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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Steve Linton.
##
## 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 declares the operations for direct product elements.
##
## Direct product elements are immutable finite type-safe lists.
##
#############################################################################
##
#V InfoDirectProductElements . . . . . . . . . . . . . . . . . . Info Class
##
DeclareInfoClass( "InfoDirectProductElements" );
#############################################################################
##
#C IsDirectProductElement( <obj> ) . . category of direct product elements
##
## <#GAPDoc Label="IsDirectProductElement">
## <ManSection>
## <Filt Name="IsDirectProductElement" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsDirectProductElement"/> is a subcategory of the meet of
## <Ref Filt="IsDenseList"/>,
## <Ref Filt="IsMultiplicativeElementWithInverse"/>,
## <Ref Filt="IsAdditiveElementWithInverse"/>,
## and <Ref Filt="IsCopyable"/>,
## where the arithmetic operations (addition, zero, additive inverse,
## multiplication, powering, one, inverse) are defined componentwise.
## <P/>
## Note that each of these operations will cause an error message if
## its result for at least one component cannot be formed.
## <P/>
## For an object in the filter <Ref Filt="IsDirectProductElement"/>,
## <Ref Oper="ShallowCopy"/> returns a mutable plain list with the same
## entries.
## The sum and the product of a direct product element and a list in
## <Ref Filt="IsListDefault"/> is the list of sums and products,
## respectively.
## The sum and the product of a direct product element and an object
## that is neither a list nor a collection
## is the direct product element of componentwise sums and products,
## respectively.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsDirectProductElement",
IsDenseList
and IsSmallList
and IsCopyable
and IsMultiplicativeElementWithInverse
and IsAdditiveElementWithInverse );
#############################################################################
##
#C IsDirectProductElementFamily( <family> ) . . . category of direct product
#C element families
##
## <ManSection>
## <Filt Name="IsDirectProductElementFamily" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryFamily( "IsDirectProductElement" );
#############################################################################
##
#C IsDirectProductElementCollection( <coll> ) . category of direct product
#C element collections
##
## <ManSection>
## <Filt Name="IsDirectProductElementCollection" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryCollections( "IsDirectProductElement" );
#############################################################################
##
#O DirectProductElementsFamily( <famlist> ) . . . . family of direct product
#O elements
##
## <ManSection>
## <Oper Name="DirectProductElementsFamily" Arg='famlist'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareOperation( "DirectProductElementsFamily", [ IsCollection ] );
#############################################################################
##
#A ComponentsOfDirectProductElementsFamily( <fam> ) . . component families
##
## <ManSection>
## <Attr Name="ComponentsOfDirectProductElementsFamily" Arg='fam'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "ComponentsOfDirectProductElementsFamily",
IsDirectProductElementFamily );
#############################################################################
##
#R IsDefaultDirectProductElementRep( <obj> ) . . . . . representation as a
#R component object
##
DeclareRepresentation( "IsDefaultDirectProductElementRep",
IsPositionalObjectRep and IsDirectProductElement, [] );
#############################################################################
##
#V EmptyDirectProductElementsFamily
##
BindGlobal( "EmptyDirectProductElementsFamily",
NewFamily( "DirectProductElementsFamily([])", IsDirectProductElement,
IsDirectProductElement ) );
#############################################################################
##
#O DirectProductElement( [<fam>, ]<objlist> )
#O DirectProductElementNC( <fam>, <objlist> ) . . . . omits check on object
#O families and objlist length
##
## <ManSection>
## <Oper Name="DirectProductElement" Arg='[fam, ]objlist'/>
## <Oper Name="DirectProductElementNC" Arg='fam, objlist'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareOperation( "DirectProductElement", [ IsList ]);
DeclareOperation( "DirectProductElementNC",
[ IsDirectProductElementFamily, IsList ]);
#############################################################################
##
##
## <#GAPDoc Label="DirectProductFamily">
## <ManSection>
## <Func Name="DirectProductFamily" Arg='args'/>
##
## <Description>
## <A>args</A> must be a dense list of <Ref Attr="CollectionsFamily"/>
## families, otherwise the function raises an error.
## <P/>
## <Ref Func="DirectProductFamily"/> returns <C>fam</C>, a collections
## family of <Ref Filt="IsDirectProductElement"/> objects.
## <P/>
## <C>fam</C> is the <Ref Attr="CollectionsFamily"/> of
## <Ref Filt="IsDirectProductElement"/> objects
## whose <C>i</C>-th component is in <C>ElementsFamily(args[i])</C>.
## <P/>
## Note that a collection in <C>fam</C> may not itself be a
## direct product; it just is a subcollection of a direct product.
## <P/>
## <Example><![CDATA[
## gap> D8 := DihedralGroup(IsPermGroup, 8);;
## gap> FamilyObj(D8) = CollectionsFamily(PermutationsFamily);
## true
## gap> fam := DirectProductFamily([FamilyObj(D8), FamilyObj(D8)]);;
## gap> ComponentsOfDirectProductElementsFamily(ElementsFamily(fam));
## [ <Family: "PermutationsFamily">, <Family: "PermutationsFamily"> ]
## ]]></Example>
## Also note that not all direct products in &GAP; are created via these
## families. For example if the arguments to <Ref Func="DirectProduct"/>
## are permutation groups, then it returns a permutation group as well, whose
## elements are not <Ref Filt="IsDirectProductElement"/> objects.
## <P/>
## <Example><![CDATA[
## gap> fam = FamilyObj(DirectProduct(D8, D8));
## false
## gap> D4 := DihedralGroup(IsPcGroup, 4);;
## gap> fam2 := DirectProductFamily([FamilyObj(D8), FamilyObj(D4)]);;
## gap> fam2 = FamilyObj(DirectProduct(D8, D4));
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "DirectProductFamily" );
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