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##
#W rws.gd automgrp package Yevgen Muntyan
#W Dmytro Savchuk
##
#Y Copyright (C) 2003 - 2018 Yevgen Muntyan, Dmytro Savchuk
##
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##
#O AG_UseRewritingSystem( <G>[, <setting>] )
##
## Tells whether computations in the group <G> should use a rewriting system.
## <setting> defaults to `true' if omitted. This function initially only
## tries to find involutions in <G>. See `AG_AddRelators' ("AG_AddRelators")
## and `AG_UpdateRewritingSystem' ("AG_UpdateRewritingSystem") for the ways
## to add more relators.
##
## \beginexample
## gap> G := AutomatonGroup("a=(1,1)(1,2),b=(a,c),c=(a,d),d=(1,b)");
## < a, b, c, d >
## gap> Comm(a*b, b*a);
## b^-1*a^-2*b^-1*a*b^2*a
## gap> AG_UseRewritingSystem(G);
## gap> Comm(a*b, b*a);
## 1
## gap> AG_UseRewritingSystem(G, false);
## gap> Comm(a*b, b*a);
## b^-1*a^-2*b^-1*a*b^2*a
## \endexample
##
DeclareOperation("AG_UseRewritingSystem", [IsObject]);
DeclareOperation("AG_UseRewritingSystem", [IsObject, IsBool]);
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##
#O AG_AddRelators( <G>, <relators> )
##
## Adds relators from the list <relators> to the rewriting system used in
## <G>.
##
## \beginexample
## gap> G := AutomatonGroup("a=(1,1)(1,2),b=(a,c),c=(a,d),d=(1,b)");
## < a, b, c, d >
## gap> AG_UseRewritingSystem(G);
## gap> b*c;
## b*c
## gap> AG_AddRelators(G, [b*c*d]);
## gap> b*c;
## d
## \endexample
##
## In some cases it's hard to find relations directly from the wreath
## recursion of a self-similar group (at least, there is no general agorithm).
## This function provides possibility to add relators manually. After that
## one can use `AG_UpdateRewritingSystem' (see "AG_UpdateRewritingSystem")
## and `AG_UseRewritingSystem' (see "AG_UseRewritingSystem") to use these
## relators in computations. In the example below we consider a finite group
## $H$, in which $a=b$, but the standard algorithm is unable to solve the
## word problem. There are two solutions for that. One can manually add a
## relator, or one can ask if the group is finite (which does not stop
## generally if the group is infinite).
## \beginexample
## gap> H := SelfSimilarGroup("a=(a*b,1)(1,2), b=(1,b*a^-1)(1,2), c=(b, a*b)");
## < a, b, c >
## gap> AG_AddRelators(H, [a*b^-1]);
## gap> AG_UseRewritingSystem(H);
## gap> Order(a*c);
## 4
## \endexample
##
DeclareOperation("AG_AddRelators", [IsObject, IsList]);
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##
#O AG_UpdateRewritingSystem( <G>, <maxlen> )
##
## Tries to find new relators of length up to <maxlen> and adds them into
## the rewriting system. It can also be used after introducing new relators
## via `AG_AddRelators' (see "AG_AddRelators").
##
## \beginexample
## gap> G := AutomatonGroup("a=(1,1)(1,2),b=(a,c),c=(a,d),d=(1,b)");
## < a, b, c, d >
## gap> AG_UseRewritingSystem(G);
## gap> b*c;
## b*c
## gap> AG_UpdateRewritingSystem(G, 3);
## gap> b*c;
## d
## \endexample
##
DeclareOperation("AG_UpdateRewritingSystem", [IsObject]);
DeclareOperation("AG_UpdateRewritingSystem", [IsObject, IsPosInt]);
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##
#O AG_RewritingSystem( <G> )
##
## Returns the rewriting system object. See also `AG_UseRewritingSystem'
## ("AG_UseRewritingSystem").
##
DeclareOperation("AG_RewritingSystem", [IsObject]);
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##
#O AG_RewritingSystemRules( <G> )
##
## Returns the list of rules used in the rewriting system of group <G>.
## \beginexample
## gap> G := AutomatonGroup("a=(1,1)(1,2),b=(a,c),c=(a,d),d=(1,b)");
## < a, b, c, d >
## gap> AG_UseRewritingSystem(G);
## gap> AG_RewritingSystemRules(G);
## [ [ a^2, <identity ...> ], [ b^2, <identity ...> ], [ c^2, <identity ...> ],
## [ d^2, <identity ...> ], [ A, a ], [ B, b ], [ C, c ], [ D, d ] ]
## \endexample
##
DeclareOperation("AG_RewritingSystemRules", [IsObject]);
DeclareOperation("AG_ReducedForm", [IsObject]);
DeclareOperation("AG_ReducedForm", [IsObject, IsObject]);
#E
[ Dauer der Verarbeitung: 0.128 Sekunden
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2026-03-28
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