Quelle isorms.xml Sprache: XML

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#W isorms.xml
#Y Copyright (C) 2015 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
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<#GAPDoc Label="IsRMSIsoByTriple">
 <ManSection>
 <Filt Name = "IsRMSIsoByTriple" Type = "Category"/>
 <Filt Name = "IsRZMSIsoByTriple" Type = "Category"/>
 <Description>
 The isomorphisms between finite Rees matrix or 0-matrix semigroups
 <C>S</C> and <C>T</C> over groups <C>G</C> and <C>H</C>, respectively,
 specified by a triple consisting of:

 <Enum>

 <Item>
 an isomorphism of the underlying graph of <C>S</C> to the underlying
 graph of of <C>T</C>
 </Item>

 <Item>
 an isomorphism from <C>G</C> to <C>H</C>
 </Item>

 <Item>
 a function from <C>Rows(S)</C> union <C>Columns(S)</C> to <C>H</C>
 </Item>

 </Enum>

 belong to the categories <C>IsRMSIsoByTriple</C> and
 <C>IsRZMSIsoByTriple</C>. Basic operators for such isomorphism are given
 in <Ref Subsect =
 "Operators for isomorphisms of Rees (0-)matrix semigroups"/>,
 and basic operations are:

 <Ref Attr = "Range" BookName = "ref"/>,
 <Ref Attr = "Source" BookName = "ref"/>,
 <Ref Oper = "ELM_LIST" Label = "for IsRMSIsoByTriple"/>,
 <Ref Func = "CompositionMapping" BookName = "ref"/>,
 <Ref Oper = "ImagesElm" Label = "for IsRMSIsoByTriple"/>,
 <Ref Oper = "ImagesRepresentative" Label = "for IsRMSIsoByTriple"/>,
 <Ref Attr = "InverseGeneralMapping" BookName = "ref"/>,
 <Ref Oper = "PreImagesRepresentative" BookName = "ref"/>,
 <Ref Prop = "IsOne" BookName = "ref"/>.

 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="RMSIsoByTriple">
 <ManSection>
 <Oper Name = "RMSIsoByTriple" Arg = "R1, R2, triple"/>
 <Oper Name = "RZMSIsoByTriple" Arg = "R1, R2, triple"/>
 <Returns>An isomorphism.</Returns>
 <Description>
 If <A>R1</A> and <A>R2</A> are isomorphic regular Rees 0-matrix semigroups
 whose underlying semigroups are groups then <C>RZMSIsoByTriple</C> returns
 the isomorphism between <A>R1</A> and <A>R2</A> defined by <A>triple</A>,
 which should be a list consisting of the following:

 <List>
 <Item>
 <C><A>triple</A>[1]</C> should be a permutation describing an
 isomorphism from the graph of <A>R1</A> to the graph of <A>R2</A>,
 i.e. it should satisfy
 <C>OnDigraphs(RZMSDigraph(<A>R1</A>), <A>triple</A>[1])
 = RZMSDigraph(<A>R2</A>)</C>.
 </Item>
 <Item>
 <C><A>triple</A>[2]</C> should be an isomorphism from the underlying
 group of <A>R1</A> to the underlying group of <A>R2</A> (see
 <Ref Attr="UnderlyingSemigroup"
 Label="for a Rees 0-matrix semigroup"
 BookName="ref"/>).
 </Item>
 <Item>
 <C><A>triple</A>[3]</C> should be a list of elements from the
 underlying group of <A>R2</A>. If the
 <Ref Attr="Matrix" BookName="ref"/>
 of <A>R1</A> has <M>m</M> columns
 and <M>n</M> rows, then the list should have length <M>m + n</M>,
 where the first <M>m</M> entries should correspond to the columns of
 <A>R1</A>'s matrix, and the last n entries should correspond to
 the rows. These column and row entries should correspond to the
 <M>u_i</M> and <M>v_\lambda</M> elements in Theorem 3.4.1 of
 <Cite Key = "Howie1995aa"/>.
 </Item>
 </List>

 If <A>triple</A> describes a valid isomorphism from <A>R1</A> to <A>R2</A>
 then this will return an object in the category
 <Ref Filt="IsRZMSIsoByTriple"/>; otherwise an error will be returned. 

 If <A>R1</A> and <A>R2</A> are instead Rees matrix semigroups (without
 zero) then <C>RMSIsoByTriple</C> should be used instead. This operation
 is used in the same way, but it should be noted that since an RMS's graph
 is a complete bipartite graph, <C><A>triple</A>[1]</C> can be any
 permutation on <C>[1 .. m + n]</C>, so long as no point in <C>[1 .. m]</C>
 is mapped to a point in <C>[m + 1 .. m + n]</C>. 

