element <A>obj</A> in the category <Ref Filt="IsCommutativeElement"/>
## knows that the multiplication of any elements in the family of <A>obj</A>
## is commutative.
## For example, each finite field element and each rational number lies in
## this category.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsCommutativeElement", IsMultiplicativeElement );
DeclareCategoryCollections( "IsCommutativeElement" );
DeclareCategoryCollections( "IsCommutativeElementCollection" );
#############################################################################
##
#C IsFiniteOrderElement( <obj> )
#C IsFiniteOrderElementCollection( <obj> )
#C IsFiniteOrderElementCollColl( <obj> )
##
## <#GAPDoc Label="IsFiniteOrderElement">
## <ManSection>
## <Filt Name="IsFiniteOrderElement" Arg='obj' Type='Category'/>
## <Filt Name="IsFiniteOrderElementCollection" Arg='obj' Type='Category'/>
## <Filt Name="IsFiniteOrderElementCollColl" Arg='obj' Type='Category'/>
##
## <Description>
## An element <A>obj</A> in the category <Ref Filt="IsFiniteOrderElement"/>
## knows that it has finite multiplicative order.
## For example, each finite field element and each permutation lies in
## this category.
## However the value may be <K>false</K> even if <A>obj</A> has finite
## order, but if this was not known when <A>obj</A> was constructed.
## <P/>
## Although it is legal to set this filter for any object with finite order,
## this is really useful only in the case that all elements of a family are
## known to have finite order.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsFiniteOrderElement",
IsMultiplicativeElementWithInverse );
DeclareCategoryCollections( "IsFiniteOrderElement" );
DeclareCategoryCollections( "IsFiniteOrderElementCollection" );
#############################################################################
##
#C IsJacobianElement( <obj> ) . elements belong. to fam. with Jacobi ident.
#C IsJacobianElementCollection( <obj> )
#C IsJacobianElementCollColl( <obj> )
##
## <#GAPDoc Label="IsJacobianElement">
## <ManSection>
## <Filt Name="IsJacobianElement" Arg='obj' Type='Category'/>
## <Filt Name="IsJacobianElementCollection" Arg='obj' Type='Category'/>
## <Filt Name="IsJacobianElementCollColl" Arg='obj' Type='Category'/>
## <Filt Name="IsRestrictedJacobianElement" Arg='obj' Type='Category'/>
## <Filt Name="IsRestrictedJacobianElementCollection" Arg='obj' Type='Category'/>
## <Filt Name="IsRestrictedJacobianElementCollColl" Arg='obj' Type='Category'/>
##
## <Description>
## An element <A>obj</A> in the category <Ref Filt="IsJacobianElement"/>
## knows that the multiplication of any elements in the family <M>F</M>
## of <A>obj</A> satisfies the Jacobi identity, that is,
## <M>x * y * z + z * x * y + y * z * x</M> is zero
## for all <M>x</M>, <M>y</M>, <M>z</M> in <M>F</M>.
## <P/>
## For example, each Lie matrix (see <Ref Filt="IsLieMatrix"/>)
## lies in this category.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsJacobianElement", IsRingElement );
DeclareCategoryCollections( "IsJacobianElement" );
DeclareCategoryCollections( "IsJacobianElementCollection" );
DeclareCategory( "IsRestrictedJacobianElement", IsJacobianElement );
DeclareCategoryCollections( "IsRestrictedJacobianElement" );
DeclareCategoryCollections( "IsRestrictedJacobianElementCollection" );
#############################################################################
##
#C IsZeroSquaredElement( <obj> ) . . . elements belong. to zero squared fam.
#C IsZeroSquaredElementCollection( <obj> )
#C IsZeroSquaredElementCollColl( <obj> )
