Quelle HomalgRing.gd Sprache: unbekannt

# SPDX-License-Identifier: GPL-2.0-or-later
# MatricesForHomalg: Matrices for the homalg project
#
# Declarations
#

##
DeclareGlobalVariable( "CommonHomalgTableForRings" );

####################################
#
# categories:
#
####################################

# three new GAP-categories:

## <#GAPDoc Label="IsHomalgRing">
## <ManSection>
## <Filt Type="Category" Arg="R" Name="IsHomalgRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; category of &homalg; rings. 
## (It is a subcategory of the &GAP; categories <C>IsStructureObject</C>
## and <C>IsHomalgRingOrModule</C>.)
## <Listing Type="Code"><![CDATA[
DeclareCategory( "IsHomalgRing",
 IsStructureObject and
 IsRingWithOne and
 IsHomalgRingOrModule );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>

## <#GAPDoc Label="IsPreHomalgRing">
## <ManSection>
## <Filt Type="Category" Arg="R" Name="IsPreHomalgRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; category of pre &homalg; rings. 
## (It is a subcategory of the &GAP; category <C>IsHomalgRing</C>.) 
## These are rings with an incomplete <C>homalgTable</C>.
## They provide flexibility for developers to support a wider class of rings,
## as was necessary for the development of the &LocalizeRingForHomalg; package.
## They are not suited for direct usage.
## <Listing Type="Code"><![CDATA[
DeclareCategory( "IsPreHomalgRing",
 IsHomalgRing );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>

## <#GAPDoc Label="IsHomalgRingElement">
## <ManSection>
## <Filt Type="Category" Arg="r" Name="IsHomalgRingElement"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; category of elements of &homalg; rings which are not GAP4 built-in.
## <Listing Type="Code"><![CDATA[
DeclareCategory( "IsHomalgRingElement",
 IsExtAElement and
 IsExtLElement and
 IsExtRElement and
 IsAdditiveElementWithInverse and
 IsMultiplicativeElementWithInverse and
 IsAssociativeElement and
 IsAdditivelyCommutativeElement and
 ## all the above guarantees IsHomalgRingElement => IsRingElement (in GAP4)
 IsAttributeStoringRep );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>

####################################
#
# properties:
#
####################################

## properties listed alphabetically (ignoring left/right):

## <#GAPDoc Label="IsZero:rings">
## <ManSection>
## <Prop Arg="R" Name="IsZero" Label="for rings"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the ring <A>R</A> is the zero ring, i.e., if <C>One</C><M>(</M><A>R</A><M>)=</M><C>Zero</C><M>(</M><A>R</A><M>)</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsZero",
 IsHomalgRing );

## <#GAPDoc Label="IsNonZeroRing">
## <ManSection>
## <Prop Arg="R" Name="IsNonZeroRing" Label="for rings"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the ring <A>R</A> is not the zero ring, i.e., if <C>One</C><M>(</M><A>R</A><M>)</M> is different from <C>Zero</C><M>(</M><A>R</A><M>)</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsNonZeroRing",
 IsHomalgRing );

## <#GAPDoc Label="ContainsAField">
## <ManSection>
## <Prop Arg="R" Name="ContainsAField"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "ContainsAField",
 IsHomalgRing );

## <#GAPDoc Label="IsRationalsForHomalg">
## <ManSection>
## <Prop Arg="R" Name="IsRationalsForHomalg"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRationalsForHomalg",
 IsHomalgRing );

## <#GAPDoc Label="IsFieldForHomalg">
## <ManSection>
## <Prop Arg="R" Name="IsFieldForHomalg"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFieldForHomalg",
 IsHomalgRing );

## <#GAPDoc Label="IsDivisionRingForHomalg">
## <ManSection>
## <Prop Arg="R" Name="IsDivisionRingForHomalg"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsDivisionRingForHomalg",
 IsHomalgRing );

## <#GAPDoc Label="IsIntegersForHomalg">
## <ManSection>
## <Prop Arg="R" Name="IsIntegersForHomalg"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsIntegersForHomalg",
 IsHomalgRing );

## <#GAPDoc Label="IsResidueClassRingOfTheIntegers">
## <ManSection>
## <Prop Arg="R" Name="IsResidueClassRingOfTheIntegers"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsResidueClassRingOfTheIntegers",
 IsHomalgRing );

## <#GAPDoc Label="IsBezoutRing">
## <ManSection>
## <Prop Arg="R" Name="IsBezoutRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsBezoutRing",
 IsHomalgRing );

## <#GAPDoc Label="IsIntegrallyClosedDomain">
## <ManSection>
## <Prop Arg="R" Name="IsIntegrallyClosedDomain"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsIntegrallyClosedDomain",
 IsHomalgRing );

