<Chapter Label =
"Semigroup homomorphisms">
<Heading>
Semigroup homomorphisms
</Heading>
In this chapter we describe the various ways to define a homomorphism from a
semigroup to another semigroup. <P/>
<!--**********************************************************************-->
<!--**********************************************************************-->
<Section>
<Heading>
Homomorphisms of arbitrary semigroups
</Heading>
<#
Include Label =
"SemigroupHomomorphismByImages">
<#
Include Label =
"SemigroupHomomorphismByFunction">
The following methods relate to semigroup homomorphisms by images or
by function:
<List>
<Item>
<Ref Attr =
"Range" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"Image" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"Images" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"ImageElm" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"PreImage" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"PreImages" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"PreImagesRepresentative" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"PreImagesRange" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"PreImagesElm" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"PreImagesSet" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"IsSurjective" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"IsInjective" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"IsBijective" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"Source" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"Range" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"ImagesSource" BookName =
"ref"/>,
</Item>
<Item>
<Ref Attr =
"KernelOfSemigroupHomomorphism"/>.
</Item>
</List>
<#
Include Label =
"IsSemigroupHomomorphismByImages">
<#
Include Label =
"IsSemigroupHomomorphismByFunction">
<#
Include Label =
"AsSemigroupHomomorphismByImages">
<#
Include Label =
"AsSemigroupHomomorphismByFunction">
<#
Include Label =
"KernelOfSemigroupHomomorphism">
</Section>
<!--**********************************************************************-->
<!--**********************************************************************-->
<Section>
<Heading>
Isomorphisms of arbitrary semigroups
</Heading>
<#
Include Label =
"IsIsomorphicSemigroup">
<#
Include Label =
"SmallestMultiplicationTable">
<#
Include Label =
"CanonicalMultiplicationTable">
<#
Include Label =
"CanonicalMultiplicationTablePerm">
<#
Include Label =
"OnMultiplicationTable">
<#
Include Label =
"IsomorphismSemigroups">
<#
Include Label =
"AutomorphismGroup">
<#
Include Label =
"SemigroupIsomorphismByImages">
<#
Include Label =
"SemigroupIsomorphismByFunction">
<#
Include Label =
"IsSemigroupIsomorphismByFunction">
<#
Include Label =
"AsSemigroupIsomorphismByFunction">
<#
Include Label =
"SmallerDegreeTransformationRepresentation">
<#
Include Label =
"MinimalFaithfulTransformationDegree">
</Section>
<!--**********************************************************************-->
<!--**********************************************************************-->
<Section>
<Heading>
Isomorphisms of Rees (
0-)matrix semigroups
</Heading>
An isomorphism between two regular finite Rees (
0-)matrix semigroups whose
underlying semigroups are groups can be described by a triple defined in
terms of the matrices and underlying groups of the semigroups. For a full
description of the theory involved, see Section
3.
4 of
<Cite Key =
"Howie1995aa"/>. <P/>
An isomorphism described in this way can be constructed
using <Ref Oper=
"RMSIsoByTriple"/> or <Ref Oper=
"RZMSIsoByTriple"/>, and
will satisfy the filter <Ref Filt=
"IsRMSIsoByTriple"/> or
<Ref Filt=
"IsRZMSIsoByTriple"/>. <P/>
<#
Include Label =
"IsRMSIsoByTriple">
<#
Include Label =
"RMSIsoByTriple">
<#
Include Label =
"ELM_LIST">
<#
Include Label =
"CompositionMapping2">
<#
Include Label =
"ImagesElm">
<#
Include Label =
"CanonicalReesZeroMatrixSemigroup">
<!--********************************************************************-->
<Subsection Label =
"Operators for isomorphisms of Rees (0-)matrix semigroups">
<Heading>
Operators for isomorphisms of Rees (
0-)matrix semigroups
</Heading>
<List>
<Mark><C><A>map</A>[<A>i</A>]</C></Mark>
<Item>
<Index Key =
"ELM_LIST"><C>ELM_LIST</C> (for Rees (
0-)matrix semigroup
isomorphisms by triples)
</Index>
<C><A>map</A>[i]</C> returns the <A>i</A>th component of the Rees
(
0-)matrix semigroup isomorphism by triple <A>map</A> when
<C><A>i</A> =
1,
2,
3</C>; see
<Ref Oper =
"ELM_LIST" Label =
"for IsRMSIsoByTriple"/>.
</Item>
<Mark><C><A>map1</A> * <A>map2</A></C></Mark>
<Item>
<Index Key =
"*"><C> * </C>
(for Rees (
0-)matrix semigroup isomorphisms by triples)
</Index>
returns the composition of <A>map2</A> and <A>map1</A>;
see <Ref Oper =
"CompositionMapping2" Label =
"for IsRMSIsoByTriple"/>.
</Item>
<Mark><C><A>map1</A> < <A>map2</A></C></Mark>
<Item>
<Index Key =
"<"><C><</C>
(for Rees (
0-)matrix semigroup isomorphisms by triples)
</Index>
returns <K>true</K> if <A>map1</A> is lexicographically less than
<A>map2</A>.
</Item>
<Mark><C><A>map1</A> = <A>map2</A></C></Mark>
<Item>
<Index Key =
"="><C> = </C>
(for Rees (
0-)matrix semigroup isomorphisms by triples)
</Index>
returns <K>true</K> if the Rees (
0-)matrix semigroup
isomorphisms by triple <A>map1</A> and <A>map2</A> have equal source and
range, and are equal as functions, and <K>false</K> otherwise. <P/>
It is possible for <A>map1</A> and <A>map2</A> to be equal but to
have distinct components.
</Item>
<Mark><C><A>pt</A> ^ <A>map</A></C></Mark>
<Item>
<Index Key =
"^"><C> ^ </C>
(for Rees (
0-)matrix semigroup isomorphisms by triples)
</Index>
returns the image of the
element <A>pt</A> of the source of
<A>map</A> under the isomorphism <A>map</A>; see
<Ref Oper =
"ImagesElm" Label =
"for IsRMSIsoByTriple"/>.
</Item>
</List>
</Subsection>
</Section>
<!--**********************************************************************-->
<!--**********************************************************************-->
</Chapter>
