## File: semirel.tst

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gap 4r4p12-2
 `123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127` ``````############################################################################# ## #W semirel.tst GAP library Robert F. Morse ## #H \$Id: semirel.tst,v 1.4.2.4 2005/08/29 14:50:35 gap Exp \$ ## #Y Copyright (C) 1996, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany ## ## To be listed in testall.g ## gap> START_TEST("\$Id: semirel.tst,v 1.4.2.4 2005/08/29 14:50:35 gap Exp \$"); ## ## 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,i->Size(i))) = Set(List(grp1,i->Size(i))); true gap> glp := EquivalenceRelationPartition(GreensLRelation(s1));; gap> glp1 := EquivalenceRelationPartition(GreensLRelation(t1));; gap> Set(List(glp,i->Size(i))) = Set(List(glp1,i->Size(i))); true gap> gjp := EquivalenceRelationPartition(GreensJRelation(s1));; gap> gjp1 := EquivalenceRelationPartition(GreensJRelation(t1));; gap> Set(List(gjp,i->Size(i))) = Set(List(gjp1,i->Size(i))); 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) > = KernelOfTransformation(Representative(i)))); true gap> ForAll(GreensLClasses(t3), > i->ForAll(AsSSortedList(i),j->ImageSetOfTransformation(j) > = ImageSetOfTransformation(Representative(i)))); true gap> ForAll(GreensJClasses(t3), > i->ForAll(AsSSortedList(i),j->RankOfTransformation(j) > = RankOfTransformation(Representative(i)))); true gap> ForAll(GreensHClasses(t3), > i->ForAll(AsSSortedList(i),j->ImageSetOfTransformation(j) > = ImageSetOfTransformation(Representative(i)) > and KernelOfTransformation(j) = KernelOfTransformation(Representative(i)) > )); true gap> ForAll(GreensRClasses(t4), > i->ForAll(AsSSortedList(i),j->KernelOfTransformation(j) > = KernelOfTransformation(Representative(i)))); true gap> ForAll(GreensLClasses(t4), > i->ForAll(AsSSortedList(i),j->ImageSetOfTransformation(j) > = ImageSetOfTransformation(Representative(i)))); true gap> ForAll(GreensJClasses(t4), > i->ForAll(AsSSortedList(i),j->RankOfTransformation(j) > = RankOfTransformation(Representative(i)))); true gap> ForAll(GreensHClasses(t4), > i->ForAll(AsSSortedList(i),j->ImageSetOfTransformation(j) > = ImageSetOfTransformation(Representative(i)) > and KernelOfTransformation(j) = KernelOfTransformation(Representative(i)) > )); true gap> STOP_TEST( "semirel.tst", 120100000 ); ############################################################################# ## #E ``````