## <#GAPDoc Label="IsUniqueFactorizationDomain">
## <ManSection>
## <Prop Arg="R" Name="IsUniqueFactorizationDomain"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsUniqueFactorizationDomain",
 IsHomalgRing );

## <#GAPDoc Label="IsKaplanskyHermite">
## <ManSection>
## <Prop Arg="R" Name="IsKaplanskyHermite"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsKaplanskyHermite",
 IsHomalgRing );

## <#GAPDoc Label="IsDedekindDomain">
## <ManSection>
## <Prop Arg="R" Name="IsDedekindDomain"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsDedekindDomain",
 IsHomalgRing );

## <#GAPDoc Label="IsDiscreteValuationRing">
## <ManSection>
## <Prop Arg="R" Name="IsDiscreteValuationRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsDiscreteValuationRing",
 IsHomalgRing );

## <#GAPDoc Label="IsFreePolynomialRing">
## <ManSection>
## <Prop Arg="R" Name="IsFreePolynomialRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFreePolynomialRing",
 IsHomalgRing );

## <#GAPDoc Label="IsWeylRing">
## <ManSection>
## <Prop Arg="R" Name="IsWeylRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsWeylRing",
 IsHomalgRing );

## <#GAPDoc Label="IsLocalizedWeylRing">
## <ManSection>
## <Prop Arg="R" Name="IsLocalizedWeylRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLocalizedWeylRing",
 IsHomalgRing );

## <#GAPDoc Label="IsExteriorRing">
## <ManSection>
## <Prop Arg="R" Name="IsExteriroRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsExteriorRing",
 IsHomalgRing );

## <#GAPDoc Label="IsShiftAlgebra">
## <ManSection>
## <Prop Arg="R" Name="IsShiftAlgebra"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IsRationalShiftAlgebra">
## <ManSection>
## <Prop Arg="R" Name="IsRationalShiftAlgebra"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRationalShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IsPseudoDoubleShiftAlgebra">
## <ManSection>
## <Prop Arg="R" Name="IsPseudoDoubleShiftAlgebra"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsPseudoDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IsRationalPseudoDoubleShiftAlgebra">
## <ManSection>
## <Prop Arg="R" Name="IsRationalPseudoDoubleShiftAlgebra"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRationalPseudoDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IsDoubleShiftAlgebra">
## <ManSection>
## <Prop Arg="R" Name="IsDoubleShiftAlgebra"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IsRationalDoubleShiftAlgebra">
## <ManSection>
## <Prop Arg="R" Name="IsRationalDoubleShiftAlgebra"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRationalDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IsBiasedDoubleShiftAlgebra">
## <ManSection>
## <Prop Arg="R" Name="IsBiasedDoubleShiftAlgebra"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsBiasedDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IsGlobalDimensionFinite">
## <ManSection>
## <Prop Arg="R" Name="IsGlobalDimensionFinite"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsGlobalDimensionFinite",
 IsHomalgRing );

## <#GAPDoc Label="IsLeftGlobalDimensionFinite">
## <ManSection>
## <Prop Arg="R" Name="IsLeftGlobalDimensionFinite"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftGlobalDimensionFinite",
 IsHomalgRing );

## <#GAPDoc Label="IsRightGlobalDimensionFinite">
## <ManSection>
## <Prop Arg="R" Name="IsRightGlobalDimensionFinite"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightGlobalDimensionFinite",
 IsHomalgRing );

## <#GAPDoc Label="HasInvariantBasisProperty">
## <ManSection>
## <Prop Arg="R" Name="HasInvariantBasisProperty"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "HasInvariantBasisProperty",
 IsHomalgRing );

## <#GAPDoc Label="IsLocal">
## <ManSection>
## <Prop Arg="R" Name="IsLocal"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLocal",
 IsHomalgRing );

## <#GAPDoc Label="IsSemiLocalRing">
## <ManSection>
## <Prop Arg="R" Name="IsSemiLocalRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSemiLocalRing",
 IsHomalgRing );

## <#GAPDoc Label="IsIntegralDomain">
## <ManSection>
## <Prop Arg="R" Name="IsIntegralDomain"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsIntegralDomain",
 IsHomalgRing );

## <#GAPDoc Label="IsHereditary">
## <ManSection>
## <Prop Arg="R" Name="IsHereditary"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsHereditary",
 IsHomalgRing );

## <#GAPDoc Label="IsLeftHereditary">
## <ManSection>
## <Prop Arg="R" Name="IsLeftHereditary"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftHereditary",
 IsHomalgRing );

## <#GAPDoc Label="IsRightHereditary">
## <ManSection>
## <Prop Arg="R" Name="IsRightHereditary"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightHereditary",
 IsHomalgRing );

## <#GAPDoc Label="IsHermite">
## <ManSection>
## <Prop Arg="R" Name="IsHermite"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsHermite",
 IsHomalgRing );

