utilsfrgrp.gd

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#W  utilsfrgrp.gd              automgrp package                Yevgen Muntyan
#W                                                             Dmytro Savchuk
##  automgrp v 1.3
##
#Y  Copyright (C) 2003 - 2016 Yevgen Muntyan, Dmytro Savchuk
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##  Here are utility functions dealing with free groups, words, etc.
##


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##  AG_ReducedListOfWordsByNielsen(<words_list>[, <any_string>])
##  AG_ReducedListOfWordsByNielsenBack(<words_list>[, <any_string>])
##
##  It applies Nielsen transformations to <words_list>;
##  input is a list of associative words from the same family.
##
##  Returned value is either fail in case of some error or
##  triple [<result>, <transform>, <did_something>] :
##    <result> is the list of words obtained from <list> using Nielsen
##      transformations;
##    <transform> is the list of words for obtaining <result> from
##      <list> by substituting: <transform> = [w_1, ..., w_n],
##      where w_i are words on alphabet of Length(<list>) letters such that
##      w_i(<words_list>[1], ..., <words_list>[n]) = <result>[i];
##    <did_something> is true if <result> differs from <list> and is
##      false otherwise
##
##  ReduceListOfWordsByNielsenBack is almost identical to
##    ReduceListOfWordsByNielsen: the difference is in the order of
##    comparisons performed inside of main loop;
##    it makes Mihaylov feel better?
##
##  The variants with the second string argument do the same
##    except that the lexicographic ordering on set of words is generated
##    by the following ordering: x_1 < x_1^{-1} < x_2 < x_2^{-1} < ...
##    It's intended for nice output - like "a^2, b^2" instead of "a^-2, b^-2"
##
DeclareOperation("AG_ReducedListOfWordsByNielsen", [IsAssocWordCollection]);
DeclareOperation("AG_ReducedListOfWordsByNielsenBack", [IsAssocWordCollection]);
DeclareOperation("AG_ReducedListOfWordsByNielsen", [IsAssocWordCollection, IsString]);
DeclareOperation("AG_ReducedListOfWordsByNielsenBack", [IsAssocWordCollection, IsString]);
DeclareOperation("AG_ReducedListOfWordsByNielsen", [IsAssocWordCollection, IsFunction]);
DeclareOperation("AG_ReducedListOfWordsByNielsenBack", [IsAssocWordCollection, IsFunction]);


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##  AG_ReducedByNielsen(<obj>)
##  AG_ApplyNielsen(<obj>)
##
##  It applies Nielsen transformations to <obj> or to generators of <obj> if
##  <obj> is a group etc.
##  AG_ReducedByNielsen is nondestructive variant; AG_ApplyNielsen changes object.
##  They are not necessarily implemented both in the same for given type.
##
DeclareOperation("AG_ReducedByNielsen", [IsObject]);
DeclareOperation("AG_ApplyNielsen", [IsObject]);


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##  AG_ComputeMihailovaSystemPairs(<pairs_list>)
##
##  For given list of pairs of free group words, it tries to compute the
##    Mihaylov normal form.
##
##  Returned value is either fail in case of some error or
##  triple [<result>, <transform>, <did_something>] :
##    <result> is the list of pairs representing Mihaylov normal form;
##    <transform> is the list of words for obtaining <result> from
##      <list> by substituting: <transform> = [w_1, ..., w_n],
##      where w_i are words on alphabet of Length(<pairs_list>) letters
##      such that w_i(<pairs_list>[1], ..., <pairs_list>[n]) = <result>[i];
##    <did_something> is true if <result> differs from <pairs_list> and is
##      false otherwise
##
DeclareGlobalFunction("AG_ComputeMihailovaSystemPairs");


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