In this chapter we give instructions on how to create semigroup homomorphisms using MONOID in several different ways.
In Section 7.2, we give functions for creating arbitrary semigroup homomorphism specified by a function on the elements, the images of the generators, or the images of all the semigroup elements. These functions were written to support the functions for computing the automorphism group of an arbitrary transformation semigroup and to specify isomorphisms between different classes of semigroup, such as finitely presented semigroups and transformation semigroups.
In Section 7.3, we show how to specify and compute the inner automorphisms of a transformation semigroup. The functions that can be used to find the entire automorphism group of an arbitrary transformation semigroup are given in Section 7.4. The AutomorphismGroup
(7.4-1) has an interactive mode that allows the user to decide how the computation should proceed. This can be invoked by using the command SetInfoLevel(InfoAutos, 4);
see InfoAutos
(7.1-1).
In Section 7.5, commands for creating automorphisms and finding all automorphisms of Rees matrix semigroups and Rees 0
-matrix semigroups are given.
In Section 7.6, functions for specifying the automorphisms of a zero group are given.
In the final section (7.7), functions for finding isomorphisms between various kinds of semigroups are given.
The methods behind the commands in this chapter are taken from [ABM07].
Please note: the following functions can only be used fully if GRAPE is fully installed (and loaded):
AutomorphismGroup
(7.4-1) with argument satisfying IsTransformationSemigroup
(Reference: IsTransformationSemigroup) or IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup)
RightTransStabAutoGroup
(7.5-9) with argument satisfying IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup)
RZMSGraph
(7.5-8)
RZMSInducedFunction
(7.5-6)
RZMStoRZMSInducedFunction
(7.5-7)
IsomorphismSemigroups
(7.7-5) with both arguments satisfying IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup)
Please see Chapter 1 for further details on how to obtain GRAPE.
> InfoAutos | ( info class ) |
This is the InfoClass for the functions in this chapter. Setting the value of InfoAutos
to 1, 2, 3,
or 4
using the command SetInfoLevel
(Reference: SetInfoLevel) will give different levels of information about what GAP
is doing during a computation. In particular, if the level of InfoAutos
is set to 4
, then AutomorphismGroup
(7.4-1) runs in interactive mode.
The principal functions for creating arbitrary semigroup homomorphisms are the following three.
> SemigroupHomomorphismByFunction ( S, T, func ) | ( operation ) |
> SemigroupHomomorphismByFunctionNC ( S, T, func ) | ( operation ) |
returns a semigroup homomorphism with representation IsSemigroupHomomorphismByFunctionRep
from the semigroup S
to the semigroup T
defined by the function func
.
SemigroupHomomorphismByFunction
will find an isomorphism from S
to a finitely presented semigroup or monoid (using IsomorphismFpSemigroup
(7.7-3) or IsomorphismFpMonoid
(7.7-4)) and then check that the list of values under func
of the generators of S
satisfy the relations of this presentation.
SemigroupHomomorphismByFunctionNC
does not check that func
defines a homomorphism and, in this case S
and T
can be semigroups, D-classes, H-classes or any combination of these.
gap> gens:=[ Transformation( [ 1, 4, 3, 5, 2 ] ), > Transformation( [ 2, 3, 1, 1, 2 ] ) ];; gap> S:=Semigroup(gens);; gap> gens:=[ Transformation( [ 1, 5, 1, 2, 1 ] ), > Transformation( [ 5, 1, 4, 3, 2 ] ) ];; gap> T:=Semigroup(gens);; gap> idem:=Random(Idempotents(T));; gap> hom:=SemigroupHomomorphismByFunction(S, T, x-> idem); SemigroupHomomorphism ( <semigroup with 2 generators>-><semigroup with 2 generators>) gap> hom:=SemigroupHomomorphismByFunctionNC(S, T, x-> idem); SemigroupHomomorphism ( <semigroup with 2 generators>-><semigroup with 2 generators>) |
> SemigroupHomomorphismByImagesOfGens ( S, T, list ) | ( operation ) |
> SemigroupHomomorphismByImagesOfGensNC ( S, T, list ) | ( operation ) |
returns a semigroup homomorphism with representation IsSemigroupHomomorphismByImagesOfGensRep
from S
to T
where the image of the i
th generator of S
is the i
th position in list
.
SemigroupHomomorphismByImagesOfGens
will find an isomorphism from S
to a finitely presented semigroup or monoid (using IsomorphismFpSemigroup
(7.7-3) or IsomorphismFpMonoid
(7.7-4)) and then check that list
satisfies the relations of this presentation.
SemigroupHomomorphismByImagesOfGensNC
does not check that list
induces a homomorphism.
gap> gens:=[ Transformation( [ 1, 4, 3, 5, 2 ] ), > Transformation( [ 2, 3, 1, 1, 2 ] ) ];; gap> S:=Semigroup(gens);; gap> gens:=[ Transformation( [ 1, 5, 1, 2, 1 ] ), > Transformation( [ 5, 1, 4, 3, 2 ] ) ];; gap> T:=Semigroup(gens);; gap> SemigroupHomomorphismByImagesOfGens(S, T, GeneratorsOfSemigroup(T)); fail gap> SemigroupHomomorphismByImagesOfGens(S, S, GeneratorsOfSemigroup(S)); SemigroupHomomorphismByImagesOfGens ( <trans. semigroup of size 161 with 2 generators>-><trans. semigroup of size 161 with 2 generators>) |
> SemigroupHomomorphismByImages ( S, T, list ) | ( operation ) |
> SemigroupHomomorphismByImagesNC ( S, T, list ) | ( operation ) |
returns a semigroup homomorphism with representation IsSemigroupHomomorphismByImagesRep
from S
to T
where the image of the i
th element of S
is the i
th position in list
.
SemigroupHomomorphismByImages
will find an isomorphism from S
to a finitely presented semigroup or monoid (using IsomorphismFpSemigroup
(7.7-3) or IsomorphismFpMonoid
(7.7-4)) and then check that list
satisfies the relations of this presentation.
SemigroupHomomorphismByImagesNC
does not check that list
induces a homomorphism.
gap> gens:=[ Transformation( [ 2, 3, 4, 2, 4 ] ), > Transformation( [ 3, 4, 2, 1, 4 ] ) ];; gap> S:=Semigroup(gens);; gap> gens:=[ Transformation( [ 2, 4, 4, 1, 2 ] ), > Transformation( [ 5, 1, 1, 5, 1 ] ) ];; gap> T:=Semigroup(gens);; gap> idem:=Transformation( [ 5, 5, 5, 5, 5 ] );; gap> list:=List([1..Size(S)], x-> idem);; gap> hom:=SemigroupHomomorphismByImages(S, T, list); SemigroupHomomorphismByImagesOfGens ( <trans. semigroup of size 164 with 2 generators>-><trans. semigroup with 2 generators>) gap> SemigroupHomomorphismByImagesNC(S, T, list); SemigroupHomomorphismByImages ( <trans. semigroup of size 164 with 2 generators>-><trans. semigroup with 2 generators>) |
> InnerAutomorphismOfSemigroup ( S, perm ) | ( operation ) |
> InnerAutomorphismOfSemigroupNC ( S, perm ) | ( operation ) |
returns the inner automorphism of the transformation semigroup S
given by the permutation perm
. The degree of perm
should be at most the degree of S
.
