Quelle semirel.tst Sprache: unbekannt

#@local H,L,R,S,a,b,f,gjp,gjp1,glp,glp1,grp,grp1,rels,s1,s2,s3,sc,t1,t2,t20
#@local t3,t4,t5,D
gap> START_TEST("semirel.tst");

##
##  Three non-commutative finite semigroups
##
gap> f := FreeSemigroup(2);;
gap> a := GeneratorsOfSemigroup(f)[1];; b := GeneratorsOfSemigroup(f)[2];;
gap> rels := [[a^2,a],[b^2,b],[(a*b)^10,a*b],[(b*a)^10,b*a]];;
gap> s1 := f/rels;; Size(s1);
38
gap> t1 := Range(IsomorphismTransformationSemigroup(s1));;
gap> Size(t1);
38
gap> f := FreeSemigroup(2);;
gap> a := GeneratorsOfSemigroup(f)[1];; b := GeneratorsOfSemigroup(f)[2];;
gap> rels := [[a^5,a],[b^5,b],[(a*b)^5,a*b],[(b*a)^5,b*a],[a*b^2,a*b],[a^2*b,a*b]];;
gap> s2 := f/rels;;
gap> t2 := Range(IsomorphismTransformationSemigroup(s2));;
gap> Size(t2);
108
gap> f := FreeSemigroup(2);;
gap> a := GeneratorsOfSemigroup(f)[1];; b := GeneratorsOfSemigroup(f)[2];;
gap> rels := [[a^4,a],[b^4,b],[(a*b)^2,a^2*b^2],[(b*a)^2,b^2*a^2]];;
gap> s3 := f/rels;;
gap> t3 := Range(IsomorphismTransformationSemigroup(s3));;
gap> Size(t3);
294

##
##  A commutative finite semigroup
##
gap> f := FreeSemigroup(2);;
gap> a := GeneratorsOfSemigroup(f)[1];; b := GeneratorsOfSemigroup(f)[2];;
gap> rels := [[a*b,b*a],[a^4,a],[b^4,b],[(a*b)^2,a^2*b^2],[(b*a)^2,b^2*a^2]];;
gap> sc := f/rels;;
gap> t4 := Range(IsomorphismTransformationSemigroup(sc));;
gap> Size(t4);
15

##
##  Full transformation semigroup of elements of 3, 4, and 5
##
gap> t3 := FullTransformationSemigroup(3);;
gap> t4 := FullTransformationSemigroup(4);;
gap> t5 := FullTransformationSemigroup(5);;

##
##  Size is known no computation required
##
gap> t20 := FullTransformationSemigroup(20);;
gap> Size(t20);
104857600000000000000000000

##
##  Green's relations
##
gap> grp := EquivalenceRelationPartition(GreensRRelation(s1));;
gap> grp1 := EquivalenceRelationPartition(GreensRRelation(t1));;
gap> Set(List(grp,Size)) = Set(List(grp1,Size));
true
gap> glp := EquivalenceRelationPartition(GreensLRelation(s1));;
gap> glp1 := EquivalenceRelationPartition(GreensLRelation(t1));;
gap> Set(List(glp,Size)) = Set(List(glp1,Size));
true
gap> gjp := EquivalenceRelationPartition(GreensJRelation(s1));;
gap> gjp1 := EquivalenceRelationPartition(GreensJRelation(t1));;
gap> Set(List(gjp,Size)) = Set(List(gjp1,Size));
true