##
## <#GAPDoc Label="IsZeroSquaredElement">
## <ManSection>
## <Filt Name="IsZeroSquaredElement" Arg='obj' Type='Category'/>
## <Filt Name="IsZeroSquaredElementCollection" Arg='obj' Type='Category'/>
## <Filt Name="IsZeroSquaredElementCollColl" Arg='obj' Type='Category'/>
##
## <Description>
## An element <A>obj</A> in the category <Ref Filt="IsZeroSquaredElement"/>
## knows that <C><A>obj</A>^2 = Zero( <A>obj</A> )</C>.
## For example, each Lie matrix (see <Ref Filt="IsLieMatrix"/>)
## lies in this category.
## <P/>
## Although it is legal to set this filter for any zero squared object,
## this is really useful only in the case that all elements of a family are
## known to have square zero.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsZeroSquaredElement", IsRingElement );
DeclareCategoryCollections( "IsZeroSquaredElement" );
DeclareCategoryCollections( "IsZeroSquaredElementCollection" );
#############################################################################
##
#P IsZero( <elm> ) . . . . . . . . . . . . . . . . . . test for zero element
##
## <#GAPDoc Label="IsZero">
## <ManSection>
## <Prop Name="IsZero" Arg='elm'/>
##
## <Description>
## is <K>true</K> if <C><A>elm</A> = Zero( <A>elm</A> )</C>,
## and <K>false</K> otherwise.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsZero", IsAdditiveElementWithZero );
#############################################################################
##
#P IsOne( <elm> ) . . . . . . . . . . . . . . . . test for identity element
##
## <#GAPDoc Label="IsOne">
## <ManSection>
## <Prop Name="IsOne" Arg='elm'/>
##
## <Description>
## is <K>true</K> if <C><A>elm</A> = One( <A>elm</A> )</C>,
## and <K>false</K> otherwise.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsOne", IsMultiplicativeElementWithOne );
#############################################################################
##
#A ZeroImmutable( <obj> ) . . additive neutral of an element/domain/family
#A Zero( <obj> ) synonym of ZeroImmutable
#O ZeroMutable( <obj> ) . . . . . . mutable additive neutral of an element
#O ZeroOp( <obj> ) synonym of ZeroMutable
#O ZeroSameMutability( <obj> ) mutability preserving zero (0*<obj>)
