  
  [1X15 [33X[0;0YFinitely presented semigroups and Tietze transformations[133X[101X
  
  [33X[0;0YIn  this  chapter  we  describe the functions implemented in [5XSemigroups[105X that
  extend  the  features  available  in [5XGAP[105X for dealing with finitely presented
  semigroups and monoids.[133X
  
  [33X[0;0YSection  [14X15.1[114X  (written by Maria Tsalakou and Murray Whyte) and Section [14X15.2[114X
  (written  by  Luke  Elliott)  demonstrate  a number of helper functions that
  allow  the  user  to  convert between different representations of words and
  relations.[133X
  
  [33X[0;0YIn  the  later  sections,  written  and  implemented  by  Ben Spiers and Tom
  Conti-Leslie,  we  describe  how  to  change  the  relations  of  a finitely
  presented   semigroup   either   manually   or  automatically  using  Tietze
  transformations (which is abbreviated to [12XStz[112X.[133X
  
  
  [1X15.1 [33X[0;0YChanging representation for words and strings[133X[101X
  
  [33X[0;0YThis  section  contains  various  methods  for dealing with words, which for
  these purposes are lists of positive integers.[133X
  
  [1X15.1-1 WordToString[101X
  
  [33X[1;0Y[29X[2XWordToString[102X( [3XA[103X, [3Xw[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA string.[133X
  
  [33X[0;0YReturns  the  word  [3Xw[103X, in the form of a string of letters of the alphabet [3XA[103X.
  The alphabet is given as a string containing its members.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XWordToString("abcd", [4, 2, 3, 1, 1, 4, 2, 3]);[127X[104X
    [4X[28X"dbcaadbc"[128X[104X
  [4X[32X[104X
  
  [1X15.1-2 RandomWord[101X
  
  [33X[1;0Y[29X[2XRandomWord[102X( [3Xl[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA word.[133X
  
  [33X[0;0YReturns a random word of length [3Xl[103X over [3Xn[103X letters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomWord(8, 5);[127X[104X
    [4X[28X[ 2, 4, 3, 4, 5, 3, 3, 2 ][128X[104X
    [4X[25Xgap>[125X [27XRandomWord(8, 5);[127X[104X
    [4X[28X[ 3, 3, 5, 5, 5, 4, 4, 5 ][128X[104X
    [4X[25Xgap>[125X [27XRandomWord(8, 4);[127X[104X
    [4X[28X[ 1, 4, 1, 1, 3, 3, 4, 4 ][128X[104X
  [4X[32X[104X
  
  [1X15.1-3 StandardiseWord[101X
  
  [33X[1;0Y[29X[2XStandardiseWord[102X( [3Xw[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XStandardizeWord[102X( [3Xw[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of positive integers.[133X
  
  [33X[0;0YThis function takes a word [3Xw[103X, consisting of [10Xn[110X distinct positive integers and
  returns a word [10Xs[110X where the characters of [10Xs[110X correspond to those of [3Xw[103X in order
  of first appearence.[133X
  
  [33X[0;0YThe word [3Xw[103X is changed in-place into word [10Xs[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xw := [3, 1, 2];[127X[104X
    [4X[28X[ 3, 1, 2 ][128X[104X
    [4X[25Xgap>[125X [27XStandardiseWord(w);[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xw;[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xw := [4, 2, 10, 2];[127X[104X
    [4X[28X[ 4, 2, 10, 2 ][128X[104X
    [4X[25Xgap>[125X [27XStandardiseWord(w);[127X[104X
    [4X[28X[ 1, 2, 3, 2 ][128X[104X
  [4X[32X[104X
  
  [1X15.1-4 StringToWord[101X
  
  [33X[1;0Y[29X[2XStringToWord[102X( [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of positive integers.[133X
  