##
##  See that Green's classes for full transformation semigroups
##  are of the proper form
##
gap> ForAll(GreensRClasses(t3),
> i->ForAll(AsSSortedList(i),j->KernelOfTransformation(j,3)
> = KernelOfTransformation(Representative(i),3)));
true
gap> ForAll(GreensLClasses(t3),
> i->ForAll(AsSSortedList(i),j->ImageSetOfTransformation(j,3)
> = ImageSetOfTransformation(Representative(i),3)));
true
gap> ForAll(GreensJClasses(t3),
> i->ForAll(AsSSortedList(i),j->RankOfTransformation(j,3)
> = RankOfTransformation(Representative(i),3)));
true
gap> ForAll(GreensHClasses(t3),
> i->ForAll(AsSSortedList(i),j->ImageSetOfTransformation(j,3)
> = ImageSetOfTransformation(Representative(i),3)
> and KernelOfTransformation(j,3) = KernelOfTransformation(Representative(i),3)
> ));
true
gap> ForAll(GreensRClasses(t4),
> i->ForAll(AsSSortedList(i),j->KernelOfTransformation(j,4)
> = KernelOfTransformation(Representative(i),4)));
true
gap> ForAll(GreensLClasses(t4),
> i->ForAll(AsSSortedList(i),j->ImageSetOfTransformation(j,4)
> = ImageSetOfTransformation(Representative(i),4)));
true
gap> ForAll(GreensJClasses(t4),
> i->ForAll(AsSSortedList(i),j->RankOfTransformation(j,4)
> = RankOfTransformation(Representative(i),4)));
true
gap> ForAll(GreensHClasses(t4),
> i->ForAll(AsSSortedList(i),j->ImageSetOfTransformation(j,4)
> = ImageSetOfTransformation(Representative(i),4)
> and KernelOfTransformation(j,4) = KernelOfTransformation(Representative(i),4)
> ));
true

# Issue 395 (recursion depth trap in DClassOfLClass)
gap> S := Semigroup(Transformation([2, 4, 3, 4]),
>                   Transformation([3, 3, 2, 3]));
<transformation semigroup of degree 4 with 2 generators>
gap> L := GreensLClassOfElement(S, S.1);
<Green's L-class: Transformation( [ 2, 4, 3, 4 ] )>
gap> DClassOfLClass(L);
<Green's D-class: Transformation( [ 2, 4, 3, 4 ] )>
gap> R := GreensRClassOfElement(S, S.1);
<Green's R-class: Transformation( [ 2, 4, 3, 4 ] )>
gap> DClassOfRClass(R);
<Green's D-class: Transformation( [ 2, 4, 3, 4 ] )>
gap> H := GreensHClassOfElement(S, S.1);
<Green's H-class: Transformation( [ 2, 4, 3, 4 ] )>
gap> DClassOfHClass(H);
<Green's D-class: Transformation( [ 2, 4, 3, 4 ] )>
gap> LClassOfHClass(H);
<Green's L-class: Transformation( [ 2, 4, 3, 4 ] )>
gap> RClassOfHClass(H);
<Green's R-class: Transformation( [ 2, 4, 3, 4 ] )>

# GreensXClasses for a GreensClass
gap> S := Semigroup([Transformation([1, 1, 1, 1]),
>                    Transformation([1, 1, 1, 2]),
>                    Transformation([1, 1, 1, 3])]);;
gap> D := GreensDClasses(S);
[ <Green's D-class: Transformation( [ 1, 1, 1, 1 ] )>,
  <Green's D-class: Transformation( [ 1, 1, 1, 2 ] )>,
  <Green's D-class: Transformation( [ 1, 1, 1, 3 ] )> ]
gap> L := GreensLClasses(S);
[ <Green's L-class: Transformation( [ 1, 1, 1, 1 ] )>,
  <Green's L-class: Transformation( [ 1, 1, 1, 2 ] )>,
  <Green's L-class: Transformation( [ 1, 1, 1, 3 ] )> ]
gap> R := GreensRClasses(S);
[ <Green's R-class: Transformation( [ 1, 1, 1, 1 ] )>,
  <Green's R-class: Transformation( [ 1, 1, 1, 2 ] )>,
  <Green's R-class: Transformation( [ 1, 1, 1, 3 ] )> ]
gap> H := GreensHClasses(S);
[ <Green's H-class: Transformation( [ 1, 1, 1, 1 ] )>,
  <Green's H-class: Transformation( [ 1, 1, 1, 2 ] )>,
  <Green's H-class: Transformation( [ 1, 1, 1, 3 ] )> ]
gap> Concatenation(List(D, GreensLClasses)) = L;
true
gap> Concatenation(List(D, GreensRClasses)) = R;
true
gap> Concatenation(List(D, GreensHClasses)) = H;
true
gap> Concatenation(List(L, GreensHClasses)) = H;
true
gap> Concatenation(List(R, GreensHClasses)) = H;
true

#
gap> STOP_TEST("semirel.tst");

Quelle semirel.tst Sprache: unbekannt

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