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18 Semigroup homomorphisms
 18.1 Isomorphisms of arbitrary semigroups
 18.2 Isomorphisms of Rees (0-)matrix semigroups

18 Semigroup homomorphisms

In this chapter we describe the various ways to define a homomorphism from a semigroup to another semigroup.

18.1 Isomorphisms of arbitrary semigroups

18.1-1 IsIsomorphicSemigroup
‣ IsIsomorphicSemigroup( S, T )( operation )

Returns: true or false.

This operation attempts to determine if the semigroups S and T are isomorphic, it returns true if they are and false if they are not.

At present this only works for rather small semigroups, with approximately 50 elements or less.

Please note: the Grape package version 4.5 or higher must be available and compiled installed for this function to work.

gap> S := Semigroup(PartialPerm([1, 2, 4], [3, 5, 1]), 
>                   PartialPerm([1, 3, 5], [4, 3, 2]));;
gap> Size(S);
11
gap> T := SemigroupByMultiplicationTable(
> SmallestMultiplicationTable(S));;
gap> IsIsomorphicSemigroup(S, T);
true

18.1-2 SmallestMultiplicationTable
‣ SmallestMultiplicationTable( S )( attribute )

Returns: The lex-least multiplication table of a semigroup.

This function returns the lex-least multiplication table of a semigroup isomorphic to the semigroup S. SmallestMultiplicationTable is an isomorphism invariant of semigroups, and so it can, for example, be used to check if two semigroups are isomorphic.

Due to the high complexity of computing the smallest multiplication table of a semigroup, this function only performs well for semigroups with at most approximately 50 elements.

SmallestMultiplicationTable is based on the function IdSmallSemigroup (Smallsemi: IdSmallSemigroup) by Andreas Distler.

Please note: the Grape package version 4.5 or higher must be loaded for this function to work.

gap> S := Semigroup(
> Bipartition([[1, 2, 3, -1, -3], [-2]]), 
> Bipartition([[1, 2, 3, -1], [-2], [-3]]), 
> Bipartition([[1, 2, 3], [-1], [-2, -3]]), 
> Bipartition([[1, 2, -1], [3, -2], [-3]]));;
gap> Size(S);
8
gap> SmallestMultiplicationTable(S);
[ [ 1, 1, 3, 4, 5, 6, 7, 8 ], [ 1, 1, 3, 4, 5, 6, 7, 8 ], 
  [ 1, 1, 3, 4, 5, 6, 7, 8 ], [ 1, 3, 3, 4, 5, 6, 7, 8 ], 
  [ 5, 5, 6, 7, 5, 6, 7, 8 ], [ 5, 5, 6, 7, 5, 6, 7, 8 ], 
  [ 5, 6, 6, 7, 5, 6, 7, 8 ], [ 5, 6, 6, 7, 5, 6, 7, 8 ] ]

18.1-3 IsomorphismSemigroups
‣ IsomorphismSemigroups( S, T )( operation )

Returns: An isomorphism or fail.

This operation returns an isomorphism from the semigroup S and to the semigroup T if it exists, and it returns fail if it does not.

At present this only works for Rees matrix semigroups and Rees 0-matrix semigroups.

Please note: the Grape package version 4.5 or higher must be available and compiled for this function to work, when the argument R is a Rees 0-matrix semigroup.

gap> D := DClass(FullTransformationMonoid(5), 
>                Transformation([1, 2, 3, 4, 1]));;
gap> S := PrincipalFactor(D);;
gap> StructureDescription(UnderlyingSemigroup(S));
"S4"
gap> S;
<Rees 0-matrix semigroup 10x5 over S4>
gap> D := DClass(PartitionMonoid(5), 
>                Bipartition([[1], [2, -2], [3, -3], 
>                             [4, -4], [5, -5], [-1]]));;
gap> T := PrincipalFactor(D);;
gap> StructureDescription(UnderlyingSemigroup(T));
"S4"
gap> T;
<Rees 0-matrix semigroup 15x15 over S4>
gap> IsomorphismSemigroups(S, T);
fail
gap> I := SemigroupIdeal(FullTransformationMonoid(5), 
>                        Transformation([1, 1, 2, 3, 4]));;
gap> T := PrincipalFactor(DClass(I, I.1));;
gap> StructureDescription(UnderlyingSemigroup(T));
"S4"
gap> T;
<Rees 0-matrix semigroup 10x5 over S4>
gap> IsomorphismSemigroups(S, T) <> fail;
true

18.2 Isomorphisms of Rees (0-)matrix semigroups

PREAMBLE

18.2-1 Operators for isomorphisms of Rees (0-)matrix semigroup by triples
map[i]

map[i] returns the ith component of the Rees (0-)matrix semigroup isomorphism by triple map when i = 1, 2, 3; see ELM_LIST (???).

map1 * map2

returns the composition of map2 and map1; see CompositionMapping2 (???).

map1 < map2

returns true if map1 is lexicographically less than map2.

map1 = map2

returns true if the Rees (0-)matrix semigroup isomorphisms by triple map1 and map2 have equal source and range, and are equal as functions, and false otherwise.

It is possible for map1 and map2 to be equal but to have distinct components.

pt ^ map

returns the image of the element pt of the source of map under the isomorphism map; see ImagesElm (???).

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