The notion of inner automorphisms of semigroups differs from the notion of the same name for groups. Indeed, if S
is a semigroup of transformations of degree n
, then g
in the symmetric group S_n
induces an inner automorphism of S
if the mapping that takes s
to g^-1sg
for all s
in S
is an automorphism of S
.
InnerAutomorphismOfSemigroup
checks that the mapping induced by perm
is an automorphism and InnerAutomorphismOfSemigroupNC
only creates the appropriate object without performing a check that the permutation actually induces an automorphism.
gap> gens:=[ Transformation( [ 6, 2, 7, 5, 3, 5, 4 ] ), > Transformation( [ 7, 7, 5, 7, 2, 4, 3 ] ) ];; gap> S:=Monoid(gens);; gap> InnerAutomorphismOfSemigroup(S, (1,2,3,4,5)); fail gap> InnerAutomorphismOfSemigroupNC(S, (1,2,3,4,5)); ^(1,2,3,4,5) gap> InnerAutomorphismOfSemigroup(S, ()); ^() |
> ConjugatorOfInnerAutomorphismOfSemigroup ( f ) | ( attribute ) |
returns the permutation perm
used to construct the inner automorphism f
of a semigroup; see InnerAutomorphismOfSemigroup
(7.3-1) for further details.
gap> S:=RandomSemigroup(3,8);; gap> f:=InnerAutomorphismOfSemigroupNC(S, (1,2)(3,4)); ^(1,2)(3,4) gap> ConjugatorOfInnerAutomorphismOfSemigroup(f); (1,2)(3,4) |
> IsInnerAutomorphismOfSemigroup ( f ) | ( property ) |
returns true
if the general mapping f
is an inner automorphism of a semigroup; see InnerAutomorphismOfSemigroup
(7.3-1) for further details.
gap> S:=RandomSemigroup(2,9);; gap> f:=InnerAutomorphismOfSemigroupNC(S, (1,2)(3,4)); ^(1,2)(3,4) gap> IsInnerAutomorphismOfSemigroup(f); true |
> InnerAutomorphismsOfSemigroup ( S ) | ( attribute ) |
InnerAutomorphismsOfSemigroup
returns the group of inner automorphisms of the transformation semigroup S
.
The same result can be obtained by applying InnerAutomorphismsAutomorphismGroup
(7.3-6) to the result of AutomorphismGroup
(7.4-1) of S
. It is possible that the inner automorphism of S
have been calculated at the same time as the entire automorphism group of S
but it might not be. If the degree of S
is high, then this function may take a long time to return a value.
The notion of inner automorphisms of semigroups differs from the notion of the same name for groups. Indeed, if S
is a semigroup of transformations of degree n
, then g
in the symmetric group S_n
induces an inner automorphism of S
if the mapping that takes s
to g^-1sg
for all s
in S
is an automorphism of S
.
gap> x:=Transformation([2,3,4,5,6,7,8,9,1]);; gap> y:=Transformation([4,2,3,4,5,6,7,8,9]);; gap> S:=Semigroup(x,y);; gap> G:=InnerAutomorphismsOfSemigroup(S); <group of size 54 with 2 generators> |
> InnerAutomorphismsOfSemigroupInGroup ( S, G[, bval] ) | ( operation ) |
InnerAutomorphismsOfSemigroupInGroup
returns the group of inner automorphisms of the transformation semigroup S
that also belong to the group G
. The default setting is that the inner automorphisms of S
are calculated first, then filtered to see which elements also belong to G
.
If the optional argument bval
is present and true
, then the filtering is done as the inner automorphisms are found rather than after they have all been found. Otherwise, then this is equivalent to doing InnerAutomorphismsOfSemigroupInGroup(S, G)
.
If InfoAutos
(7.1-1) is set to level 4
, then a prompt will appear during the procedure to let you decide when the filtering should be done. In this case the value of bval
is irrelevant.
gap> gens:=[ Transformation( [1,8,11,2,5,16,13,14,3,6,15,10,7,4,9,12 ] ), > Transformation( [1,16,9,6,5,8,13,12,15,2,3,4,7,10,11,14] ), > Transformation( [1,3,7,9,1,15,5,11,13,11,13,3,5,15,7,9 ] ) ];; gap> S:=Semigroup(gens);; gap> InnerAutomorphismsOfSemigroup(S); <group of size 16 with 3 generators> gap> G:=Group(SemigroupHomomorphismByImagesOfGensNC(S, S, gens)); <group with 1 generators> gap> InnerAutomorphismsOfSemigroupInGroup(S, G); <group of size 1 with 1 generators> gap> InnerAutomorphismsOfSemigroupInGroup(S, G, true); <group of size 1 with 1 generators> gap> InnerAutomorphismsOfSemigroupInGroup(S, G, false); <group of size 1 with 1 generators> |
> InnerAutomorphismsAutomorphismGroup ( autgroup ) | ( attribute ) |
If autgroup
satisfies IsAutomorphismGroupOfSemigroup
(7.4-3) then, this attribute stores the subgroup of inner automorphisms of the original semigroup.
It is possible that the inner automorphisms of autgroup
have been calculated at the same time as autgroup
was calculated but they might not be. If the degree of underlying semigroup is high, then this function may take a long time to return a value.
The notion of inner automorphisms of semigroups differs from the notion of the same name for groups. Indeed, if S
is a semigroup of transformations of degree n
, then g
in the symmetric group S_n
induces an inner automorphism of S
if the mapping that takes s
to g^-1sg
for all s
in S
is an automorphism of S
.
If autgroup
satisfies IsAutomorphismGroupOfZeroGroup
(7.4-5), then InnerAutomorphismsAutomorphismGroup
returns the subgroup of inner automorphisms inside the automorphism group of the zero group by computing the inner automorphisms of the underlying group. Note that in this case the notion of inner automorphisms corresponds to that of the group theoretic notion.
gap> g1:=Transformation([3,3,2,6,2,4,4,6]);; gap> g2:=Transformation([5,1,7,8,7,5,8,1]);; gap> m6:=Semigroup(g1,g2);; gap> A:=AutomorphismGroup(m6); <group of size 12 with 2 generators> gap> InnerAutomorphismsAutomorphismGroup(A); <group of size 12 with 2 generators> gap> last=InnerAutomorphismsOfSemigroup(m6); |
> IsInnerAutomorphismsOfSemigroup ( G ) | ( property ) |
returns true
if G
is the inner automorphism group of a transformation semigroup.
The notion of inner automorphisms of semigroups differs from the notion of the same name for groups. Indeed, if S
is a semigroup of transformations of degree n
, then g
in the symmetric group S_n
induces an inner automorphism of S
if the mapping that takes s
to g^-1sg
for all s
in S
is an automorphism of S
.