## <#GAPDoc Label="IsLeftHermite">
## <ManSection>
## <Prop Arg="R" Name="IsLeftHermite"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftHermite",
 IsHomalgRing );

## <#GAPDoc Label="IsRightHermite">
## <ManSection>
## <Prop Arg="R" Name="IsRightHermite"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightHermite",
 IsHomalgRing );

## <#GAPDoc Label="IsNoetherian">
## <ManSection>
## <Prop Arg="R" Name="IsNoetherian"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsNoetherian",
 IsHomalgRing );

## <#GAPDoc Label="IsLeftNoetherian">
## <ManSection>
## <Prop Arg="R" Name="IsLeftNoetherian"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftNoetherian",
 IsHomalgRing );

## <#GAPDoc Label="IsRightNoetherian">
## <ManSection>
## <Prop Arg="R" Name="IsRightNoetherian"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightNoetherian",
 IsHomalgRing );

## <#GAPDoc Label="IsCohenMacaulay">
## <ManSection>
## <Prop Arg="R" Name="IsCohenMacaulay"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsCohenMacaulay",
 IsHomalgRing );

## <#GAPDoc Label="IsGorenstein">
## <ManSection>
## <Prop Arg="R" Name="IsGorenstein"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsGorenstein",
 IsHomalgRing );

## <#GAPDoc Label="IsKoszul">
## <ManSection>
## <Prop Arg="R" Name="IsKoszul"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsKoszul",
 IsHomalgRing );

## <#GAPDoc Label="IsArtinian">
## <ManSection>
## <Prop Arg="R" Name="IsArtinian" Label="for rings"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsArtinian",
 IsHomalgRing );

## <#GAPDoc Label="IsLeftArtinian">
## <ManSection>
## <Prop Arg="R" Name="IsLeftArtinian"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftArtinian",
 IsHomalgRing );

## <#GAPDoc Label="IsRightArtinian">
## <ManSection>
## <Prop Arg="R" Name="IsRightArtinian"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightArtinian",
 IsHomalgRing );

## <#GAPDoc Label="IsOreDomain">
## <ManSection>
## <Prop Arg="R" Name="IsOreDomain"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsOreDomain",
 IsHomalgRing );

## <#GAPDoc Label="IsLeftOreDomain">
## <ManSection>
## <Prop Arg="R" Name="IsLeftOreDomain"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftOreDomain",
 IsHomalgRing );

## <#GAPDoc Label="IsRightOreDomain">
## <ManSection>
## <Prop Arg="R" Name="IsRightOreDomain"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightOreDomain",
 IsHomalgRing );

## <#GAPDoc Label="IsPrincipalIdealRing">
## <ManSection>
## <Prop Arg="R" Name="IsPrincipalIdealRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsPrincipalIdealRing",
 IsHomalgRing );

## <#GAPDoc Label="IsLeftPrincipalIdealRing">
## <ManSection>
## <Prop Arg="R" Name="IsLeftPrincipalIdealRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftPrincipalIdealRing",
 IsHomalgRing );

## <#GAPDoc Label="IsRightPrincipalIdealRing">
## <ManSection>
## <Prop Arg="R" Name="IsRightPrincipalIdealRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightPrincipalIdealRing",
 IsHomalgRing );

## <#GAPDoc Label="IsRegular">
## <ManSection>
## <Prop Arg="R" Name="IsRegular"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRegular",
 IsHomalgRing );

## <#GAPDoc Label="IsFiniteFreePresentationRing">
## <ManSection>
## <Prop Arg="R" Name="IsFiniteFreePresentationRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFiniteFreePresentationRing",
 IsHomalgRing );

## <#GAPDoc Label="IsLeftFiniteFreePresentationRing">
## <ManSection>
## <Prop Arg="R" Name="IsLeftFiniteFreePresentationRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftFiniteFreePresentationRing",
 IsHomalgRing );

## <#GAPDoc Label="IsRightFiniteFreePresentationRing">
## <ManSection>
## <Prop Arg="R" Name="IsRightFiniteFreePresentationRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightFiniteFreePresentationRing",
 IsHomalgRing );

## <#GAPDoc Label="IsSimpleRing">
## <ManSection>
## <Prop Arg="R" Name="IsSimpleRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSimpleRing",
 IsHomalgRing );

## <#GAPDoc Label="IsSemiSimpleRing">
## <ManSection>
## <Prop Arg="R" Name="IsSemiSimpleRing"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSemiSimpleRing",
 IsHomalgRing );

## <#GAPDoc Label="IsSuperCommutative">
## <ManSection>
## <Prop Arg="R" Name="IsSuperCommutative"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSuperCommutative",
 IsHomalgRing );

