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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 implementation of magmas, monoids, and groups from
## a multiplication table.
##
#############################################################################
##
#F MagmaByMultiplicationTableCreator( <A>, <domconst> )
##
## <ManSection>
## <Func Name="MagmaByMultiplicationTableCreator" Arg='A, domconst'/>
##
## <Description>
## This is a utility for the uniform construction of a magma,
## a magma-with-one, or a magma-with-inverses from a multiplication table.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "MagmaByMultiplicationTableCreator" );
DeclareGlobalFunction( "MagmaByMultiplicationTableCreatorNC" );
#############################################################################
##
#F MagmaByMultiplicationTable( <A> )
##
## <#GAPDoc Label="MagmaByMultiplicationTable">
## <ManSection>
## <Func Name="MagmaByMultiplicationTable" Arg='A'/>
##
## <Description>
## For a square matrix <A>A</A> with <M>n</M> rows such that all entries of
## <A>A</A> are in the range <M>[ 1 .. n ]</M>,
## <Ref Func="MagmaByMultiplicationTable"/> returns a magma
## <M>M</M> with multiplication <C>*</C> defined by <A>A</A>.
## That is, <M>M</M> consists of the elements <M>m_1, m_2, \ldots, m_n</M>,
## and <M>m_i * m_j = m_k</M>, with <M>k =</M> <A>A</A><M>[i][j]</M>.
## <P/>
## The ordering of elements is defined by
## <M>m_1 < m_2 < \cdots < m_n</M>,
## so <M>m_i</M> can be accessed as
## <C>MagmaElement( <A>M</A>, <A>i</A> )</C>,
## see <Ref Func="MagmaElement"/>.
## <Example><![CDATA[
## gap> MagmaByMultiplicationTable([[1,2,3],[2,3,1],[1,1,1]]);
## <magma with 3 generators>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MagmaByMultiplicationTable" );
#############################################################################
##
#F MagmaWithOneByMultiplicationTable( <A> )
##
## <#GAPDoc Label="MagmaWithOneByMultiplicationTable">
## <ManSection>
## <Func Name="MagmaWithOneByMultiplicationTable" Arg='A'/>
##
## <Description>
## The only differences between <Ref Func="MagmaByMultiplicationTable"/> and
## <Ref Func="MagmaWithOneByMultiplicationTable"/> are that the latter
## returns a magma-with-one (see <Ref Func="MagmaWithOne"/>)
## if the magma described by the matrix <A>A</A> has an identity,
## and returns <K>fail</K> if not.
## <Example><![CDATA[
## gap> MagmaWithOneByMultiplicationTable([[1,2,3],[2,3,1],[3,1,1]]);
## <magma-with-one with 3 generators>
## gap> MagmaWithOneByMultiplicationTable([[1,2,3],[2,3,1],[1,1,1]]);
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MagmaWithOneByMultiplicationTable" );
#############################################################################
##
#F MagmaWithInversesByMultiplicationTable( <A> )
##
## <#GAPDoc Label="MagmaWithInversesByMultiplicationTable">
## <ManSection>
## <Func Name="MagmaWithInversesByMultiplicationTable" Arg='A'/>
##
## <Description>
## <Ref Func="MagmaByMultiplicationTable"/> and
## <Ref Func="MagmaWithInversesByMultiplicationTable"/>
## differ only in that the latter returns
## magma-with-inverses (see <Ref Func="MagmaWithInverses"/>)
## if each element in the magma described by the matrix <A>A</A>
## has an inverse,
## and returns <K>fail</K> if not.
## <Example><![CDATA[
## gap> MagmaWithInversesByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);
## <magma-with-inverses with 3 generators>
## gap> MagmaWithInversesByMultiplicationTable([[1,2,3],[2,3,1],[3,2,1]]);
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MagmaWithInversesByMultiplicationTable" );