  [33X[0;0YThis  function  takes a string [3Xs[103X, consisting of [10Xn[110X distinct positive integers
  and returns a word [10Xw[110X (i.e. a list of positive integers) over the alphabet [10X[1
  ..  n][110X.  The  positive  integers  of [10Xw[110X correspond to the characters of [3Xs[103X, in
  order of first appearence.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xw := "abac";[127X[104X
    [4X[28X"abac"[128X[104X
    [4X[25Xgap>[125X [27XStringToWord(w);[127X[104X
    [4X[28X[ 1, 2, 1, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xw := "ccala";[127X[104X
    [4X[28X"ccala"[128X[104X
    [4X[25Xgap>[125X [27XStringToWord(w);[127X[104X
    [4X[28X[ 1, 1, 2, 3, 2 ][128X[104X
    [4X[25Xgap>[125X [27Xw := "a1b5";[127X[104X
    [4X[28X"a1b5"[128X[104X
    [4X[25Xgap>[125X [27XStringToWord(w);[127X[104X
    [4X[28X[ 1, 2, 3, 4 ][128X[104X
  [4X[32X[104X
  
  
  [1X15.2 [33X[0;0YHelper functions[133X[101X
  
  [33X[0;0YThis  section  describes  operations  implemented  in  [5XSemigroups[105X  that  are
  designed  to  interact  with  standard  [5XGAP[105X  methods  for  creating finitely
  presented   semigroups  and  monoids  (see  [14X'Reference:  Finitely  Presented
  Semigroups and Monoids'[114X).[133X
  
  [1X15.2-1 ParseRelations[101X
  
  [33X[1;0Y[29X[2XParseRelations[102X( [3Xgens[103X, [3Xrels[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of pairs of semigroup elements.[133X
  
  [33X[0;0Y[10XParseRelations[110X converts a string describing relations for a semigroup to the
  list  of pairs of semigroup elements it represents. Any white space given is
  ignored. The output list is then compatible with other GAP functions. In the
  below  examples  we  see  free  semigroups  being directly quotiented by the
  output of the [10XParseRelations[110X function.[133X
  
  [33X[0;0Y[3Xgens[103X  must be a list of generators for a free semigroup, each being a single
  English  letter (in upper or lower case). [3Xrels[103X must be string that lists the
  equalities desired.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := FreeSemigroup("x", "y", "z");;[127X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables(f);[127X[104X
    [4X[25Xgap>[125X [27XParseRelations([x, y, z], "  x=(y^2z) ^2x, y=xxx , z=y^3");[127X[104X
    [4X[28X[ [ x, (y^2*z)^2*x ], [ y, x^3 ], [ z, y^3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xr := ParseRelations([x, y, z], "  x=(y^2z)^2x, y=xxx=z , z=y^3");[127X[104X
    [4X[28X[ [ x, (y^2*z)^2*x ], [ y, x^3 ], [ x^3, z ], [ z, y^3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xf / r;[127X[104X
    [4X[28X<fp semigroup with 3 generators and 4 relations of length 23>[128X[104X
    [4X[25Xgap>[125X [27Xf2 := FreeSemigroup("a");[127X[104X
    [4X[28X<free semigroup on the generators [ a ]>[128X[104X
    [4X[25Xgap>[125X [27Xf2 / ParseRelations(GeneratorsOfSemigroup(f2), "a = a^2");[127X[104X
    [4X[28X<fp semigroup with 1 generator and 1 relation of length 4>[128X[104X
  [4X[32X[104X
  
  [1X15.2-2 ElementOfFpSemigroup[101X
  
  [33X[1;0Y[29X[2XElementOfFpSemigroup[102X( [3XS[103X, [3Xword[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn element of the fp semigroup [3XS[103X.[133X
  
  [33X[0;0YWhen  [3XS[103X is a finitely presented semigroup and [3Xword[103X is an associative word in
  the  associated  free  semigroup (see [2XIsAssocWord[102X ([14XReference: IsAssocWord[114X)),
  this returns the fp semigroup element with representative [3Xword[103X.[133X
  