## <#GAPDoc Label="BasisAlgorithmRespectsPrincipalIdeals">
## <ManSection>
## <Prop Arg="R" Name="BasisAlgorithmRespectsPrincipalIdeals"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "BasisAlgorithmRespectsPrincipalIdeals",
 IsHomalgRing );

## <#GAPDoc Label="AreUnitsCentral">
## <ManSection>
## <Prop Arg="R" Name="AreUnitsCentral"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <A>R</A> is a ring for &homalg;.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "AreUnitsCentral",
 IsHomalgRing );

## <#GAPDoc Label="IsMinusOne">
## <ManSection>
## <Prop Arg="r" Name="IsMinusOne"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the ring element <A>r</A> is the additive inverse of one.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsMinusOne",
 IsRingElement );

## <#GAPDoc Label="IsMonic:ringelement">
## <ManSection>
## <Prop Arg="r" Name="IsMonic" Label="for homalg ring elements"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; ring element <A>r</A> is monic.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsMonic",
 IsHomalgRingElement );

## <#GAPDoc Label="IsMonicUptoUnit:ringelement">
## <ManSection>
## <Prop Arg="r" Name="IsMonicUptoUnit" Label="for homalg ring elements"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if leading coefficient of the &homalg; ring element <A>r</A> is a unit.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsMonicUptoUnit",
 IsHomalgRingElement );

## <#GAPDoc Label="IsLeftRegular:ringelement">
## <ManSection>
## <Prop Arg="r" Name="IsLeftRegular" Label="for homalg ring elements"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; ring element <A>r</A> is left regular.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftRegular",
 IsHomalgRingElement );

## <#GAPDoc Label="IsRightRegular:ringelement">
## <ManSection>
## <Prop Arg="r" Name="IsRightRegular" Label="for homalg ring elements"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; ring element <A>r</A> is right regular.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightRegular",
 IsHomalgRingElement );

## <#GAPDoc Label="IsRegular:ringelement">
## <ManSection>
## <Prop Arg="r" Name="IsRegular" Label="for homalg ring elements"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; ring element <A>r</A> is regular, i.e. left and right regular.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRegular",
 IsHomalgRingElement );

## <#GAPDoc Label="IsIrreducibleHomalgRingElement:ringelement">
## <ManSection>
## <Prop Arg="r" Name="IsIrreducibleHomalgRingElement" Label="for homalg ring elements"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; ring element <A>r</A> is irreducible.
## The short operation name is <C>IsIrreducible</C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsIrreducibleHomalgRingElement",
 IsHomalgRingElement );

####################################
#
# attributes:
#
####################################

## <#GAPDoc Label="Inverse:ring_element">
## <ManSection>
## <Attr Arg="r" Name="Inverse" Label="for homalg ring elements"/>
## <Returns>a &homalg; ring element or fail</Returns>
## <Description>
## The inverse of the &homalg; ring element <A>r</A>.
## <Example><![CDATA[
## gap> zz := HomalgRingOfIntegers( );;
## gap> R := zz / 2^8;
## Z/( 256 )
## gap> r := (1/3*One(R)+1/5)+3/7;
## |[ 157 ]|
## gap> 1 / r; ## = r^-1;
## |[ 181 ]|
## gap> s := (1/3*One(R)+2/5)+3/7;
## |[ 106 ]|
## gap> s^(-1);
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>

##
DeclareOperation( "MinusOneMutable",
 [ IsHomalgRingElement ] );

##
DeclareAttribute( "EvalRingElement",
 IsHomalgRingElement );

DeclareAttribute( "CoefficientsOfUnivariatePolynomial",
 IsHomalgRingElement );

DeclareAttribute( "MaximalDegreePart",
 IsHomalgRingElement );

DeclareAttribute( "IrreducibleFactors",
 IsRingElement );

## <#GAPDoc Label="Zero:ring">
## <ManSection>
## <Attr Arg="R" Name="Zero" Label="for homalg rings"/>
## <Returns>a &homalg; ring element</Returns>
## <Description>
## The zero of the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
#DeclareAttribute( "Zero",
# IsHomalgRing );

## <#GAPDoc Label="One:ring">
## <ManSection>
## <Attr Arg="R" Name="One" Label="for homalg rings"/>
## <Returns>a &homalg; ring element</Returns>
## <Description>
## The one of the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
#DeclareAttribute( "One",
# IsHomalgRing );

## <#GAPDoc Label="MinusOne">
## <ManSection>
## <Attr Arg="R" Name="MinusOne"/>
## <Returns>a &homalg; ring element</Returns>
## <Description>
## The minus one of the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "MinusOne",
 IsHomalgRing );

## <#GAPDoc Label="ProductOfIndeterminates">
## <ManSection>
## <Attr Arg="R" Name="ProductOfIndeterminates"/>
## <Returns>a &homalg; ring element</Returns>
## <Description>
## The product of indeterminates of the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ProductOfIndeterminates",
 IsHomalgRing );