Note that this property is set to true
when the computation of the inner automorphisms is performed. Otherwise, there is no method to check if an arbitrary group satisfies this property.
gap> S:=RandomSemigroup(5,5); <semigroup with 5 generators> gap> I:=InnerAutomorphismsOfSemigroup(S);; gap> IsInnerAutomorphismsOfSemigroup(I); true |
> IsInnerAutomorphismsOfZeroGroup ( G ) | ( property ) |
returns true
if G
is the inner automorphism group of a zero group. This property is set to true
when the computation of the inner automorphism group of the zero group is performed. Otherwise, there is no method to check if an arbitrary group satisfies this property.
Every inner automorphism of a zero group is just an inner automorphism of the underlying group that fixes the zero element. So, this notion of inner automorphism corresponds to the notion of inner automorphisms of a group.
gap> zg:=ZeroGroup(CyclicGroup(70)); <zero group with 4 generators> gap> I:=InnerAutomorphismsAutomorphismGroup(AutomorphismGroup(zg)); <group of size 1 with 1 generators> gap> IsInnerAutomorphismsOfZeroGroup(I); true |
> AutomorphismGroup ( S ) | ( attribute ) |
AutomorphismGroup
returns the group of automorphisms of the transformation semigroup, zero group, zero semigroup, Rees matrix semigroup, or Rees 0-matrix semigroup S
; that is, semigroups satisfying the properties IsTransformationSemigroup
(Reference: IsTransformationSemigroup), IsZeroGroup
(5.2-15), IsZeroSemigroup
(5.2-14), IsReesMatrixSemigroup
(Reference: IsReesMatrixSemigroup), or IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup).
If S
is a transformation semigroup, then AutomorphismGroup
computes the automorphism group of S
using the algorithm described in [ABM07].
If S
is a (completely) simple transformation semigroup, then the automorphism group is computed by passing to an isomorphic Rees matrix semigroup. If S
is a transformation group, then the automorphism group is computed by passing to an isomorphic permutation group. If S
has order <10
and knows its Cayley table (MultiplicationTable
(Reference: MultiplicationTable)), then the automorphism group is calculated by finding the setwise stabilizer of the Cayley table in the symmetric group of degree |S|
under the action on the Cayley table.
If S
is a zero group, then AutomorphismGroup
computes the automorphism group of the underlying group. Obviously, every automorphism of a zero group is the extension of an automorphism of the underlying group that fixes the zero element.
If S
is a zero semigroup, then every permutation of the elements of S
that fixes the zero element is an automorphism. Thus the automorphism group of a zero semigroup of order n
is isomorphic to the symmetric group on n-1
elements.
If S
is a Rees matrix semigroup or a Rees 0-matrix semigroup, then the automorphism group of S
is calculated using the algorithm described in [ABM07, Section 2]. In this case, the returned group has as many generators as elements. This may be changed in the future.
If InfoAutos
(7.1-1) is set to level 4
, then prompts will appear during the procedure to allow you interactive control over the computation.
Please note: if grape is not loaded, then this function will not work when S
satisfies IsTransformationSemigroup
(Reference: IsTransformationSemigroup) or IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup).
gap> g1:=Transformation([5,4,4,2,1]);; gap> g2:=Transformation([2,5,5,4,1]);; gap> m2:=Monoid(g1,g2);; gap> IsTransformationSemigroup(m2); true gap> AutomorphismGroup(m2); <group of size 24 with 5 generators> gap> IsAutomorphismGroupOfSemigroup(last); true gap> zg:=ZeroGroup(CyclicGroup(70)); <zero group with 4 generators> gap> IsZeroGroup(zg); true gap> AutomorphismGroup(zg); <group with 3 generators> gap> IsAutomorphismGroupOfZeroGroup(last); true gap> InnerAutomorphismsOfSemigroup(zg); <group of size 1 with 1 generators> gap> InnerAutomorphismsAutomorphismGroup(AutomorphismGroup(zg)); <group of size 1 with 1 generators> gap> last2=InnerAutomorphismsAutomorphismGroup(AutomorphismGroup(zg)); true gap> S:=ZeroSemigroup(10); <zero semigroup with 10 elements> gap> Size(S); 10 gap> Elements(S); [ 0, z1, z2, z3, z4, z5, z6, z7, z8, z9 ] gap> A:=AutomorphismGroup(S); <group with 2 generators> gap> IsAutomorphismGroupOfZeroSemigroup(A); true gap> Factorial(9)=Size(A); true gap> G:=Group([ (2,5)(3,4) ]);; gap> mat:=[ [ (), (), (), (), () ], > [ (), (), (2,5)(3,4), (2,5)(3,4), () ], > [ (), (), (), (2,5)(3,4), (2,5)(3,4) ], > [ (), (2,5)(3,4), (), (2,5)(3,4), () ], > [ (), (2,5)(3,4), (), (2,5)(3,4), () ] ];; gap> rms:=ReesMatrixSemigroup(G, mat); Rees Matrix Semigroup over Group([ (2,5)(3,4) ]) gap> A:=AutomorphismGroup(rms); <group of size 12 with 12 generators> gap> IsAutomorphismGroupOfRMS(A); true gap> G:=ZeroGroup(Group([ (1,3)(2,5), (1,3,2,5) ]));; gap> elts:=Elements(G);; gap> mat:=[ [ elts[7], elts[1], elts[9], elts[1], elts[1] ], > [ elts[1], elts[1], elts[1], elts[9], elts[1] ], > [ elts[9], elts[1], elts[1], elts[4], elts[9] ], > [ elts[1], elts[1], elts[1], elts[1], elts[1] ], > [ elts[1], elts[5], elts[1], elts[1], elts[1] ] ];; gap> rzms:=ReesZeroMatrixSemigroup(G, mat);; gap> AutomorphismGroup(rzms); gap> IsAutomorphismGroupOfRZMS(A); true <group of size 512 with 512 generators> |
> AutomorphismsSemigroupInGroup ( S, G[, bvals] ) | ( operation ) |
AutomorphismsSemigroupInGroup
returns the group of automorphisms of the transformation semigroup S
that also belong to the group G
. If the value of G
is fail
, then AutomorphismsSemigroupInGroup
returns the same value as AutomorphismGroup
(7.4-1). The default setting is that the automorphisms of S
are calculated first, then filtered to see which elements also belong to G
.
The optional argument bvals
is a list of 5
Boolean variables that correspond to the following options:
if bvals[1]
is true
, then GAP will run a cheap check to see if all the automorphisms are inner. Note that this can return false
when all the automorphisms are inner, that is the condition is sufficient but not necessary. The default setting is false
.
if bvals[2]
is true
, then GAP will try to compute the inner automorphisms of S
before computing the entire automorphism group. For semigroups of large degree this may not be sensible. The default setting is false
.
if bvals[3]
is true
, then GAP will test elements in the inner automorphism search space to see if they are in G
as the inner automorphisms are found rather than after they have all been found. The default setting is false
.
if bvals[4]
is true
, then GAP will test elements in the outer (i.e. not inner) automorphism search space to see if they are in G
as they are found rather than after they have all been found. The default setting is false
.
if bvals[5]
is true
, then GAP will keep track of non-automorphisms in the search for outer automorphisms. The default setting is false
.