  [33X[0;0YThis  function  is  just  a  short form of the [5XGAP[105X library implementation of
  [2XElementOfFpSemigroup[102X   ([14XReference:   ElementOfFpSemigroup[114X)  which  does  not
  require retrieving an element family.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := FreeSemigroup("x", "y");;[127X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables(f);[127X[104X
    [4X[25Xgap>[125X [27Xs := f / [[x * x, x], [y * y, y]];[127X[104X
    [4X[28X<fp semigroup with 2 generators and 2 relations of length 8>[128X[104X
    [4X[25Xgap>[125X [27Xa := ElementOfFpSemigroup(s, x * y);[127X[104X
    [4X[28Xx*y[128X[104X
    [4X[25Xgap>[125X [27Xb := ElementOfFpSemigroup(s, x * y * y);[127X[104X
    [4X[28Xx*y^2[128X[104X
    [4X[25Xgap>[125X [27Xa in s;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xa = b;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X15.2-3 ElementOfFpMonoid[101X
  
  [33X[1;0Y[29X[2XElementOfFpMonoid[102X( [3XM[103X, [3Xword[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn element of the fp monoid [3XM[103X.[133X
  
  [33X[0;0YWhen [3XM[103X is a finitely presented monoid and [3Xword[103X is an associative word in the
  associated  free  monoid  (see  [2XIsAssocWord[102X  ([14XReference: IsAssocWord[114X)), this
  returns the fp monoid element with representative [3Xword[103X.[133X
  
  [33X[0;0YThis is analogous to [2XElementOfFpSemigroup[102X ([14X15.2-2[114X).[133X
  
  
  [1X15.3 [33X[0;0YCreating Tietze transformation objects[133X[101X
  
  [33X[0;0YIt  is  possible  to use [5XGAP[105X to create finitely presented semigroups without
  the  [5XSemigroups[105X package, by creating a free semigroup, then quotienting by a
  list  of  relations.  This is described in the reference manual ([14X'Reference:
  Finitely Presented Semigroups and Monoids'[114X).[133X
  
  [33X[0;0YHowever,  finitely  presented semigroups do not allow for their relations to
  be  simplified,  so in the following sections, we describe how to create and
  modify  the  semigroup Tietze ([2XIsStzPresentation[102X ([14X15.3-2[114X)) object associated
  with  an  fp  semigroup. This object can be automatically simplified, or the
  user can manually apply Tietze transformations to add or remove relations or
  generators in the presentation.[133X
  
  [33X[0;0YThis    object    is    analogous    to    [2XPresentationFpGroup[102X   ([14XReference:
  PresentationFpGroup[114X)  implemented for fp groups in the main [5XGAP[105X distribution
  ([14X'Reference:  Presentations  and  Tietze Transformations'[114X), but its features
  are  semigroup-specific.  Most  of  the  functions  used to create, view and
  manipulate semigroup Tietze objects are prefixed with [12XStz[112X.[133X
  
  [1X15.3-1 StzPresentation[101X
  
  [33X[1;0Y[29X[2XStzPresentation[102X( [3XS[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA semigroup Tietze (Stz) object.[133X
  
  [33X[0;0YIf  [3Xs[103X  is  an  fp semigroup ([2XIsFpSemigroup[102X ([14XReference: IsFpSemigroup[114X)), then
  this  function  returns  a modifiable object representing the generators and
  relations of [3Xs[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");;[127X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables(F);;[127X[104X
    [4X[25Xgap>[125X [27XS := F / [[a * b, c], [b * c, a], [c * a, b]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 12>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(S);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 12>[128X[104X
  [4X[32X[104X
  
  [1X15.3-2 IsStzPresentation[101X
  
  [33X[1;0Y[29X[2XIsStzPresentation[102X( [3Xstz[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YEvery   semigroup   Tietze   object   is   an   element   of   the  category
  [10XIsStzPresentation[110X. Internally, each Stz object contains a list of generators
  (each  represented as a string) and a list of relations (each represented as
  a    pair   of   [10XLetterRep[110X   words,   see   [2XLetterRepAssocWord[102X   ([14XReference:
  LetterRepAssocWord[114X)).  These  generator  and  relation lists can be modified
  using Tietze transformations ([14X15.5[114X).[133X
  