## <#GAPDoc Label="ProductOfIndeterminatesOverBaseRing">
## <ManSection>
## <Attr Arg="R" Name="ProductOfIndeterminatesOverBaseRing"/>
## <Returns>a &homalg; ring element</Returns>
## <Description>
## The product of indeterminates of the &homalg; ring <A>R</A> over its base ring.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ProductOfIndeterminatesOverBaseRing",
 IsHomalgRing );

## <#GAPDoc Label="homalgTable">
## <ManSection>
## <Attr Arg="R" Name="homalgTable"/>
## <Returns>a &homalg; table</Returns>
## <Description>
## The &homalg; table of <A>R</A> is a ring dictionary, i.e. the translator between &homalg;
## and the (specific implementation of the) ring. 
## Every &homalg; ring has a &homalg; table.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "homalgTable",
 IsHomalgRing );

## <#GAPDoc Label="RingElementConstructor">
## <ManSection>
## <Attr Arg="R" Name="RingElementConstructor"/>
## <Returns>a function</Returns>
## <Description>
## The constructor of ring elements in the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RingElementConstructor",
 IsHomalgRing );

## <#GAPDoc Label="TypeOfHomalgMatrix">
## <ManSection>
## <Attr Arg="R" Name="TypeOfHomalgMatrix"/>
## <Returns>a type</Returns>
## <Description>
## The &GAP4;-type of &homalg; matrices over the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "TypeOfHomalgMatrix",
 IsHomalgRing );

## <#GAPDoc Label="ConstructorForHomalgMatrices">
## <ManSection>
## <Attr Arg="R" Name="ConstructorForHomalgMatrices"/>
## <Returns>a type</Returns>
## <Description>
## The constructor for &homalg; matrices over the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ConstructorForHomalgMatrices",
 IsHomalgRing );

## <#GAPDoc Label="RationalParameters">
## <ManSection>
## <Attr Arg="R" Name="RationalParameters"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of rational parameters of the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "RationalParameters",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminatesOfPolynomialRing">
## <ManSection>
## <Attr Arg="R" Name="IndeterminatesOfPolynomialRing"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminates of the &homalg; polynomial ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "IndeterminatesOfPolynomialRing",
 IsHomalgRing );

## <#GAPDoc Label="RelativeIndeterminatesOfPolynomialRing">
## <ManSection>
## <Attr Arg="R" Name="RelativeIndeterminatesOfPolynomialRing"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminates of the &homalg; polynomial ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "RelativeIndeterminatesOfPolynomialRing",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateCoordinatesOfRingOfDerivations">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateCoordinatesOfRingOfDerivations"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate coordinates of the &homalg; Weyl ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateCoordinatesOfRingOfDerivations",
 IsHomalgRing );

## <#GAPDoc Label="RelativeIndeterminateCoordinatesOfRingOfDerivations">
## <ManSection>
## <Attr Arg="R" Name="RelativeIndeterminateCoordinatesOfRingOfDerivations"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate coordinates of the &homalg; Weyl ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeIndeterminateCoordinatesOfRingOfDerivations",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateDerivationsOfRingOfDerivations">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateDerivationsOfRingOfDerivations"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate derivations of the &homalg; Weyl ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateDerivationsOfRingOfDerivations",
 IsHomalgRing );

## <#GAPDoc Label="RelativeIndeterminateDerivationsOfRingOfDerivations">
## <ManSection>
## <Attr Arg="R" Name="RelativeIndeterminateDerivationsOfRingOfDerivations"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate derivations of the &homalg; Weyl ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeIndeterminateDerivationsOfRingOfDerivations",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateAntiCommutingVariablesOfExteriorRing">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateAntiCommutingVariablesOfExteriorRing"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of anti-commuting indeterminates of the &homalg; exterior ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateAntiCommutingVariablesOfExteriorRing",
 IsHomalgRing );