##
## <#GAPDoc Label="ZeroImmutable">
## <ManSection>
## <Attr Name="ZeroImmutable" Arg='obj'/>
## <Attr Name="Zero" Arg='obj'/>
## <Oper Name="ZeroMutable" Arg='obj'/>
## <Oper Name="ZeroOp" Arg='obj'/>
## <Oper Name="ZeroSameMutability" Arg='obj'/>
##
## <Description>
## <Ref Attr="ZeroImmutable"/>, <Ref Oper="ZeroMutable"/>,
## and <Ref Oper="ZeroSameMutability"/> all
## return the additive neutral element of the additive element <A>obj</A>.
## <P/>
## They differ only w.r.t. the mutability of the result.
## <Ref Attr="ZeroImmutable"/> is an attribute and hence returns an
## immutable result.
## <Ref Oper="ZeroMutable"/> is guaranteed to return a new <E>mutable</E>
## object whenever a mutable version of the required element exists in &GAP;
## (see <Ref Filt="IsCopyable"/>).
## <Ref Oper="ZeroSameMutability"/> returns a result that is mutable if
## <A>obj</A> is mutable and if a mutable version of the required element
## exists in &GAP;;
## for lists, it returns a result of the same immutability level as
## the argument. For instance, if the argument is a mutable matrix
## with immutable rows, it returns a similar object.
## <P/>
## <C>ZeroSameMutability( <A>obj</A> )</C> is equivalent to
## <C>0 * <A>obj</A></C>.
## <P/>
## <Ref Attr="Zero"/> is a synonym of <Ref Attr="ZeroImmutable"/>.
## <Ref Oper="ZeroOp"/> is a synonym of <Ref Oper="ZeroMutable"/>.
## <P/>
## If <A>obj</A> is a domain or a family then <Ref Attr="Zero"/> is defined
## as the zero element of all elements in <A>obj</A>,
## provided that all these elements have the same zero.
## For example, the family of all cyclotomics has the zero element <C>0</C>,
## but a collections family (see <Ref Attr="CollectionsFamily"/>) may
## contain matrices of all dimensions and then it cannot have a unique
## zero element.
## Note that <Ref Attr="Zero"/> is applicable to a domain only if it is an
## additive magma-with-zero
## (see <Ref Filt="IsAdditiveMagmaWithZero"/>);
## use <Ref Attr="AdditiveNeutralElement"/> otherwise.
## <P/>
## The default method of <Ref Attr="Zero"/> for additive elements calls
## <Ref Oper="ZeroMutable"/>
## (note that methods for <Ref Oper="ZeroMutable"/> must <E>not</E> delegate
## to <Ref Attr="Zero"/>);
## so other methods to compute zero elements need to be installed only for
## <Ref Oper="ZeroMutable"/> and (in the case of copyable objects)
## <Ref Oper="ZeroSameMutability"/>.
## <P/>
## For domains, <Ref Attr="Zero"/> may call <Ref Attr="Representative"/>,
## but <Ref Attr="Representative"/> is allowed to fetch the zero of a domain
## <A>D</A> only if <C>HasZero( <A>D</A> )</C> is <K>true</K>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ZeroImmutable", IsAdditiveElementWithZero );
DeclareAttribute( "ZeroImmutable", IsFamily );
DeclareSynonymAttr( "Zero", ZeroImmutable );
DeclareOperationKernel( "ZeroMutable", [ IsAdditiveElementWithZero ],
ZERO_MUT );
DeclareSynonym( "ZeroOp", ZeroMutable );
DeclareOperationKernel( "ZeroSameMutability", [ IsAdditiveElementWithZero ],
ZERO_SAMEMUT );
#############################################################################
##
#O `<elm1>+<elm2>' . . . . . . . . . . . . . . . . . . . sum of two elements
##
DeclareOperationKernel( "+", [ IsExtAElement, IsExtAElement ], SUM );