#############################################################################
##
#F MagmaElement( <M>, <i> ) . . . . . . . . . . <i>-th element of magma <M>
##
## <#GAPDoc Label="MagmaElement">
## <ManSection>
## <Func Name="MagmaElement" Arg='M, i'/>
##
## <Description>
## For a magma <A>M</A> and a positive integer <A>i</A>,
## <Ref Func="MagmaElement"/> returns the <A>i</A>-th element of <A>M</A>,
## w.r.t. the ordering <C><</C>.
## If <A>M</A> has less than <A>i</A> elements then <K>fail</K> is returned.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MagmaElement" );
#############################################################################
##
#F SemigroupByMultiplicationTable( <A> )
##
## <#GAPDoc Label="SemigroupByMultiplicationTable">
## <ManSection>
## <Func Name="SemigroupByMultiplicationTable" Arg='A'/>
##
## <Description>
## returns the semigroup whose multiplication is defined by the square
## matrix <A>A</A> (see <Ref Func="MagmaByMultiplicationTable"/>)
## if such a semigroup exists.
## Otherwise <K>fail</K> is returned.
## <Example><![CDATA[
## gap> SemigroupByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);
## <semigroup of size 3, with 3 generators>
## gap> SemigroupByMultiplicationTable([[1,2,3],[2,3,1],[3,2,1]]);
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SemigroupByMultiplicationTable" );
#############################################################################
##
#F MonoidByMultiplicationTable( <A> )
##
## <#GAPDoc Label="MonoidByMultiplicationTable">
## <ManSection>
## <Func Name="MonoidByMultiplicationTable" Arg='A'/>
##
## <Description>
## returns the monoid whose multiplication is defined by the square
## matrix <A>A</A> (see <Ref Func="MagmaByMultiplicationTable"/>)
## if such a monoid exists.
## Otherwise <K>fail</K> is returned.
## <Example><![CDATA[
## gap> MonoidByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);
## <monoid of size 3, with 3 generators>
## gap> MonoidByMultiplicationTable([[1,2,3],[2,3,1],[1,3,2]]);
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MonoidByMultiplicationTable" );
#############################################################################
##
#F GroupByMultiplicationTable( <A> )
##
## <ManSection>
## <Func Name="GroupByMultiplicationTable" Arg='A'/>
##
## <Description>
## returns the group whose multiplication is defined by the square
## matrix <A>A</A> (see <Ref Func="MagmaByMultiplicationTable"/>)
## if such a group exists.
## Otherwise <K>fail</K> is returned.
## <Example><![CDATA[
## gap> GroupByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);
## <group of size 3 with 3 generators>
## gap> GroupByMultiplicationTable([[1,2,3],[2,3,1],[3,2,1]]);
## fail
## ]]></Example>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "GroupByMultiplicationTable" );
#############################################################################
##
#A MultiplicationTable( <elms> )
#A MultiplicationTable( <M> )
##
## <#GAPDoc Label="MultiplicationTable">
## <ManSection>
## <Heading>MultiplicationTable</Heading>
## <Attr Name="MultiplicationTable" Arg='elms'
## Label="for a list of elements"/>
## <Attr Name="MultiplicationTable" Arg='M' Label="for a magma"/>
##
## <Description>
## For a list <A>elms</A> of elements that form a magma <M>M</M>,
## <Ref Attr="MultiplicationTable" Label="for a list of elements"/> returns
## a square matrix <M>A</M> of positive integers such that
## <M>A[i][j] = k</M> holds if and only if
## <A>elms</A><M>[i] *</M> <A>elms</A><M>[j] =</M> <A>elms</A><M>[k]</M>.
## This matrix can be used to construct a magma isomorphic to <M>M</M>,
## using <Ref Func="MagmaByMultiplicationTable"/>.
## <P/>
## For a magma <A>M</A>,
## <Ref Attr="MultiplicationTable" Label="for a magma"/> returns
## the multiplication table w.r.t. the sorted list of elements of
## <A>M</A>.
## <P/>
## <Example><![CDATA[
## gap> l:= [ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ];;
## gap> a:= MultiplicationTable( l );
## [ [ 1, 2, 3, 4 ], [ 2, 1, 4, 3 ], [ 3, 4, 1, 2 ], [ 4, 3, 2, 1 ] ]
## gap> m:= MagmaByMultiplicationTable( a );
## <magma with 4 generators>
## gap> One( m );
## m1
## gap> elm:= MagmaElement( m, 2 ); One( elm ); elm^2;
## m2
## m1
## m1
## gap> Inverse( elm );
## m2
## gap> AsGroup( m );
## <group of size 4 with 2 generators>
## gap> a:= [ [ 1, 2 ], [ 2, 2 ] ];
## [ [ 1, 2 ], [ 2, 2 ] ]
## gap> m:= MagmaByMultiplicationTable( a );
## <magma with 2 generators>
## gap> One( m ); Inverse( MagmaElement( m, 2 ) );
## m1
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "MultiplicationTable", IsHomogeneousList );
DeclareAttribute( "MultiplicationTable", IsMagma );
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