## <#GAPDoc Label="RelativeIndeterminateAntiCommutingVariablesOfExteriorRing">
## <ManSection>
## <Attr Arg="R" Name="RelativeIndeterminateAntiCommutingVariablesOfExteriorRing"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of anti-commuting relative indeterminates of the &homalg; exterior ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeIndeterminateAntiCommutingVariablesOfExteriorRing",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminatesOfExteriorRing">
## <ManSection>
## <Attr Arg="R" Name="IndeterminatesOfExteriorRing"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of all indeterminates (commuting and anti-commuting) of the &homalg; exterior ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminatesOfExteriorRing",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateCoordinatesOfShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateCoordinatesOfShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate coordinates of the &homalg; shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateCoordinatesOfShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="ParametersOfRationalShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="ParametersOfRationalShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate coordinates of the &homalg; shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ParametersOfRationalShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="RelativeIndeterminateCoordinatesOfShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="RelativeIndeterminateCoordinatesOfShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate coordinates of the &homalg; shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeIndeterminateCoordinatesOfShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="RelativeParametersOfRationalShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="RelativeParametersOfRationalShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate coordinates of the &homalg; shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeParametersOfRationalShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateShiftsOfShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateShiftsOfShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate shifts of the &homalg; shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateShiftsOfShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateShiftsOfRationalShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateShiftsOfRationalShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate shifts of the &homalg; shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateShiftsOfRationalShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateCoordinatesOfPseudoDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateCoordinatesOfPseudoDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate coordinates of the &homalg; pseudo double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateCoordinatesOfPseudoDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="ParametersOfRationalPseudoDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="ParametersOfRationalPseudoDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate coordinates of the &homalg; pseudo double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ParametersOfRationalPseudoDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="RelativeIndeterminateCoordinatesOfPseudoDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="RelativeIndeterminateCoordinatesOfPseudoDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate coordinates of the &homalg; pseudo double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeIndeterminateCoordinatesOfPseudoDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="RelativeParametersOfRationalPseudoDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="RelativeParametersOfRationalPseudoDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate coordinates of the &homalg; pseudo double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeParametersOfRationalPseudoDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateShiftsOfPseudoDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateShiftsOfPseudoDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate shifts of the &homalg; pseudo double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateShiftsOfPseudoDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateShiftsOfRationalPseudoDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateShiftsOfRationalPseudoDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate shifts of the &homalg; pseudo double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateShiftsOfRationalPseudoDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateCoordinatesOfDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateCoordinatesOfDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate coordinates of the &homalg; double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateCoordinatesOfDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="ParametersOfRationalDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="ParametersOfRationalDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate coordinates of the &homalg; double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ParametersOfRationalDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="RelativeIndeterminateCoordinatesOfDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="RelativeIndeterminateCoordinatesOfDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate coordinates of the &homalg; double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeIndeterminateCoordinatesOfDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="RelativeParametersOfRationalDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="RelativeParametersOfRationalDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate coordinates of the &homalg; double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeParametersOfRationalDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateShiftsOfDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateShiftsOfDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate shifts of the &homalg; double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateShiftsOfDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateShiftsOfRationalDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateShiftsOfRationalDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate shifts of the &homalg; double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateShiftsOfRationalDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateCoordinatesOfBiasedDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateCoordinatesOfBiasedDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate coordinates of the &homalg; biased double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateCoordinatesOfBiasedDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="RelativeIndeterminateCoordinatesOfBiasedDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="RelativeIndeterminateCoordinatesOfBiasedDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of relative indeterminate coordinates of the &homalg; biased double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RelativeIndeterminateCoordinatesOfBiasedDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="IndeterminateShiftsOfBiasedDoubleShiftAlgebra">
## <ManSection>
## <Attr Arg="R" Name="IndeterminateShiftsOfBiasedDoubleShiftAlgebra"/>
## <Returns>a list of &homalg; ring elements</Returns>
## <Description>
## The list of indeterminate shifts of the &homalg; biased double-shift algebra <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndeterminateShiftsOfBiasedDoubleShiftAlgebra",
 IsHomalgRing );

## <#GAPDoc Label="CoefficientsRing">
## <ManSection>
## <Attr Arg="R" Name="CoefficientsRing"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## The ring of coefficients of the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CoefficientsRing",
 IsHomalgRing );

DeclareAttribute( "CoefficientsRing",
 IsHomalgRingElement );

## <#GAPDoc Label="BaseRing">
## <ManSection>
## <Attr Arg="R" Name="BaseRing"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## In case <A>R</A> was constructed as a polynomial or exterior ring over a base ring <M>T</M>,
## and only in this case, the &homalg; ring <M>T</M> is returned.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "BaseRing",
 IsHomalgRing );

## <#GAPDoc Label="KrullDimension">
## <ManSection>
## <Attr Arg="R" Name="KrullDimension"/>
## <Returns>a non-negative integer</Returns>
## <Description>
## The Krull dimension of the commutative &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "KrullDimension",
 IsHomalgRing );

## <#GAPDoc Label="DegreeOverPrimeField">
## <ManSection>
## <Attr Arg="R" Name="DegreeOverPrimeField"/>
## <Returns>a positive integer</Returns>
## <Description>
## The degree over the prime field over the &homalg; field <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "DegreeOverPrimeField",
 IsHomalgRing );