#############################################################################
##
#A AdditiveInverseImmutable( <elm> ) . . . . additive inverse of an element
#A AdditiveInverse( <elm> ) . . . . additive inverse of an element
#O AdditiveInverseMutable( <elm> ) . mutable additive inverse of an element
#O AdditiveInverseOp( <elm> ) . mutable additive inverse of an element
#O AdditiveInverseSameMutability( <elm> ) . additive inverse of an element
##
## <#GAPDoc Label="AdditiveInverseImmutable">
## <ManSection>
## <Attr Name="AdditiveInverseImmutable" Arg='elm'/>
## <Attr Name="AdditiveInverse" Arg='elm'/>
## <Oper Name="AdditiveInverseMutable" Arg='elm'/>
## <Oper Name="AdditiveInverseOp" Arg='elm'/>
## <Oper Name="AdditiveInverseSameMutability" Arg='elm'/>
##
## <Description>
## <Ref Attr="AdditiveInverseImmutable"/>,
## <Ref Oper="AdditiveInverseMutable"/>, and
## <Ref Oper="AdditiveInverseSameMutability"/> all return the
## additive inverse of <A>elm</A>.
## <P/>
## They differ only w.r.t. the mutability of the result.
## <Ref Attr="AdditiveInverseImmutable"/> is an attribute and hence returns
## an immutable result.
## <Ref Oper="AdditiveInverseMutable"/> is guaranteed to return a new
## <E>mutable</E> object whenever a mutable version of the required element
## exists in &GAP; (see <Ref Filt="IsCopyable"/>).
## <Ref Oper="AdditiveInverseSameMutability"/> returns a result that is
## mutable if <A>elm</A> is mutable and if a mutable version of the required
## element exists in &GAP;;
## for lists, it returns a result of the same immutability level as
## the argument. For instance, if the argument is a mutable matrix
## with immutable rows, it returns a similar object.
## <P/>
## <C>AdditiveInverseSameMutability( <A>elm</A> )</C> is equivalent to
## <C>-<A>elm</A></C>.
## <P/>
## <Ref Attr="AdditiveInverse"/> is a synonym of
## <Ref Attr="AdditiveInverseImmutable"/>.
## <Ref Oper="AdditiveInverseOp"/> is a synonym of
## <Ref Oper="AdditiveInverseMutable"/>.
## <P/>
## The default method of <Ref Attr="AdditiveInverse"/> calls
## <Ref Oper="AdditiveInverseMutable"/>
## (note that methods for <Ref Oper="AdditiveInverseMutable"/>
## must <E>not</E> delegate to <Ref Attr="AdditiveInverse"/>);
## so other methods to compute additive inverses need to be installed only
## for <Ref Oper="AdditiveInverseMutable"/> and (in the case of copyable
## objects) <Ref Oper="AdditiveInverseSameMutability"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AdditiveInverseImmutable", IsAdditiveElementWithInverse );
DeclareSynonymAttr( "AdditiveInverse", AdditiveInverseImmutable );
DeclareOperationKernel( "AdditiveInverseMutable",
[ IsAdditiveElementWithInverse ], AINV_MUT);
DeclareSynonym( "AdditiveInverseOp", AdditiveInverseMutable);
DeclareOperationKernel( "AdditiveInverseSameMutability",
[ IsAdditiveElementWithInverse ], AINV_SAMEMUT );
#############################################################################
##
#O `<elm1>-<elm2>' . . . . . . . . . . . . . . . difference of two elements
##
DeclareOperationKernel( "-",
[ IsExtAElement, IsNearAdditiveElementWithInverse ], DIFF );
#############################################################################
##
#O `<elm1>*<elm2>' . . . . . . . . . . . . . . . . . product of two elements
##
DeclareOperationKernel( "*", [ IsExtRElement, IsExtLElement ], PROD );
#############################################################################
##
#A OneImmutable( <obj> ) multiplicative neutral of an element/domain/family
#A One( <obj> )
#A Identity( <obj> )
#O OneMutable( <obj> ) . . . . . . . . multiplicative neutral of an element
#O OneOp( <obj> )
#O OneSameMutability( <obj> )