  [33X[0;0YWhen  a [10XIsStzPresentation[110X object [3Xstz[103X is created from an fp semigroup [10Xs[110X using
  [10Xstz := StzPresentation(s)[110X, the generators and relations of [3Xstz[103X are initially
  equal  to  the generators and relations of [10Xs[110X. However, as the Stz object [3Xstz[103X
  is  modified,  these lists may change, and their current state can be viewed
  using  [2XGeneratorsOfStzPresentation[102X  ([14X15.3-3[114X)  and [2XRelationsOfStzPresentation[102X
  ([14X15.3-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");;[127X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables(F);;[127X[104X
    [4X[25Xgap>[125X [27XS := F / [[a * b, c], [b * c, a], [c * a, b]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 12>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(S);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 12>[128X[104X
    [4X[25Xgap>[125X [27XIsStzPresentation(stz);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X15.3-3 GeneratorsOfStzPresentation[101X
  
  [33X[1;0Y[29X[2XGeneratorsOfStzPresentation[102X( [3Xstz[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of strings.[133X
  
  [33X[0;0YIf     [3Xstz[103X     is     an     [2XStzPresentation[102X     ([14X15.3-1[114X)    object,    then
  [10XGeneratorsOfStzPresentation[110X  will  return (as strings) the generators of the
  fp  semigroup that the presentation was created from. In the [2XStzPresentation[102X
  ([14X15.3-1[114X) object, it is only necessary to know how many generators there are,
  but  for  the  purposes  of  representing  generators  and  relations of the
  presentation  object  and  building  a new fp semigroup from the object, the
  strings representing the generators are stored.[133X
  
  [33X[0;0YAs  Tietze transformations are performed on [3Xstz[103X, the generators will change,
  but  the  labels will remain as close to the original labels as possible, so
  that  if  a  generator in the fp semigroup obtained from the presentation is
  the  same as a generator in the original fp semigroup, then they should have
  the same label.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 19>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 19>[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfStzPresentation(stz);[127X[104X
    [4X[28X[ "a", "b", "c" ][128X[104X
  [4X[32X[104X
  
  [1X15.3-4 RelationsOfStzPresentation[101X
  
  [33X[1;0Y[29X[2XRelationsOfStzPresentation[102X( [3Xstz[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA   list  of  pairs  of  words  in  [10XLetterRep[110X  ([2XLetterRepAssocWord[102X
            ([14XReference: LetterRepAssocWord[114X)) form.[133X
  
  [33X[0;0YIf     [3Xstz[103X     is     an     [2XStzPresentation[102X     ([14X15.3-1[114X)    object,    then
  [10XRelationsOfStzPresentation[110X  will  return  in  letter  rep  form  the current
  relations  of the presentation object. When the presentation object is first
  created,  these  will  be  the  [10XLetterRep[110X  forms  of the relations of the fp
  semigroup  object  that is used to create [3Xstz[103X. As Tietze transformations are
  performed  on  the  presentation  object,  the  relations  returned  by this
  function will change to reflect the transformations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 19>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 19>[128X[104X
    [4X[25Xgap>[125X [27XRelationsOfStzPresentation(stz);[127X[104X
    [4X[28X[ [ [ 1 ], [ 2, 2, 2, 2, 2, 3 ] ], [128X[104X
    [4X[28X  [ [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X15.3-5 UnreducedFpSemigroup[101X
  
  [33X[1;0Y[29X[2XUnreducedFpSemigroup[102X( [3Xstz[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YAn fp semigroup.[133X
  
  [33X[0;0YIf [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object, then [10XUnreducedFpSemigroup[110X will
  return  the  fp  semigroup that was used to create [3Xstz[103X using [2XStzPresentation[102X
  ([14X15.3-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 19>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 19>[128X[104X
    [4X[25Xgap>[125X [27XUnreducedFpSemigroup(stz) = T;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X15.3-6 Length[101X
  
  [33X[1;0Y[29X[2XLength[102X( [3Xstz[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA non-negative integer.[133X
  