 <Example><![CDATA[
gap> g := SymmetricGroup(3);;
gap> mat := [[0, 0, (1, 3)], [(1, 2, 3), (), (2, 3)], [0, 0, ()]];;
gap> R := ReesZeroMatrixSemigroup(g, mat);;
gap> id := IdentityMapping(g);;
gap> g_elms_list := [(), (), (), (), (), ()];;
gap> RZMSIsoByTriple(R, R, [(), id, g_elms_list]);
((), IdentityMapping( SymmetricGroup( [ 1 .. 3 ] ) ),
[ (), (), (), (), (), () ])
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="ELM_LIST">
 <ManSection>
 <Oper Name = "ELM_LIST" Label = "for IsRMSIsoByTriple"
 Arg = "map, pos"/>
 <Returns>
 A component of an isomorphism of Rees (0-)matrix semigroups by triple.
 </Returns>
 <Description>
 <C>ELM_LIST(<A>map</A>, <A>i</A>)</C> returns the <C>i</C>th component of
 the Rees (0-)matrix semigroup isomorphism by triple <A>map</A> when
 <C>i = 1, 2, 3</C>.
 

 The components of an isomorphism of Rees (0-)matrix semigroups by triple
 are:
 <Enum>
 <Item>
 An isomorphism of the underlying graphs of the source and range of
 <A>map</A>, respectively.
 </Item>

 <Item>
 An isomorphism of the underlying groups of the source and range of
 <A>map</A>, respectively.
 </Item>

 <Item>
 An function from the union of the rows and columns of the source of
 <A>map</A> to the underlying group of the range of <A>map</A>.
 </Item>
 </Enum>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="CompositionMapping2">
 <ManSection>
 <Oper Name = "CompositionMapping2" Label = "for IsRMSIsoByTriple"
 Arg = "map1, map2"/>
 <Oper Name = "CompositionMapping2" Label = "for IsRZMSIsoByTriple"
 Arg = "map1, map2"/>
 <Returns>
 A Rees (0-)matrix semigroup by triple.
 </Returns>
 <Description>
 If <A>map1</A> and <A>map2</A> are isomorphisms of Rees matrix or
 0-matrix semigroups specified by triples and the range of <A>map2</A> is
 contained in the source of <A>map1</A>, then
 <C>CompositionMapping2(<A>map1</A>, <A>map2</A>)</C> returns the
 isomorphism from <C>Source(<A>map2</A>)</C> to <C>Range(<A>map1</A>)</C>
 specified by the triple with components:

 <Enum>
 <Item>
 <C><A>map1</A>[1] * <A>map2</A>[1]</C>
 </Item>

 <Item>
 <C><A>map1</A>[2] * <A>map2</A>[2]</C>
 </Item>

 <Item>
 the componentwise product of <C><A>map1</A>[1] * <A>map2</A>[3]</C>
 and <C><A>map1</A>[3] * <A>map2</A>[2]</C>.
 </Item>
 </Enum>
 <Example><![CDATA[
gap> R := ReesZeroMatrixSemigroup(Group([(1, 2, 3, 4)]),
> [[(1, 3)(2, 4), (1, 4, 3, 2), (), (1, 2, 3, 4), (1, 3)(2, 4), 0],
> [(1, 4, 3, 2), 0, (), (1, 4, 3, 2), (1, 2, 3, 4), (1, 2, 3, 4)],
> [(), (), (1, 4, 3, 2), (1, 2, 3, 4), 0, (1, 2, 3, 4)],
> [(1, 2, 3, 4), (1, 4, 3, 2), (1, 2, 3, 4), 0, (), (1, 2, 3, 4)],
> [(1, 3)(2, 4), (1, 2, 3, 4), 0, (), (1, 4, 3, 2), (1, 2, 3, 4)],
> [0, (1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4), ()]]);
<Rees 0-matrix semigroup 6x6 over Group([ (1,2,3,4) ])>
gap> G := AutomorphismGroup(R);;
gap> G.2;
((), IdentityMapping( Group( [ (1,2,3,4) ] ) ),
[ (), (), (), (), (), (), (), (), (), (), (), () ])
gap> G.3;
(( 2, 4)( 3, 5)( 8,10)( 9,11), GroupHomomorphismByImages( Group(
[ (1,2,3,4) ] ), Group( [ (1,2,3,4) ] ), [ (1,2,3,4) ],
[ (1,2,3,4) ] ), [ (), (1,3)(2,4), (1,3)(2,4), (1,3)(2,4),
 (1,3)(2,4), (1,3)(2,4), (), (1,3)(2,4), (1,3)(2,4), (1,3)(2,4),
 (1,3)(2,4), (1,3)(2,4) ])
gap> CompositionMapping2(G.2, G.3);
(( 2, 4)( 3, 5)( 8,10)( 9,11), GroupHomomorphismByImages( Group(
[ (1,2,3,4) ] ), Group( [ (1,2,3,4) ] ), [ (1,2,3,4) ],
[ (1,2,3,4) ] ), [ (), (1,3)(2,4), (1,3)(2,4), (1,3)(2,4),
 (1,3)(2,4), (1,3)(2,4), (), (1,3)(2,4), (1,3)(2,4), (1,3)(2,4),
 (1,3)(2,4), (1,3)(2,4) ])]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="ImagesElm">
 <ManSection>
 <Oper Name = "ImagesElm" Label = "for IsRMSIsoByTriple" Arg = "map, pt"/>
 <Oper Name = "ImagesRepresentative" Label = "for IsRMSIsoByTriple"
 Arg = "map, pt"/>
 <Returns>
 An element of a Rees (0-)matrix semigroup or a list containing such an
 element.
 </Returns>