##
## <#GAPDoc Label="OneImmutable">
## <ManSection>
## <Attr Name="OneImmutable" Arg='obj'/>
## <Attr Name="One" Arg='obj'/>
## <Attr Name="Identity" Arg='obj'/>
## <Oper Name="OneMutable" Arg='obj'/>
## <Oper Name="OneOp" Arg='obj'/>
## <Oper Name="OneSameMutability" Arg='obj'/>
##
## <Description>
## <Ref Attr="OneImmutable"/>, <Ref Oper="OneMutable"/>,
## and <Ref Oper="OneSameMutability"/> return the multiplicative neutral
## element of the multiplicative element <A>obj</A>.
## <P/>
## They differ only w.r.t. the mutability of the result.
## <Ref Attr="OneImmutable"/> is an attribute and hence returns an immutable
## result.
## <Ref Oper="OneMutable"/> is guaranteed to return a new <E>mutable</E>
## object whenever a mutable version of the required element exists in &GAP;
## (see <Ref Filt="IsCopyable"/>).
## <Ref Oper="OneSameMutability"/> returns a result that is mutable if
## <A>obj</A> is mutable
## and if a mutable version of the required element exists in &GAP;;
## for lists, it returns a result of the same immutability level as
## the argument. For instance, if the argument is a mutable matrix
## with immutable rows, it returns a similar object.
## <P/>
## If <A>obj</A> is a multiplicative element then
## <C>OneSameMutability( <A>obj</A> )</C>
## is equivalent to <C><A>obj</A>^0</C>.
## <P/>
## <Ref Attr="One"/> and <Ref Attr="Identity"/> are
## synonyms of <C>OneImmutable</C>.
## <Ref Oper="OneOp"/> is a synonym of <Ref Oper="OneMutable"/>.
## <P/>
## If <A>obj</A> is a domain or a family then <Ref Attr="One"/> is defined
## as the identity element of all elements in <A>obj</A>,
## provided that all these elements have the same identity.
## For example, the family of all cyclotomics has the identity element
## <C>1</C>,
## but a collections family (see <Ref Attr="CollectionsFamily"/>)
## may contain matrices of all dimensions and then it cannot have a unique
## identity element.
## Note that <Ref Attr="One"/> is applicable to a domain only if it is a
## magma-with-one (see <Ref Filt="IsMagmaWithOne"/>);
## use <Ref Attr="MultiplicativeNeutralElement"/> otherwise.
## <P/>
## The identity of an object need not be distinct from its zero,
## so for example a ring consisting of a single element can be regarded as a
## ring-with-one (see <Ref Chap="Rings"/>).
## This is particularly useful in the case of finitely presented algebras,
## where any factor of a free algebra-with-one is again an algebra-with-one,
## no matter whether or not it is a zero algebra.
## <P/>
## The default method of <Ref Attr="One"/> for multiplicative elements calls
## <Ref Oper="OneMutable"/> (note that methods for <Ref Oper="OneMutable"/>
## must <E>not</E> delegate to <Ref Attr="One"/>);
## so other methods to compute identity elements need to be installed only
## for <Ref Oper="OneOp"/> and (in the case of copyable objects)
## <Ref Oper="OneSameMutability"/>.
## <P/>
## For domains, <Ref Attr="One"/> may call <Ref Attr="Representative"/>,
## but <Ref Attr="Representative"/> is allowed to fetch the identity of a
## domain <A>D</A> only if <C>HasOne( <A>D</A> )</C> is <K>true</K>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "OneImmutable", IsMultiplicativeElementWithOne );
DeclareAttribute( "OneImmutable", IsFamily );
DeclareSynonymAttr( "One", OneImmutable );
DeclareSynonymAttr( "Identity", OneImmutable );
DeclareOperationKernel( "OneMutable", [ IsMultiplicativeElementWithOne ],
ONE );
DeclareSynonym( "OneOp", OneMutable);
DeclareOperationKernel( "OneSameMutability",
[ IsMultiplicativeElementWithOne ], ONE_SAMEMUT );
#############################################################################
##
#A InverseImmutable( <elm> ) . . . . multiplicative inverse of an element
#A Inverse( <elm> )
#O InverseMutable( <elm> )
#O InverseOp( <elm> )
#O InverseSameMutability( <elm> ) . . multiplicative inverse of an element
##
## <#GAPDoc Label="InverseImmutable">
## <ManSection>
## <Attr Name="InverseImmutable" Arg='elm'/>
## <Attr Name="Inverse" Arg='elm'/>
## <Oper Name="InverseMutable" Arg='elm'/>
## <Oper Name="InverseOp" Arg='elm'/>
## <Oper Name="InverseSameMutability" Arg='elm'/>
##
## <Description>
## <Ref Attr="InverseImmutable"/>, <Ref Oper="InverseMutable"/>, and
## <Ref Oper="InverseSameMutability"/>
## all return the multiplicative inverse of an element <A>elm</A>,
## that is, an element <A>inv</A> such that
## <C><A>elm</A> * <A>inv</A> = <A>inv</A> * <A>elm</A>
## = One( <A>elm</A> )</C> holds;
## if <A>elm</A> is not invertible then <K>fail</K>
## (see <Ref Sect="Fail"/>) is returned.
## <P/>
## Note that the above definition implies that a (general) mapping
## is invertible in the sense of <Ref Attr="Inverse"/> only if its source
## equals its range
## (see <Ref Sect="Technical Matters Concerning General Mappings"/>).
## For a bijective mapping <M>f</M> whose source and range differ,
## <Ref Attr="InverseGeneralMapping"/> can be used
## to construct a mapping <M>g</M> with the property
## that <M>f</M> <C>*</C> <M>g</M> is the identity mapping on the source of
## <M>f</M> and <M>g</M> <C>*</C> <M>f</M> is the identity mapping on the
## range of <M>f</M>.
## <P/>
## The operations differ only w.r.t. the mutability of the result.
## <Ref Attr="InverseImmutable"/> is an attribute and hence returns an
## immutable result.
## <Ref Oper="InverseMutable"/> is guaranteed to return a new <E>mutable</E>
## object whenever a mutable version of the required element exists in &GAP;.
## <Ref Oper="InverseSameMutability"/> returns a result that is mutable if
## <A>elm</A> is mutable and if a mutable version of the required element
## exists in &GAP;;
## for lists, it returns a result of the same immutability level as
## the argument. For instance, if the argument is a mutable matrix
## with immutable rows, it returns a similar object.
## <P/>
## <C>InverseSameMutability( <A>elm</A> )</C> is equivalent to
## <C><A>elm</A>^-1</C>.
## <P/>
## <Ref Attr="Inverse"/> is a synonym of <Ref Attr="InverseImmutable"/>.
## <Ref Oper="InverseOp"/> is a synonym of <Ref Oper="InverseMutable"/>.
## <P/>
## The default method of <Ref Attr="InverseImmutable"/> calls
## <Ref Oper="InverseMutable"/> (note that methods
## for <Ref Oper="InverseMutable"/> must <E>not</E> delegate to
## <Ref Attr="InverseImmutable"/>);
## other methods to compute inverses need to be installed only for
## <Ref Oper="InverseMutable"/> and (in the case of copyable objects)
## <Ref Oper="InverseSameMutability"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "InverseImmutable", IsMultiplicativeElementWithInverse );
DeclareSynonymAttr( "Inverse", InverseImmutable );
DeclareOperationKernel( "InverseMutable",
[ IsMultiplicativeElementWithInverse ], INV );
DeclareSynonym( "InverseOp", InverseMutable );
DeclareOperationKernel( "InverseSameMutability",
[ IsMultiplicativeElementWithInverse ], INV_SAMEMUT );
#############################################################################
##
#O `<elm1>/<elm2>' . . . . . . . . . . . . . . . . quotient of two elements
##
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