Please note: if grape is not loaded, then this function will not work when S
satisfies IsTransformationSemigroup
(Reference: IsTransformationSemigroup) or IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup).
gap> g1:=Transformation([5,4,4,2,1]);; gap> g2:=Transformation([2,5,5,4,1]);; gap> m2:=Monoid(g1,g2);; gap> A:=AutomorphismsSemigroupInGroup(m2, fail, > [false, true, true, false, true]); <group of size 24 with 3 generators> gap> g1:=Transformation([3,3,2,6,2,4,4,6,3,4,6]);; gap> g2:=Transformation([4,4,6,1,3,3,3,3,11,11,11]);; gap> m7:=Monoid(g1,g2);; gap> A:=AutomorphismsSemigroupInGroup(m7, fail, > [false, true, false, false, true]); <group of size 2 with 2 generators> gap> imgs:=[ [ Transformation( [ 1, 1, 5, 4, 3, 6, 7, 8, 9, 10, 11, 12 ] ), > Transformation( [ 1, 1, 5, 7, 4, 3, 6, 8, 9, 10, 11, 12 ] ), > Transformation( [ 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 8 ] ) ], > [ Transformation( [ 1, 1, 5, 4, 3, 6, 7, 8, 9, 10, 11, 12 ] ), > Transformation( [ 1, 1, 5, 3, 7, 4, 6, 8, 9, 10, 11, 12 ] ), > Transformation( [ 1, 2, 3, 4, 5, 6, 7, 11, 12, 8, 9, 10 ] ) ] ];; gap> gens:=List(imgs, x-> SemigroupHomomorphismByImagesOfGensNC(S, S, x));; gap> G:=Group(gens); <group with 2 generators> gap> A:=AutomorphismsSemigroupInGroup(S, G, > [false, false, false, true, false]); <group of size 48 with 4 generators> gap> Size(G); 48 gap> A:=AutomorphismsSemigroupInGroup(S, G); <group of size 48 with 4 generators> gap> gens:=[ Transformation( [ 1, 1, 4, 3, 5, 6, 7, 8, 9, 10, 11, 12 ] ), > Transformation( [ 1, 1, 4, 5, 6, 7, 3, 8, 9, 10, 11, 12 ] ), > Transformation( [ 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 8 ] ) ];; gap> S:=Semigroup(gens);; gap> A:=AutomorphismsSemigroupInGroup(S, G); <group of size 48 with 4 generators> gap> HasAutomorphismGroup(S); true gap> AutomorphismGroup(S); <group of size 480 with 7 generators> |
> IsAutomorphismGroupOfSemigroup ( G ) | ( property ) |
returns true
if G
is the automorphism group of a semigroup. Note that this property is set to true
when the computation of the automorphism group is performed. Otherwise, there is no method to check if an arbitrary group satisfies this property; see AutomorphismGroup
(7.4-1) for an example of the usage of this command.
> IsAutomorphismGroupOfSimpleSemigp ( G ) | ( property ) |
returns true
if G
is the automorphism group of a simple transformation semigroup. This property is set to true
when the computation of the automorphism group of the simple transformation semigroup is performed. Otherwise, there is no method to check if an arbitrary group satisfies this property; see AutomorphismGroup
(7.4-1) for an example of the usage of this command.
> IsAutomorphismGroupOfZeroGroup ( G ) | ( property ) |
returns true
if G
is the automorphism group of a zero group. This property is set to true
when the computation of the automorphism group of the zero group is performed. Otherwise, there is no method to check if an arbitrary group satisfies this property; see AutomorphismGroup
(7.4-1) for an example of the usage of this command.
Every automorphism of a zero group is just an automorphism of the underlying group that fixes the zero element.
> IsAutomorphismGroupOfZeroSemigroup ( G ) | ( property ) |
returns true
if G
is the automorphism group of a zero semigroup. This property is set to true
when the computation of the automorphism group of the zero semigroup is performed. Otherwise, there is no method to check if an arbitrary group satisfies this property; see AutomorphismGroup
(7.4-1) for an example of the usage of this command.
Every permutation of a zero semigroup that fixes the zero element is an automorphism. Thus the automorphism group of a zero semigroup of order n
is isomorphic to the symmetric group on n-1
elements.
> IsAutomorphismGroupOfRMS ( G ) | ( property ) |
returns true
if G
is the automorphism group of a Rees matrix semigroup; that is, a semigroup created using the command ReesMatrixSemigroup
(Reference: ReesMatrixSemigroup) and/or satisfying IsReesMatrixSemigroup
(Reference: IsReesMatrixSemigroup).
Note that this property is set to true
when the computation of the automorphism group is performed. Otherwise, there is no method to check if an arbitrary group satisfies this property; see AutomorphismGroup
(7.4-1) for an example of the usage of this command.
> IsAutomorphismGroupOfRZMS ( G ) | ( property ) |
returns true
if G
is the automorphism group of a Rees matrix semigroup; that is, a semigroup created using the command ReesZeroMatrixSemigroup
(Reference: ReesZeroMatrixSemigroup) and/or satisfying IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup).
Note that this property is set to true
when the computation of the automorphism group is performed. Otherwise, there is no method to check if an arbitrary group satisfies this property; see AutomorphismGroup
(7.4-1) for an example of the usage of this command.
> RMSIsoByTriple ( rms1, rms2, triple ) | ( function ) |
this is a function to create an isomorphism between the Rees matrix semigroups rms1
and rms2
defined by triple
. The first component of triple
should be an isomorphism from the underlying group of rms1
to the underlying group of rms2
, the second component should be an isomorphism from the graph associated to the matrix of rms1
to the graph associated with the matrix of rms2
, and the third component should be a function (given as a list of image elements) from the index sets of rms1
to the underlying group of rms2
; see [ABM07, Section 2] for further details.