## <#GAPDoc Label="LeftGlobalDimension">
## <ManSection>
## <Attr Arg="R" Name="LeftGlobalDimension"/>
## <Returns>a non-negative integer</Returns>
## <Description>
## The left global dimension of the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "LeftGlobalDimension",
 IsHomalgRing );

## <#GAPDoc Label="RightGlobalDimension">
## <ManSection>
## <Attr Arg="R" Name="RightGlobalDimension"/>
## <Returns>a non-negative integer</Returns>
## <Description>
## The right global dimension of the &homalg; ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RightGlobalDimension",
 IsHomalgRing );

## <#GAPDoc Label="GlobalDimension">
## <ManSection>
## <Attr Arg="R" Name="GlobalDimension"/>
## <Returns>a non-negative integer</Returns>
## <Description>
## The global dimension of the &homalg; ring <A>R</A>.
## The global dimension is defined, only if the left and right
## global dimensions coincide.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "GlobalDimension",
 IsHomalgRing );

## <#GAPDoc Label="GeneralLinearRank">
## <ManSection>
## <Attr Arg="R" Name="GeneralLinearRank"/>
## <Returns>a non-negative integer</Returns>
## <Description>
## The general linear rank of the &homalg; ring <A>R</A> (<Cite Key="McCRob"/>, 11.1.14).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "GeneralLinearRank",
 IsHomalgRing );

## <#GAPDoc Label="ElementaryRank">
## <ManSection>
## <Attr Arg="R" Name="ElementaryRank"/>
## <Returns>a non-negative integer</Returns>
## <Description>
## The elementary rank of the &homalg; ring <A>R</A> (<Cite Key="McCRob"/>, 11.3.10).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ElementaryRank",
 IsHomalgRing );

## <#GAPDoc Label="StableRank">
## <ManSection>
## <Attr Arg="R" Name="StableRank"/>
## <Returns>a non-negative integer</Returns>
## <Description>
## The stable rank of the &homalg; ring <A>R</A> (<Cite Key="McCRob"/>, 11.3.4).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "StableRank",
 IsHomalgRing );

## <#GAPDoc Label="AssociatedGradedRing">
## <ManSection>
## <Attr Arg="R" Name="AssociatedGradedRing"/>
## <Returns>a homalg ring</Returns>
## <Description>
## The graded ring associated to the filtered ring <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AssociatedGradedRing",
 IsHomalgRing );

## <#GAPDoc Label="AssociatedPolynomialRing">
## <ManSection>
## <Attr Arg="R" Name="AssociatedPolynomialRing"/>
## <Returns>a homalg ring</Returns>
## <Description>
## The polynomial ring associated to the ring <A>R</A>
## which is an algebra over a rational function field.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AssociatedPolynomialRing",
 IsHomalgRing );

## <#GAPDoc Label="PrimitiveElement">
## <ManSection>
## <Attr Arg="R" Name="PrimitiveElement"/>
## <Returns>a homalg ring element</Returns>
## <Description>
## The primitive element of the &homalg; field <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "PrimitiveElement",
 IsHomalgRing );

####################################
#
# global functions and operations:
#
####################################

# constructors:

DeclareGlobalFunction( "CreateHomalgRing" );

DeclareGlobalFunction( "HomalgRingOfIntegers" );

DeclareOperation( "HomalgRingOfIntegersInUnderlyingCAS",
 [ IsInt, IsHomalgRing ] );

DeclareGlobalFunction( "HomalgRingElement" );

DeclareOperation( "AddRationalParameters",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "/",
 [ IsRingElement, IsHomalgRing ] );

DeclareOperation( "/",
 [ IsString, IsHomalgRing ] );

DeclareGlobalFunction( "StringToElementStringList" );

DeclareGlobalFunction( "_CreateHomalgRingToTestProperties" );

DeclareOperation( "ParseListOfIndeterminates",
 [ IsList ] );

DeclareGlobalFunction( "_PrepareInputForPolynomialRing" );

#DeclareOperation( "PolynomialRing",
# [ IsHomalgRing, IsList ] );

DeclareOperation( "*",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "\[\]",
 [ IsHomalgRing, IsList ] );

DeclareAttribute( "PolynomialRingWithDegRevLexOrdering",
 IsHomalgRing );

DeclareOperation( "PolynomialRingWithLexicographicOrdering",
 [ IsHomalgRing, IsList ] );

DeclareAttribute( "PolynomialRingWithLexicographicOrdering",
 IsHomalgRing );

DeclareOperation( "PolynomialRingWithProductOrdering",
 [ IsHomalgRing, IsList ] );

DeclareAttribute( "PolynomialRingWithProductOrdering",
 IsHomalgRing );

DeclareOperation( "PolynomialRingWithWeightedOrdering",
 [ IsHomalgRing, IsList ] );

DeclareAttribute( "PolynomialRingWithWeightedOrdering",
 IsHomalgRing );