  [33X[0;0YIf  [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object, then the [10XLength[110X of the object
  is defined as the number of generators plus the lengths of each word in each
  relation of [3Xstz[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XLength(stz);[127X[104X
    [4X[28X22[128X[104X
  [4X[32X[104X
  
  
  [1X15.4 [33X[0;0YPrinting Tietze transformation objects[133X[101X
  
  [33X[0;0YSince  the  relations are stored as flat lists of numbers, there are several
  methods installed to print the presentations in more user-friendly forms.[133X
  
  [33X[0;0YAll  printing  methods  in  this  section are displayed as information ([2XInfo[102X
  ([14XReference:  Info[114X))  in  the  class  [10XInfoFpSemigroup[110X  at  level  1.  Setting
  [10XSetInfoLevevl(InfoFpSemigroup,  0)[110X  will  suppress  the  messages, while any
  higher number will display them.[133X
  
  [1X15.4-1 StzPrintRelations[101X
  
  [33X[1;0Y[29X[2XStzPrintRelations[102X( [3Xstz[103X[, [3Xlist[103X] ) [32X operation[133X
  
  [33X[0;0YIf  [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object and [3Xlist[103X is a list of positive
  integers,  then [10XStzPrintRelations[110X prints for each [3Xi[103X in [3Xlist[103X the [3Xi[103Xth relation
  to the console in terms of the stored labels for the generators (that is, as
  words over the alphabet consisting of the generators of [3Xstz[103X).[133X
  
  [33X[0;0YIf  [3Xlist[103X is not specified then [10XStzPrintRelations[110X prints all relations of [3Xstz[103X
  in order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XStzPrintRelations(stz, [2, 3]);[127X[104X
    [4X[28X#I  2. b^6 = b^3[128X[104X
    [4X[28X#I  3. b^2 = b[128X[104X
    [4X[25Xgap>[125X [27XStzPrintRelations(stz);[127X[104X
    [4X[28X#I  1. a = b^5*c[128X[104X
    [4X[28X#I  2. b^6 = b^3[128X[104X
    [4X[28X#I  3. b^2 = b[128X[104X
  [4X[32X[104X
  
  [1X15.4-2 StzPrintRelation[101X
  
  [33X[1;0Y[29X[2XStzPrintRelation[102X( [3Xstz[103X, [3Xint[103X ) [32X operation[133X
  
  [33X[0;0YIf  [3Xstz[103X  is  an [2XStzPresentation[102X ([14X15.3-1[114X) object, then [10XStzPrintRelation[110X calls
  [10XStzPrintRelations[110X with parameters [3Xstz[103X and [10X[[3Xint[103X[10X][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XStzPrintRelation(stz, 2);[127X[104X
    [4X[28X#I  2. b^6 = b^3[128X[104X
  [4X[32X[104X
  
  [1X15.4-3 StzPrintGenerators[101X
  
  [33X[1;0Y[29X[2XStzPrintGenerators[102X( [3Xstz[103X[, [3Xlist[103X] ) [32X operation[133X
  
  [33X[0;0YIf  [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object and [3Xlist[103X is a list of positive
  integers,  then  [10XStzPrintGenerators[110X for each [3Xi[103X in [3Xlist[103X the [3Xi[103Xth generator and
  the  number  of occurrences of that generator in the relations is printed to
  the screen.[133X
  
  [33X[0;0YIf  [3Xlist[103X  is  not specified then [10XStzPrintGenerators[110X prints all generators of
  [3Xstz[103X in order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XStzPrintGenerators(stz, [1, 2]);[127X[104X
    [4X[28X#I  1.  a  1 occurrences[128X[104X
    [4X[28X#I  2.  b  17 occurrences[128X[104X
    [4X[25Xgap>[125X [27XStzPrintGenerators(stz);[127X[104X
    [4X[28X#I  1.  a  1 occurrences[128X[104X
    [4X[28X#I  2.  b  17 occurrences[128X[104X
    [4X[28X#I  3.  c  1 occurrences[128X[104X
  [4X[32X[104X
  
