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##############################################################################
##
#W example3.tst KBMag Package Derek Holt
##
gap> START_TEST( "KBMag package: example3.tst" );
gap> fsa_infolevel_saved := InfoLevel( InfoFSA );;
gap> SetInfoLevel( InfoFSA, 0 );;
gap> rws_infolevel_saved := InfoLevel( InfoRWS );;
gap> SetInfoLevel( InfoRWS, 0 );;
gap> ## SubSection 1.9, Example 3
gap> ## The Heisenberg group - that is, the free 2-generator nilpotent
gap> ## group of class 2. For this to complete, we need to use the recursive
gap> ## ordering, and reverse our initial order of generators.
gap> ## (Alternatively, we could avoid this reversal, by using a “wreathprod”
gap> ## ordering, and setting the levels of the generators to be 6,5,4,3,2,1.)
gap> F := FreeGroup( "x", "y", "z" );;
gap> x:=F.1;; y:=F.2;; z:=F.3;;
gap> G := F/[Comm(y,x)*z^-1, Comm(z,x), Comm(z,y)];;
gap> R := KBMAGRewritingSystem( G );
rec(
isRWS := true,
silent := true,
generatorOrder := [_g1,_g2,_g3,_g4,_g5,_g6],
inverses := [_g2,_g1,_g4,_g3,_g6,_g5],
ordering := "shortlex",
equations := [
[_g4*_g2*_g3,_g5*_g2],
[_g6*_g2,_g2*_g6],
[_g6*_g4,_g4*_g6]
]
)
gap> SetOrderingOfKBMAGRewritingSystem( R, "recursive" );
gap> ReorderAlphabetOfKBMAGRewritingSystem( R, (1,6)(2,5)(3,4) );
gap> R;
rec(
isRWS := true,
silent := true,
generatorOrder := [_g6,_g5,_g4,_g3,_g2,_g1],
inverses := [_g5,_g6,_g3,_g4,_g1,_g2],
ordering := "recursive",
equations := [
[_g4*_g2*_g3,_g5*_g2],
[_g6*_g2,_g2*_g6],
[_g6*_g4,_g4*_g6]
]
)
gap> SetInfoLevel( InfoRWS, 1 );
gap> KnuthBendix( R );
#I Calling external Knuth-Bendix program.
#I External Knuth-Bendix program complete.
#I System computed is confluent.
true
gap> R;
rec(
isRWS := true,
isConfluent := true,
generatorOrder := [_g6,_g5,_g4,_g3,_g2,_g1],
inverses := [_g5,_g6,_g3,_g4,_g1,_g2],
ordering := "recursive",
equations := [
[_g6*_g5,IdWord],
[_g5*_g6,IdWord],
[_g4*_g3,IdWord],
[_g3*_g4,IdWord],
[_g2*_g1,IdWord],
[_g1*_g2,IdWord],
[_g6*_g2,_g2*_g6],
[_g6*_g4,_g4*_g6],
[_g4*_g2,_g2*_g4*_g5],
[_g5*_g2,_g2*_g5],
[_g6*_g1,_g1*_g6],
[_g5*_g4,_g4*_g5],
[_g6*_g3,_g3*_g6],
[_g3*_g1,_g1*_g3*_g5],
[_g4*_g1,_g1*_g4*_g6],
[_g3*_g2,_g2*_g3*_g6],
[_g5*_g1,_g1*_g5],
[_g5*_g3,_g3*_g5]
]
)
gap> Size( R );
infinity
gap> IsReducedWord( R, z*y*x );
false
gap> ReducedForm( R, z*y*x );
x*y*z^2
gap> IsReducedForm( R, x*y*z^2 );
true
gap> ## SetInfoLevel( InfoFSA, fsa_infolevel_saved );;
gap> ## SetInfoLevel( InfoRWS, rws_infolevel_saved );;
gap> STOP_TEST( "example3.tst", 10000 );
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