DeclareOperation( "RingOfDerivations",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "RingOfDerivations",
 [ IsHomalgRing, IsList, IsList ] );

DeclareOperation( "RingOfDerivations",
 [ IsHomalgRing ] );

DeclareOperation( "ExteriorRing",
 [ IsHomalgRing, IsHomalgRing, IsHomalgRing, IsList ] );

DeclareOperation( "ExteriorRing",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "KoszulDualRing",
 [ IsHomalgRing, IsHomalgRing, IsList ] );

DeclareOperation( "KoszulDualRing",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "KoszulDualRing",
 [ IsHomalgRing ] );

DeclareOperation( "ShiftAlgebra",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "RationalShiftAlgebra",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "PseudoDoubleShiftAlgebra",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "RationalPseudoDoubleShiftAlgebra",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "DoubleShiftAlgebra",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "RationalDoubleShiftAlgebra",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "BiasedDoubleShiftAlgebra",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "UnivariatePolynomial",
 [ IsList, IsString ] );

DeclareOperation( "Homogenization",
 [ IsHomalgRingElement, IsHomalgRing ] );

DeclareOperation( "LcmOp",
 [ IsList, IsHomalgRingElement ] );

DeclareOperation( "LcmOp",
 [ IsHomalgRingElement, IsHomalgRingElement ] );

DeclareOperation( "Numerator",
 [ IsHomalgRingElement ] );

DeclareOperation( "Denominator",
 [ IsHomalgRingElement ] );

# basic operations:

DeclareOperation( "HomalgRing",
 [ IsHomalgRingElement ] );

DeclareOperation( "Factors",
 [ IsHomalgRingElement ] );

DeclareOperation( "Roots",
 [ IsHomalgRingElement ] );

DeclareOperation( "HomalgRing", ## returns itself
 [ IsHomalgRing ] );

#DeclareOperation( "Indeterminate",
# [ IsHomalgRing, IsPosInt ] );

DeclareOperation( "Indeterminates",
 [ IsHomalgRing ] );

DeclareOperation( "ExportIndeterminates",
 [ IsHomalgRing ] );

DeclareOperation( "ExportRationalParameters",
 [ IsHomalgRing ] );

DeclareOperation( "ExportVariables",
 [ IsHomalgRing ] );

#DeclareOperation( "IsUnit",
# [ IsHomalgRing, IsRingElement ] );

DeclareOperation( "IsUnit",
 [ IsHomalgRingElement ] );

DeclareOperation( "FieldOfFractions",
 [ IsHomalgRing ] );

DeclareOperation( "StandardBasisRowVectors",
 [ IsInt, IsHomalgRing ] );

DeclareOperation( "StandardBasisColumnVectors",
 [ IsInt, IsHomalgRing ] );

DeclareOperation( "RingName",
 [ IsHomalgRing ] );

DeclareOperation( "DisplayRing",
 [ IsHomalgRing ] );

DeclareOperation( "homalgRingStatistics",
 [ IsHomalgRing ] );

DeclareOperation( "IncreaseRingStatistics",
 [ IsHomalgRing, IsString ] );

DeclareOperation( "DecreaseRingStatistics",
 [ IsHomalgRing, IsString ] );

DeclareOperation( "SetRingProperties",
 [ IsHomalgRing, IsInt ] );

DeclareOperation( "SetRingProperties",
 [ IsHomalgRing, IsInt, IsInt ] );

DeclareOperation( "SetRingProperties",
 [ IsHomalgRing, IsHomalgRing, IsList ] );

DeclareOperation( "SetRingProperties",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "SetRingProperties",
 [ IsHomalgRing ] );

DeclareOperation( "EuclideanQuotient",
 [ IsHomalgRing, IsRingElement, IsRingElement ] );

DeclareOperation( "EuclideanRemainder",
 [ IsHomalgRing, IsRingElement, IsRingElement ] );

DeclareOperation( "homalgSetName",
 [ IsHomalgRingElement, IsString ] );

DeclareOperation( "homalgSetName",
 [ IsHomalgRingElement, IsString, IsHomalgRing ] );

DeclareOperation( "Random",
 [ IsHomalgRing, IsList ] );

DeclareOperation( "Random",
 [ IsHomalgRing, IsInt ] );

DeclareOperation( "UnusedVariableName",
 [ IsHomalgRing, IsString ] );

DeclareOperation( "Inequalities",
 [ IsHomalgRing ] );

DeclareOperation( "LaTeXOutput",
 [ IsHomalgRingElement ] );

DeclareOperation( "LaTeXOutput",
 [ IsHomalgRing ] );

Quelle HomalgRing.gd Sprache: unbekannt

[ Dauer der Verarbeitung: 0.32 Sekunden (vorverarbeitet) ]