  [1X15.4-4 StzPrintPresentation[101X
  
  [33X[1;0Y[29X[2XStzPrintPresentation[102X( [3Xstz[103X ) [32X operation[133X
  
  [33X[0;0YIf  [3Xstz[103X  is  an  [2XStzPresentation[102X  ([14X15.3-1[114X) object, then [10XStzPrintPresentation[110X
  prints  a comprehensive overview of [3Xstz[103X, including the generators and number
  of  occurrences  of  each generator in the relations, the relations as words
  over the generators, and the forward and backward maps that indicate how the
  unreduced semigroup maps to the semigroup currently described by [3Xstz[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XStzPrintPresentation(stz);[127X[104X
    [4X[28X#I  Current generators:[128X[104X
    [4X[28X#I  1.  a  1 occurrences[128X[104X
    [4X[28X#I  2.  b  17 occurrences[128X[104X
    [4X[28X#I  3.  c  1 occurrences[128X[104X
    [4X[28X#I  [128X[104X
    [4X[28X#I  Current relations:[128X[104X
    [4X[28X#I  1. a = b^5*c[128X[104X
    [4X[28X#I  2. b^6 = b^3[128X[104X
    [4X[28X#I  3. b^2 = b[128X[104X
    [4X[28X#I  [128X[104X
    [4X[28X#I  There are 3 generators and 3 relations of total length 22.[128X[104X
    [4X[28X#I  [128X[104X
    [4X[28X#I  Generators of original fp semigroup expressed as[128X[104X
    [4X[28X#I  combinations of generators in current presentation:[128X[104X
    [4X[28X#I  1. a = a[128X[104X
    [4X[28X#I  2. b = b[128X[104X
    [4X[28X#I  3. c = c[128X[104X
    [4X[28X#I  [128X[104X
    [4X[28X#I  Generators of current presentation expressed as[128X[104X
    [4X[28X#I  combinations of generators of original fp semigroup:[128X[104X
    [4X[28X#I  1. a = a[128X[104X
    [4X[28X#I  2. b = b[128X[104X
    [4X[28X#I  3. c = c[128X[104X
  [4X[32X[104X
  
  
  [1X15.5 [33X[0;0YChanging Tietze transformation objects[133X[101X
  
  [33X[0;0YFundamentally,  there  are  four  different  changes  that  can be made to a
  presentation  without  changing  the algebraic structure of the fp semigroup
  that  can  be  derived  from  it.  These  four  changes  are  called  Tietze
  transformations,  and  they  are  primarily  implemented  in this section as
  operations  on  an  [10XStzPresentation[110X  object  that  will  throw errors if the
  conditions have not been met to perform the Tietze transformation.[133X
  
  [33X[0;0YHowever, the checks required in order to ensure that a Tietze transformation
  is  valid  sometimes  require  verifying  equality  of  two  words  in an fp
  semigroup  (for example, to ensure that a relation we are adding to the list
  of relations can be derived from the relations already present). Since these
  checks  sometimes  do  not  terminate,  a  second  implementation  of Tietze
  transformations  assumes  good  faith and does not perform any checks to see
  whether the requested Tietze transformation actually maintains the structure
  of  the semigroup. This latter type should be used at the user's discretion.
  If  only  the  first  type  are  used,  the  presentation will always give a
  semigroup  isomorphic  to  the one used to create the object, but if instead
  one  is  not  changing  the  presentation  with the intention of maintaining
  algebraic structure, these no-check functions are available for use.[133X
  
  [33X[0;0YThe  four  Tietze  transformations  on a presentation are adding a relation,
  removing  a  relation,  adding  a  generator, and removing a generator, with
  particular  conditions  on  what  can  be added/removed in order to maintain
  structure.  More  details  on  each  transformation  and  its  arguments and
  conditions  is given in each entry below. In addition to the four elementary
  transformations, there is an additional function [10XStzSubstituteRelation[110X which
  applies multiple Tietze transformations in sequence.[133X
  
  [1X15.5-1 StzAddRelation[101X
  
  [33X[1;0Y[29X[2XStzAddRelation[102X( [3Xstz[103X, [3Xpair[103X ) [32X operation[133X
  