 <Description>
 If <A>map</A> is an isomorphism of Rees matrix or 0-matrix semigroups
 specified by a triple and <A>pt</A> is an element of the source of
 <A>map</A>, then <C>ImagesRepresentative(<A>map</A>, <A>pt</A>) =
 <A>pt</A> ^ <A>map</A></C> returns the image of <A>pt</A> under
 <A>map</A>.
 

 The image of <A>pt</A> under <A>map</A> of <C>Range(<A>map</A>)</C> is the
 element with components:

 <Enum>
 <Item>
 <C><A>pt</A>[1] ^ <A>map</A>[1]</C>
 </Item>
 <Item>
 <C>(<A>pt</A>[1] ^ <A>map</A>[3]) * (<A>pt</A>[2] ^ <A>map</A>[2])
 * (<A>pt</A>[3] ^ <A>map</A>[3]) ^ -1</C>
 </Item>
 <Item>
 <C><A>pt</A>[3] ^ <A>map</A>[1]</C>.
 </Item>
 </Enum>

 <C>ImagesElm(<A>map</A>, <A>pt</A>)</C> simply returns
 <C>[ImagesRepresentative(<A>map</A>, <A>pt</A>)]</C>.

 <Example><![CDATA[
gap> R := ReesZeroMatrixSemigroup(Group([(2, 8), (2, 8, 6)]),
> [[0, (2, 8), 0, 0, 0, (2, 8, 6)],
> [(), 0, (2, 8, 6), (2, 6), (2, 6, 8), 0],
> [(2, 8, 6), 0, (2, 6, 8), (2, 8), (), 0],
> [(2, 8, 6), 0, (2, 6, 8), (2, 8), (), 0],
> [0, (2, 8, 6), 0, 0, 0, (2, 8)],
> [(2, 8, 6), 0, (2, 6, 8), (2, 8), (), 0]]);
<Rees 0-matrix semigroup 6x6 over Group([ (2,8), (2,8,6) ])>
gap> map := RZMSIsoByTriple(R, R,
> [(), IdentityMapping(Group([(2, 8), (2, 8, 6)])),
> [(), (2, 6, 8), (), (), (), (2, 8, 6),
> (2, 8, 6), (), (), (), (2, 6, 8), ()]]);;
gap> ImagesElm(map, RMSElement(R, 1, (2, 8), 2));
[ (1,(2,8),2) ]]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="CanonicalReesZeroMatrixSemigroup">
 <ManSection>
 <Attr Name = "CanonicalReesZeroMatrixSemigroup" Arg = "S"/>
 <Attr Name = "CanonicalReesMatrixSemigroup" Arg = "S"/>

 <Returns>
 A Rees zero matrix semigroup.
 </Returns>

 <Description>
 If <A>S</A> is a Rees 0-matrix semigroup then
 <C>CanonicalReesZeroMatrixSemigroup</C> returns an isomorphic Rees
 0-matrix semigroup <C>T</C> with the same
 <Ref Attr="UnderlyingSemigroup" Label="for a Rees 0-matrix semigroup"
 BookName="ref"/>
 as <A>S</A> but the
 <Ref Attr="Matrix" BookName="ref"/>
 of <C>T</C> has been canonicalized. The output
 <C>T</C> is canonical in the sense that for any two inputs
 which are isomorphic Rees zero matrix semigroups the output of this
 function is the same.
 <C>CanonicalReesMatrixSemigroup</C> works the same but for Rees matrix
 semigroups.

 <Example><![CDATA[
gap> S := ReesZeroMatrixSemigroup(SymmetricGroup(3),
> [[(), (1, 3, 2)], [(), ()]]);;
gap> T := CanonicalReesZeroMatrixSemigroup(S);
<Rees 0-matrix semigroup 2x2 over Sym( [ 1 .. 3 ] )>
gap> Matrix(S);
[ [ (), (1,3,2) ], [ (), () ] ]
gap> Matrix(T);
[ [ (), () ], [ (), (1,2,3) ] ]]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

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