Note that this function only creates an object with representation IsRMSIsoByTripleRep
(7.5-3) and does not check that triple
actually defines an isomorphism from rms1
to rms2
or that the arguments even make sense. To create an isomorphism from rms1
to rms2
use IsomorphismSemigroups
(7.7-5).
gap> G:=Group((1,4,3,5,2));; gap> mat:=[ [ (), (), () ], [ (), (1,4,3,5,2), () ], [ (), (1,3,2,4,5), () ] ];; gap> rms:=ReesMatrixSemigroup(G, mat);; gap> l:=(4,6);; gap> g:=GroupHomomorphismByImages(G, G, [(1,4,3,5,2)], [(1,2,5,3,4)]); [ (1,4,3,5,2) ] -> [ (1,2,5,3,4) ] gap> map:=[(), (1,5,4,2,3), (), (), (), () ];; gap> RMSIsoByTriple(rms, rms, [l, g, map]); [ (4,6), GroupHomomorphismByImages( Group( [ (1,4,3,5,2) ] ), Group( [ (1,4,3,5,2) ] ), [ (1,4,3,5,2) ], [ (1,2,5,3,4) ] ), [ (), (1,5,4,2,3), (), (), (), () ] ] gap> IsRMSIsoByTripleRep(last); true gap> #the previous actually defines an automorphism of rms gap> #on the other hand, the next example is nonsense but no error gap> #is given gap> RMSIsoByTriple(rms, rms, [l, g, [()]]); [ (4,6), GroupHomomorphismByImages( Group( [ (1,4,3,5,2) ] ), Group( [ (1,4,3,5,2) ] ), [ (1,4,3,5,2) ], [ (1,2,5,3,4) ] ), [ () ] ] |
> RZMSIsoByTriple ( rzms1, rzms2, triple ) | ( function ) |
this is a function to create an isomorphism between the Rees 0-matrix semigroups rzms1
and rzms2
defined by triple
. The first component of triple
should be an isomorphism from the underlying zero group of rzms1
to the underlying zero group of rzms2
, the second component should be an isomorphism from the graph associated to the matrix of rzms1
to the graph associated with the matrix of rzms2
, and the third component should be a function (given as a list of image elements) from the index sets of rzms1
to the underlying zero group of rzms2
; see [ABM07, Section 2] for further details.
Note that this function only creates an object with representation IsRZMSIsoByTripleRep
(7.5-4) and does not check that triple
actually defines an isomorphism from rzms1
to rzms2
or that the arguments even make sense. To create an isomorphism from rzms1
to rzms2
use IsomorphismSemigroups
(7.7-5).
gap> G:=Group((1,4,3,5,2));; gap> ZG:=ZeroGroup(G); <zero group with 2 generators> gap> mat:=[ [ (), (), () ], [ (), (1,4,3,5,2), () ], [ (), (1,3,2,4,5), () ] ];; gap> mat:=List(mat, x-> List(x, ZeroGroupElt)); [ [ (), (), () ], [ (), (1,4,3,5,2), () ], [ (), (1,3,2,4,5), () ] ] gap> rms:=ReesZeroMatrixSemigroup(ZG, mat); Rees Zero Matrix Semigroup over <zero group with 2 generators> gap> l:=(4,6);; gap> g:=GroupHomomorphismByImages(G, G, [(1,4,3,5,2)], [(1,2,5,3,4)]); [ (1,4,3,5,2) ] -> [ (1,2,5,3,4) ] gap> g:=ZeroGroupAutomorphism(ZG, g); <mapping: <zero group with 2 generators> -> <zero group with 2 generators> > gap> map:=List([(), (1,5,4,2,3), (), (), (), () ], ZeroGroupElt);; gap> RZMSIsoByTriple(rms, rms, [l, g, map]); [ (4,6), <mapping: <zero group with 2 generators> -> <zero group with 2 generators> >, [ ZeroGroup(()), ZeroGroup((1,5,4,2,3)), ZeroGroup(()), ZeroGroup(()), ZeroGroup(()), ZeroGroup(()) ] ] gap> RZMSIsoByTriple(rms, rms, [l, g, [()]]); [ (4,6), <mapping: <zero group with 2 generators> -> <zero group with 2 generators> >, [ () ] ] gap> IsRZMSIsoByTripleRep(last); true |
> IsRMSIsoByTripleRep ( f ) | ( representation ) |
returns true
if the object f
is represented as an isomorphism of Rees matrix semigroups by a triple; as explained in [ABM07, Section 2]; see RMSIsoByTriple
(7.5-1) for an example of the usage of this command.
> IsRZMSIsoByTripleRep ( f ) | ( representation ) |
returns true
if the object f
is represented as an isomorphism of Rees matrix semigroups by a triple; as explained in [ABM07, Section 2]; see RZMSIsoByTriple
(7.5-2) for an example of the usage of this command.
> RMSInducedFunction ( RMS, lambda, gamma, g ) | ( operation ) |
lambda
is an automorphism of the graph associated to the Rees matrix semigroup RMS
, gamma
an automorphism of the underlying group of RMS
, and g
an element of the underlying group of RMS
. The function RMSInducedFunction
attempts to find the function determined by lambda
and gamma
from the union of the index sets I
and J
to the group G
of the Rees matrix semigroup RMS
over G
, I
, and J
with respect to P
where the first element is given by the element g
. If a conflict is found, then false
is returned together with the induced map; see [ABM07, Section 2] for further details.
gap> G:=Group([ (1,2) ]);; gap> mat:=[ [ (), (), () ], [ (), (1,2), () ], [ (), (1,2), (1,2) ], > [ (), (), () ], [ (), (1,2), () ] ];; gap> rms:=ReesMatrixSemigroup(G, mat);; gap> l:=(1,2)(4,5,6); (1,2)(4,5,6) gap> gam:=One(AutomorphismGroup(G)); IdentityMapping( Group([ (1,2) ]) ) gap> g:=(1,2); gap> RMSInducedFunction(rms, l, gam, g); [ false, [ (1,2), (), (), (), (), (1,2), (1,2), () ] ] gap> RMSInducedFunction(rms, (4,7), gam, ()); [ true, [ (), (), (), (), (), (), (), () ] ] |
> RZMSInducedFunction ( RZMS, lambda, gamma, g, comp ) | ( operation ) |
lambda
is an automorphism of the graph associated to the Rees 0- matrix semigroup RZMS
, gamma
an automorphism of the underlying zero group of RZMS
, comp
is a connected component of the graph associated to RZMS
, and g
is an element of the underlying zero group of RZMS
. The function RZMSInducedFunction
attempts to find the partial function determined by lambda
and gamma
from comp
to the zero group G^0
of G
of the Rees 0-matrix semigroup RZMS
over G^0
, I
, and J
with respect to P
where the image of the first element in comp
is given by the element g
. If a conflict is found, then fail
is returned; see [ABM07, Section 2] for further details.