  [33X[0;0YIf  [3Xstz[103X  is an [2XStzPresentation[102X ([14X15.3-1[114X) object and [3Xpair[103X is a list containing
  two  [10XLetterRep[110X  words  over  the generators of [3Xstz[103X, then [10XStzAddRelation[110X will
  perform  a Tietze transformation of the first type and add a new relation to
  [3Xstz[103X.  This  only  happens if the new relation that would be formed from [3Xpair[103X
  can  be  constructed  from  the other existing relations; that is, if we can
  perform elementary operations using the existing relations of [3Xstz[103X to convert
  [10X[3Xpair[103X[10X[1][110X into [10X[3Xpair[103X[10X[2][110X.[133X
  
  [33X[0;0YIf,  instead,  [3Xpair[103X  is a list containing two elements of the fp semigroup [10XS[110X
  that  was used to create [10Xstz[110X, and the two words are equal in that semigroup,
  then  this  function will add the [10XLetterRep[110X of these words as a new relation
  to [3Xstz[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 19>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 19>[128X[104X
    [4X[25Xgap>[125X [27Xpair := [[2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2]];;[127X[104X
    [4X[25Xgap>[125X [27XStzAddRelation(stz, pair);[127X[104X
    [4X[25Xgap>[125X [27XRelationsOfStzPresentation(stz);[127X[104X
    [4X[28X[ [ [ 1 ], [ 2, 2, 2, 2, 2, 3 ] ], [128X[104X
    [4X[28X  [ [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2 ] ], [128X[104X
    [4X[28X  [ [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xpair2 := [[1, 1], [3]];[127X[104X
    [4X[28X[ [ 1, 1 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XStzAddRelation(stz, pair2);[127X[104X
    [4X[28XError, StzAddRelation: second argument <pair> must list two[128X[104X
    [4X[28Xwords that are equal in the presentation <stz>[128X[104X
  [4X[32X[104X
  
  [1X15.5-2 StzRemoveRelation[101X
  
  [33X[1;0Y[29X[2XStzRemoveRelation[102X( [3Xstz[103X, [3Xindex[103X ) [32X operation[133X
  
  [33X[0;0YIf [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object and [3Xindex[103X is a positive integer
  less than or equal to the number of relations of [3Xstz[103X, then [10XStzRemoveRelation[110X
  will  perform  a  Tietze  transformation  of  the second type and remove the
  [3Xindex[103Xth  relation of [3Xstz[103X if that relation is such that one side of it can be
  obtained  from  the  other by a sequence of elementary operations using only
  the other relations of [3Xstz[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XStzRemoveRelation(stz, 2);[127X[104X
    [4X[25Xgap>[125X [27XRelationsOfStzPresentation(stz);[127X[104X
    [4X[28X[ [ [ 1 ], [ 2, 2, 2, 2, 2, 3 ] ], [ [ 2, 2 ], [ 2 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XStzRemoveRelation(stz, 2);[127X[104X
    [4X[28XError, StzRemoveRelation: second argument <index> must point to[128X[104X
    [4X[28Xa relation that is redundant in the presentation <stz>[128X[104X
  [4X[32X[104X
  
  [1X15.5-3 StzAddGenerator[101X
  
  [33X[1;0Y[29X[2XStzAddGenerator[102X( [3Xstz[103X, [3Xword[103X[, [3Xname[103X] ) [32X operation[133X
  
  [33X[0;0YIf  [3Xstz[103X  is  an [2XStzPresentation[102X ([14X15.3-1[114X) object and [3Xword[103X is a [10XLetterRep[110X word
  over  the  generators  of  [3Xstz[103X,  then  [10XStzAddGenerator[110X will perform a Tietze
  transformation  which  adds a new generator to [3Xstz[103X and a new relation of the
  form [10Xnewgenerator = [3Xword[103X[10X[110X.[133X
  
  [33X[0;0YIf,  instead, [3Xword[103X is a word over the fp semigroup [3XS[103X that was used to create
  [3Xstz[103X,  then  this function will add the new generator and a new relation with
  the  new  generator equal to the [10XLetterRep[110X of this word as a new relation to
  [3Xstz[103X.[133X
  
