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#############################################################################
##
#W algebra.gd The XMODALG package Zekeriya Arvasi
#W & Alper Odabas
#Y Copyright (C) 2014-2025, Zekeriya Arvasi & Alper Odabas,
##
DeclareInfoClass( "InfoXModAlg" );
############################ algebra operations ###################
DeclareProperty( "HasZeroAnnihilator", IsAlgebra and IsCommutative );
DeclareOperation( "RegularAlgebraMultiplier",
[ IsAlgebra, IsAlgebra, IsObject ] );
DeclareProperty( "IsAlgebraMultiplier", IsMapping );
DeclareOperation( "MultiplierAlgebraOfIdealBySubalgebra",
[ IsAlgebra, IsAlgebra, IsAlgebra ] );
DeclareOperation( "MultiplierAlgebraByGenerators", [ IsAlgebra, IsList ] );
DeclareAttribute( "MultiplierAlgebra", IsAlgebra, "mutable" );
DeclareAttribute( "MultiplierHomomorphism", IsAlgebra, "mutable" );
DeclareProperty( "IsMultiplierAlgebra", IsList );
############################# algebra mappings ####################
DeclareOperation( "InclusionMappingAlgebra", [ IsAlgebra, IsAlgebra ] );
DeclareOperation( "RestrictionMappingAlgebra",
[ IsAlgebraHomomorphism, IsAlgebra, IsAlgebra ] );
DeclareOperation( "AllAlgebraHomomorphisms",
[ IsAlgebra, IsAlgebra ] );
DeclareOperation( "AllBijectiveAlgebraHomomorphisms",
[ IsAlgebra, IsAlgebra ] );
DeclareOperation( "AllIdempotentAlgebraHomomorphisms",
[ IsAlgebra, IsAlgebra ] );
############################## algebra actions ####################
DeclareProperty( "IsAlgebraAction", IsAlgebraHomomorphism );
DeclareGlobalFunction( "AlgebraAction" );
DeclareAttribute( "LeftElementOfCartesianProduct", IsAlgebraAction );
DeclareAttribute( "AlgebraActionType", IsAlgebraAction );
DeclareAttribute( "AlgebraActedOn", IsAlgebraAction );
DeclareAttribute( "HasZeroModuleProduct", IsAlgebraAction );
DeclareOperation( "AlgebraActionByMultipliers",
[ IsAlgebra, IsAlgebra, IsAlgebra ] );
DeclareOperation( "AlgebraActionBySurjection", [ IsAlgebraHomomorphism ] );
DeclareOperation ( "SemidirectProductOfAlgebras",
[ IsAlgebra, IsAlgebraAction, IsAlgebra ] );
DeclareAttribute( "SemidirectProductOfAlgebrasInfo", IsAlgebra, "mutable" );
#####################################################################
################ direct sum of algebras with info #################
## section 2.4.1
#############################################################################
##
#O DirectSumOfAlgebrasWithInfo( <A> )
#A DirectSumOfAlgebrasInfo( <A> )
##
## <#GAPDoc Label="DirectSumOfAlgebrasInfo">
## <ManSection>
## <Oper Name="DirectSumOfAlgebrasWithInfo" Arg='A1 A2'/>
## <Attr Name="DirectSumOfAlgebrasInfo" Arg='A'/>
##
## <Description>
## This attribute for direct sums of algebras is missing from
## the main library, and is added here to be used in methods
## for <C>Embedding</C> and <C>Projection</C>.
## In order to construct a direct sum with this information attribute
## the operation <C>DirectSumOfAlgebrasWithInfo</C> may be used.
## This just calls <C>DirectSumOfAlgebras</C> and sets up the attribute.
## </Description>
## </ManSection>
## <Example>
## <![CDATA[
## gap> A3Rc3 := DirectSumOfAlgebrasWithInfo( A3, Rc3 );;
## gap> SetName( A3Rc3, Concatenation( Name(A3), "(+)", Name(Rc3) ) );
## gap> DirectSumOfAlgebrasInfo( A3Rc3 );
## rec( algebras := [ A3, GR(c3) ],
## embeddings :=
## [
## Basis( A3, [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
## [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
## [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ] ) -> [ v.1, v.2, v.3 ],
## CanonicalBasis( GR(c3) ) -> [ v.4, v.5, v.6 ] ], first := [ 1, 4 ],
## projections :=
## [ CanonicalBasis( A3(+)GR(c3) ) ->
## [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
## [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
## [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
## [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ],
## [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ],
## [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ],
## CanonicalBasis( A3(+)GR(c3) ) -> [ <zero> of ..., <zero> of ...,
## <zero> of ..., (1)*(), (1)*(1,2,3), (1)*(1,3,2) ] ] )
## ]]>
## </Example>
## <#/GAPDoc>
##
DeclareAttribute( "DirectSumOfAlgebrasInfo", IsAlgebra, "mutable" );
DeclareOperation( "DirectSumOfAlgebrasWithInfo", [ IsAlgebra, IsAlgebra ] );
## section 2.4.2
############################################################################
##
## Embedding( <alg> )
##
## <#GAPDoc Label="EmbeddingForDirectSumOfAlgebras">
## <ManSection>
## <Oper Name="Embedding" Label="for direct sum of algebras"
## Arg="A nr" />
## <Oper Name="Projection" Label="for direct sum of algebras"
## Arg="A nr" />
##
## <Description>
## Methods for <C>Embedding</C> and <C>Projection</C> for direct sums
## of algebras are missing from the main library, and so are included here.
## <P/>
## </Description>
## </ManSection>
## <Example>
## <![CDATA[
## gap> Embedding( A3Rc3, 1 );
## Basis( A3, [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
## [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
## [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ] ) -> [ v.1, v.2, v.3 ]
## gap> Projection( A3Rc3, 2 );
## CanonicalBasis( A3(+)GR(c3) ) -> [ <zero> of ..., <zero> of ...,
## <zero> of ..., (1)*(), (1)*(1,2,3), (1)*(1,3,2) ]
## ]]>
## </Example>
## <#/GAPDoc>
##
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