Please note: if grape is not loaded, then this function will not work.
gap> zg:=ZeroGroup(Group(()));; gap> z:=Elements(zg)[1]; 0 gap> x:=Elements(zg)[2]; () gap> mat:=[ [ z, z, z ], [ x, z, z ], [ x, x, z ] ];; gap> rzms:=ReesZeroMatrixSemigroup(zg, mat);; gap> RZMSInducedFunction(rzms, (), One(AutomorphismGroup(zg)), x, > [1,2,5,6]) [ (), (),,, (), () ] gap> RZMSInducedFunction(rzms, (), One(AutomorphismGroup(zg)), x, [3]); [ ,, () ] gap> RZMSInducedFunction(rzms, (), One(AutomorphismGroup(zg)), x, [4]); [ ,,, () ] gap> zg:=ZeroGroup(Group([ (1,5,2,3), (1,4)(2,3) ]));; gap> elts:=Elements(zg);; gap> mat:=[ [ elts[1], elts[1], elts[11], elts[1], elts[1] ], > [ elts[1], elts[13], elts[21], elts[1], elts[1] ], > [ elts[1], elts[16], elts[1], elts[16], elts[3] ], > [ elts[10], elts[17], elts[1], elts[1], elts[1] ], > [ elts[1], elts[1], elts[1], elts[4], elts[1] ] ]; gap> rzms:=ReesZeroMatrixSemigroup(zg, mat); gap> RZMSInducedFunction(rzms, (), Random(AutomorphismGroup(zg)), > Random(elts), [1..10])=fail; false |
> RZMStoRZMSInducedFunction ( RZMS1, RZMS2, lambda, gamma, elts ) | ( operation ) |
lambda
is an automorphism of the graph associated to the Rees 0- matrix semigroup RZMS1
composed with isomorphism from that graph to the graph of RZMS2
, gamma
an automorphism of the underlying zero group of RZMS1
, and elts
is a list of elements of the underlying zero group of RZMS2
. The function RZMStoRZMSInducedFunction
attempts to find the function determined by lambda
and gamma
from the union of the index sets I
and J
of RZMS1
to the zero group G^0
of the Rees 0-matrix semigroup RZMS2
over the zero group G^0
, sets I
and J
, and matrix P
where the image of the first element in the i
th connected component of the associated graph of RZMS1
is given by elts[i]
. If a conflict is found, then false
is returned; see [ABM07, Section 2] for further details.
Please note: if grape is not loaded, then this function will not work.
gap> gens:=[ Transformation( [ 4, 4, 8, 8, 8, 8, 4, 8 ] ), Transformation( [ 8, 2, 8, 2, 5, 5, 8, 8 ] ), Transformation( [ 8, 8, 3, 7, 8, 3, 7, 8 ] ), Transformation( [ 8, 6, 6, 8, 6, 8, 8, 8 ] ) ];; gap> S:=Semigroup(gens);; gap> D:=GreensDClasses(S);; gap> rms1:=Range(IsomorphismReesMatrixSemigroupOfDClass(D[1])); Rees Zero Matrix Semigroup over <zero group with 2 generators> gap> rms2:=Range(IsomorphismReesMatrixSemigroupOfDClass(D[4])); Rees Zero Matrix Semigroup over <zero group with 2 generators> gap> gam:=One(AutomorphismGroup > (UnderlyingSemigroupOfReesZeroMatrixSemigroup(Group(rms1)))); IdentityMapping( <zero group with 2 generators> ) gap> g:=One(UnderlyingSemigroupOfReesZeroMatrixSemigroup(rms2)); () gap> RZMStoRZMSInducedFunction(rms1, rms2, (2,3)(5,6), gam, [g]); [ (), (), (), (), (), () ] |
> RZMSGraph ( rzms ) | ( attribute ) |
if rzms
is a Rees 0-matrix semigroup over a zero group G^0
, 3 index sets I
and J
, and matrix P
, then RZMSGraph
returns the undirected bipartite graph with |I|+|J|
vertices and edge (i,j)
if and only if i<|I|+1
, j>|I|
and p_{j-|I|, i}
is not zero.
The returned object is a simple undirected graph created in GRAPE using the command
Graph(Group(()), [1..n+m], OnPoints, adj, true);
where adj
is true
if and only if i<|I|+1
, j>|I|
and p_{j-|I|, i}
is not zero.
Please note: if grape is not loaded, then this function will not work.
gap> zg:=ZeroGroup(Group(()));; gap> z:=Elements(zg)[1]; 0 gap> x:=Elements(zg)[2]; () gap> mat:=[ [ 0, 0, 0 ], [ (), 0, 0 ], [ (), (), 0 ] ];; gap> rzms:=ReesZeroMatrixSemigroup(zg, mat);; gap> RZMSGraph(rzms); rec( isGraph := true, order := 6, group := Group(()), schreierVector := [ -1, -2, -3, -4, -5, -6 ], adjacencies := [ [ 5, 6 ], [ 6 ], [ ], [ ], [ 1 ], [ 1, 2 ] ], representatives := [ 1, 2, 3, 4, 5, 6 ], names := [ 1, 2, 3, 4, 5, 6 ] ) gap> UndirectedEdges(last); [ [ 1, 5 ], [ 1, 6 ], [ 2, 6 ] ] |
> RightTransStabAutoGroup ( S, elts, func ) | ( operation ) |
returns a right transversal of the stabilizer w.r.t the action func
of the elements elts
in the automorphism group of the zero semigroup, Rees matrix semigroup, or Rees 0-matrix semigroup S
. That is, S
satisfying IsZeroSemigroup
(5.2-14), IsReesMatrixSemigroup
(Reference: IsReesMatrixSemigroup), or IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup).
gap> S:=ZeroSemigroup(6); <zero semigroup with 6 elements> gap> elts:=Elements(S); [ 0, z1, z2, z3, z4, z5 ] gap> Length(RightTransStabAutoGroup(S, [elts[1]], OnSets)); 1 gap> Length(RightTransStabAutoGroup(S, [elts[1], elts[2]], OnSets)); 5 gap> Length(RightTransStabAutoGroup(S, [elts[1], elts[2]], OnTuples)); 5 gap> G:=Group([ (1,2) ]);; gap> mat:=[ [ (), (), () ], [ (), (1,2), () ], [ (), (1,2), (1,2) ], > [ (), (), () ], [ (), (1,2), () ] ];; gap> rms:=ReesMatrixSemigroup(G, mat);; gap> Size(rms); 30 gap> GeneratorsOfSemigroup(rms); [ (1,(),2), (1,(),3), (1,(),4), (1,(),5), (2,(),1), (3,(),1), (1,(1,2),1) ] gap> Length(RightTransStabAutoGroup(rms, last, OnSets)); 4 gap> Length(RightTransStabAutoGroup(rms, GeneratorsOfSemigroup(rms), > OnTuples)); 8 gap> G:=ZeroGroup(Group([ (1,3) ]));; gap> z:=MultiplicativeZero(G);; x:=Elements(G)[2];; gap> mat:=[ [ z, z, z ], [ z, z, z ], [ z, z, z ], [ z, z, z ], [ z, x, z ] ];; gap> rzms:=ReesZeroMatrixSemigroup(G, mat); gap> Size(rzms); 31 gap> Size(GeneratorsOfSemigroup(rzms)); 6 gap> Length(RightTransStabAutoGroup(rzms, GeneratorsOfSemigroup(rzms), > OnSets)); 512 gap> A:=AutomorphismGroup(rzms); <group of size 3072 with 3072 generators> |
> ZeroGroupAutomorphism ( ZG, f ) | ( function ) |
converts the group automorphism f
of the underlying group of the zero group ZG
into an automorphism of the zero group ZG
.
gap> G:=Random(AllGroups(20)); <pc group of size 20 with 3 generators> gap> A:=AutomorphismGroup(G); <group with 2 generators> gap> f:=Random(A); [ f1*f2^4*f3 ] -> [ f1*f2^2 ] gap> ZG:=ZeroGroup(G); <zero group with 4 generators> gap> ZeroGroupAutomorphism(ZG, f); <mapping: <zero group with 4 generators> -> <zero group with 4 generators> > gap> IsZeroGroupAutomorphismRep(last); true gap> UnderlyingGroupAutoOfZeroGroupAuto(last2)=f; true |
> IsZeroGroupAutomorphismRep ( f ) | ( representation ) |
returns true
if the object f
is represented as an automorphism of a zero group; see ZeroGroupAutomorphism
(7.6-1) for an example of the usage of this command.
> UnderlyingGroupAutoOfZeroGroupAuto ( f ) | ( attribute ) |
returns the underlying group automorphism of the zero group automorphism f
. That is, the restriction of f
to its source without the zero; see ZeroGroupAutomorphism
(7.6-1) for an example of the usage of this command.
> IsomorphismAutomorphismGroupOfRMS ( G ) | ( attribute ) |
if G
is the automorphism group of a simple transformation semigroup, then IsomorphismAutomorphismGroupOfRMS
returns a GroupHomomorphismByImages
(Reference: GroupHomomorphismByImages) from the automorphism group of G
to the automorphism group of an isomorphic Rees matrix semigroup, obtained by using IsomorphismReesMatrixSemigroup
(7.7-7).
gap> g1:=Transformation([1,2,2,1,2]);; gap> g2:=Transformation([3,4,3,4,4]);; gap> g3:=Transformation([3,4,3,4,3]);; gap> g4:=Transformation([4,3,3,4,4]);; gap> cs5:=Semigroup(g1,g2,g3,g4);; gap> AutomorphismGroup(cs5); <group of size 16 with 3 generators> gap> IsomorphismAutomorphismGroupOfRMS(last); [ SemigroupHomomorphism ( <semigroup with 4 generators>-><semigroup with 4 generators>), SemigroupHomomorphism ( <semigroup with 4 generators>-><semigroup with 4 generators>), SemigroupHomomorphism ( <semigroup with 4 generators>-><semigroup with 4 generators>) ] -> [ [ (1,4)(2,3)(5,6), IdentityMapping( Group( [ (1,2) ] ) ), [ (), (1,2), (1,2), (), (), () ] ], [ (1,3,4,2), IdentityMapping( Group( [ (1,2) ] ) ), [ (), (), (), (), (), (1,2) ] ], [ (1,3)(2,4), IdentityMapping( Group( [ (1,2) ] ) ), [ (), (), (), (), (), (1,2) ] ] ] |
> IsomorphismPermGroup ( G ) | ( attribute ) |
if G
satisfies IsAutomorphismGroupOfSimpleSemigp
(7.4-4), then IsomorphismPermGroup
returns an isomorphism from G
to a permutation group by composing the result f
of IsomorphismAutomorphismGroupOfRMS
(7.7-1) on G
with the result of IsomorphismPermGroup
on Range(f)
.
if G
satisfies IsAutomorphismGroupOfRMS
(7.4-7) or IsAutomorphismGroupOfRZMS
(7.4-8), then IsomorphismPermGroup
returns an isomorphism from G
to a permutation group acting either on the elements of S
or on itself, whichever gives a permutation group of lower degree.
if G
is a transformation semigroup that satisfies IsGroupAsSemigroup
(5.2-3), then IsomorphismPermGroup
returns an isomorphism from G
to the permutation group obtained by applying AsPermOfRange
(2.3-2) to any element of G
.
if G
is a group H
-class of a transformation semigroup, then IsomorphismPermGroup
returns an isomorphism from G
to the permutation group obtained by applying AsPermOfRange
(2.3-2) to any element of G
.
gap> g1:=Transformation([3,3,2,6,2,4,4,6]);; gap> g2:=Transformation([5,1,7,8,7,5,8,1]);; gap> cs1:=Semigroup(g1,g2);; gap> AutomorphismGroup(cs1); <group of size 12 with 2 generators> gap> IsomorphismPermGroup(last); [ SemigroupHomomorphism ( <semigroup with 2 generators>-><semigroup with 2 generators>), SemigroupHomomorphism ( <semigroup with 2 generators>-><semigroup with 2 generators>) ] -> [ (1,11,2,12,3,10)(4,8,5,9,6,7), (1,6)(2,5)(3,4)(7,10)(8,12)(9,11) ] gap> Size(cs1); 96 gap> a:=IdempotentNC([[1,3,4],[2,5],[6],[7],[8]],[3,5,6,7,8])*(3,5);; gap> b:=IdempotentNC([[1,3,4],[2,5],[6],[7],[8]],[3,5,6,7,8])*(3,6,7,8);; gap> S:=Semigroup(a,b);; gap> IsGroupAsTransSemigroup(S); true gap> IsomorphismPermGroup(S); SemigroupHomomorphism ( <semigroup with 2 generators>->Group( [ (3,5), (3,6,7,8) ])) gap> gens:=[Transformation([3,5,3,3,5,6]), Transformation([6,2,4,2,2,6])];; gap> S:=Semigroup(gens);; gap> H:=GroupHClassOfGreensDClass(GreensDClassOfElement(S, Elements(S)[1])); {Transformation( [ 2, 2, 2, 2, 2, 6 ] )} gap> IsomorphismPermGroup(H); SemigroupHomomorphism ( {Transformation( [ 2, 2, 2, 2, 2, 6 ] )}->Group(())) |
> IsomorphismFpSemigroup ( S ) | ( attribute ) |
returns an isomorphism to a finitely presented semigroup from the transformation semigroup S
. This currently works by running the function FroidurePinExtendedAlg
(???) in the library.
If S
satisfies IsMonoid
(Reference: IsMonoid), use the command IsomorphismFpMonoid
(7.7-4) instead.
gap> gens:=[ Transformation( [1,8,11,2,5,16,13,14,3,6,15,10,7,4,9,12 ] ), > Transformation( [1,16,9,6,5,8,13,12,15,2,3,4,7,10,11,14] ), > Transformation( [1,3,7,9,1,15,5,11,13,11,13,3,5,15,7,9 ] ) ]; gap> S:=Semigroup(gens); <semigroup with 3 generators> gap> IsomorphismFpSemigroup(last); SemigroupHomomorphismByImages ( <trans. semigroup of size 16 with 3 generators>->Semigroup( [ s1, s2, s3 ] )) |
> IsomorphismFpMonoid ( S ) | ( attribute ) |
returns an isomorphism to a finitely presented monoid from the transformation monoid S
. Currently works by running the function FroidurePinExtendedAlg
(???) in the library.
If S
satisfies IsSemigroup
(Reference: IsSemigroup), use the command IsomorphismFpSemigroup
(7.7-3) instead.
gap> x:=Transformation([2,3,4,5,6,7,8,9,1]);; gap> y:=Transformation([4,2,3,4,5,6,7,8,9]);; gap> S:=Monoid(x,y);; gap> IsomorphismFpMonoid(last); SemigroupHomomorphismByImages ( <trans. semigroup of size 40266 with 3 generators>->Monoid( [ m1, m2 ], ... )) gap> Length(RelationsOfFpMonoid(Range(last))); 932 |
> IsomorphismSemigroups ( S, T ) | ( operation ) |
this operation returns an isomorphism from the semigroup S
to the semigroup T
if one exists and returns fail
otherwise.
Please note: this function currently only works for zero groups, zero semigroups, Rees matrix semigroups, and Rees 0-matrix semigroups.
Please note: if grape is not loaded, then this function will not work when S
and T
satisfy IsReesZeroMatrixSemigroup
(Reference: IsReesZeroMatrixSemigroup).
gap> ZG1:=ZeroGroup(Group((1,2,3,5,4))); <zero group with 2 generators> gap> ZG2:=ZeroGroup(Group((1,2,3,4,5))); <zero group with 2 generators> gap> IsomorphismSemigroups(ZG1, ZG2); SemigroupHomomorphismByImagesOfGens ( <zero group with 2 generators>-><zero group with 2 generators>) gap> ZG2:=ZeroGroup(Group((1,2,3,4))); <zero group with 2 generators> gap> IsomorphismSemigroups(ZG1, ZG2); fail gap> IsomorphismSemigroups(ZeroSemigroup(5),ZeroSemigroup(5)); IdentityMapping( <zero semigroup with 5 elements> ) gap> IsomorphismSemigroups(ZeroSemigroup(5),ZeroSemigroup(6)); fail gap> gens:=[ Transformation( [ 4, 4, 8, 8, 8, 8, 4, 8 ] ), > Transformation( [ 8, 2, 8, 2, 5, 5, 8, 8 ] ), > Transformation( [ 8, 8, 3, 7, 8, 3, 7, 8 ] ), > Transformation( [ 8, 6, 6, 8, 6, 8, 8, 8 ] ) ];; gap> S:=Semigroup(gens);; gap> D:=GreensDClasses(S);; gap> rms1:=Range(IsomorphismReesMatrixSemigroupOfDClass(D[1]));; gap> rms2:=Range(IsomorphismReesMatrixSemigroupOfDClass(D[4]));; gap> IsomorphismSemigroups(rms1, rms2); [ (2,3)(5,6), IdentityMapping( <zero group with 2 generators> ), [ ZeroGroup(()), ZeroGroup(()), ZeroGroup(()), ZeroGroup(()), ZeroGroup(()), ZeroGroup(()) ] ] gap> IsomorphismSemigroups(rms2, rms1); [ (2,3)(5,6), IdentityMapping( <zero group with 2 generators> ), [ ZeroGroup(()), ZeroGroup(()), ZeroGroup(()), ZeroGroup(()), ZeroGroup(()), ZeroGroup(()) ] ] gap> rms2:=Range(IsomorphismReesMatrixSemigroupOfDClass(D[2])); Group(()) gap> IsomorphismSemigroups(rms2, rms1); fail gap> rms2:=RandomReesZeroMatrixSemigroup(5,5,5); Rees Zero Matrix Semigroup over <zero group with 2 generators> gap> IsomorphismSemigroups(rms2, rms1); fail gap> rms2:=RandomReesMatrixSemigroup(5,5,5); Rees Matrix Semigroup over Group([ (1,2)(3,4,5), (2,4,3), (1,4,5,3), (1,4,5,2) ]) gap> IsomorphismSemigroups(rms2, rms1); fail gap> IsomorphismSemigroups(rms1, rms2); fail |
> IsomorphismReesMatrixSemigroupOfDClass ( D ) | ( attribute ) |
The principal factor of the D
-class D
is the semigroup with elements D
and 0
and multiplication x*y
defined to be the product xy
in the semigroup containing D
if xy
in D
and 0
otherwise.
IsomorphismReesMatrixSemigroupOfDClass
returns an isomorphism from the principal factor of the D
-class D
to a Rees matrix, Rees 0-matrix or zero semigroup, as given by the Rees-Suschewitsch Theorem; see [How95, Theorem 3.2.3].
gap> g1:=Transformation( [ 4, 6, 3, 8, 5, 6, 10, 4, 3, 7 ] );; gap> g2:=Transformation( [ 5, 6, 6, 3, 8, 6, 3, 7, 8, 4 ] );; gap> g3:=Transformation( [ 8, 6, 3, 2, 8, 10, 9, 2, 6, 2 ] );; gap> m23:=Monoid(g1,g2,g3);; gap> D:=GreensDClasses(m23)[17]; {Transformation( [ 7, 6, 6, 6, 7, 4, 8, 6, 6, 6 ] )} gap> IsomorphismReesMatrixSemigroupOfDClass(D); SemigroupHomomorphism ( {Transformation( [ 7, 6, 6, 6, 7, 4, 8, 6, 6, 6 ] )}-><zero semigroup with 3 elements>) gap> D:=GreensDClasses(m23)[77]; {Transformation( [ 6, 6, 6, 6, 6, 6, 6, 6, 6, 6 ] )} gap> IsomorphismReesMatrixSemigroupOfDClass(D); SemigroupHomomorphism ( {Transformation( [ 6, 6, 6, 6, 6, 6, 6, 6, 6, 6 ] )}->Rees Matrix Semigroup over Group(())) gap> D:=GreensDClasses(m23)[1]; {Transformation( [ 1 .. 10 ] )} gap> IsomorphismReesMatrixSemigroupOfDClass(D); SemigroupHomomorphism ( {Transformation( [ 1 .. 10 ] )}->Group(())) gap> D:=GreensDClasses(m23)[23]; {Transformation( [ 6, 7, 3, 6, 6, 6, 6, 6, 7, 6 ] )} gap> IsomorphismReesMatrixSemigroupOfDClass(D); SemigroupHomomorphism ( {Transformation( [ 6, 7, 3, 6, 6, 6, 6, 6, 7, 6 ] )}->Rees Zero Matrix Semigroup over <zero group with 3 generators>) |
> IsomorphismReesMatrixSemigroup ( S ) | ( operation ) |
returns an isomorphism from the (completely) simple transformation semigroup S
to a Rees matrix semigroup, as given by the Rees-Suschewitsch Theorem; see [How95, Theorem 3.2.3].
gap> g1:=Transformation( [ 2, 3, 4, 5, 1, 8, 7, 6, 2, 7 ] );; gap> g2:=Transformation( [ 2, 3, 4, 5, 6, 8, 7, 1, 2, 2 ] );; gap> cs2:=Semigroup(g1,g2);; gap> IsomorphismReesMatrixSemigroup(cs2); SemigroupHomomorphism ( <semigroup with 2 generators>->Rees Matrix Semigroup over Group( [ (2,5)(3,8)(4,6), (1,6,3)(5,8) ])) |
generated by GAPDoc2HTML