  [33X[0;0YA  new name for the generator is chosen and added automatically based on the
  names  of the existing generators to [2XGeneratorsOfStzPresentation[102X ([14X15.3-3[114X) if
  the argument [3Xname[103X is not included. If it is, and if [3Xname[103X is a string that is
  not  equal  to  an  existing generator, then the string added to the list of
  generators will be [3Xname[103X instead.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XStzAddGenerator(stz, [2, 2, 2]);[127X[104X
    [4X[25Xgap>[125X [27XRelationsOfStzPresentation(stz);[127X[104X
    [4X[28X[ [ [ 1 ], [ 2, 2, 2, 2, 2, 3 ] ], [128X[104X
    [4X[28X  [ [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2 ] ], [ [ 2, 2 ], [ 2 ] ], [128X[104X
    [4X[28X  [ [ 2, 2, 2 ], [ 4 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X15.5-4 StzRemoveGenerator[101X
  
  [33X[1;0Y[29X[2XStzRemoveGenerator[102X( [3Xstz[103X, [3Xgen/genname[103X[, [3Xindex[103X] ) [32X operation[133X
  
  [33X[0;0YIf  [3Xstz[103X  is an [2XStzPresentation[102X ([14X15.3-1[114X) object and [3Xgen[103X is a positive integer
  less   than   or   equal   to   the   number  of  generators  of  [3Xstz[103X,  then
  [10XStzRemoveGenerator[110X  will  perform  a Tietze transformation which removes the
  [3Xgen[103Xth generator of [3Xstz[103X.[133X
  
  [33X[0;0YThe  argument  [3Xstz[103X  must contain a relation of the form [10X[3Xgen[103X[10X = word[110X or [10Xword =
  [3Xgen[103X[10X[110X,  where [10Xword[110X contains no occurrences of the generator [3Xgen[103X being removed.
  The  generator  is  then  removed  from  the presentation by replacing every
  instance of [3Xgen[103X with a copy of [10Xword[110X.[133X
  
  [33X[0;0YIf  the  second  argument is a string [3Xgenname[103X rather than a positive integer
  [3Xgen[103X,  then  the function searches the generators of [3Xstz[103X for a generator with
  the same name and attempts to remove the generator if the same conditions as
  above are met.[133X
  
  [33X[0;0YIf  the  argument  [3Xindex[103X  is included and is a positive integer less than or
  equal  to  the number of relations, then rather than searching the relations
  for  the  first to satisfy the necessary conditions, the function checks the
  [3Xindex[103Xth  relation  to  see if it satisfies those conditions, and applies the
  Tietze transformation by removing this relation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XStzRemoveGenerator(stz, 1);[127X[104X
    [4X[25Xgap>[125X [27XRelationsOfStzPresentation(stz);[127X[104X
    [4X[28X[ [ [ 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1 ] ], [ [ 1, 1 ], [ 1 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X15.5-5 StzSubstituteRelation[101X
  
  [33X[1;0Y[29X[2XStzSubstituteRelation[102X( [3Xstz[103X, [3Xindex[103X, [3Xside[103X ) [32X operation[133X
  
  [33X[0;0YIf [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object and [3Xindex[103X is a positive integer
  less than or equal to the number of relations of [3Xstz[103X and [3Xside[103X is either [10X1[110X or
  [10X2[110X, then [10XStzRemoveGenerator[110X will perform a sequence of Tietze transformations
  in  order  to replace, for the [3Xindex[103Xth relation (say [10X[u, v][110X), to replace all
  instances  of  the [3Xside[103Xth word of the relation in all other relations by the
  other side (so, for [10X[3Xside[103X[10X=1[110X, all instances of [10Xu[110X in all other relations of [3Xstz[103X
  are  replaced  by  [10Xv[110X). This requires two Tietze transformations per relation
  containing [10Xu[110X, one to add a new redundant relation with each [10Xu[110X replaced by [10Xv[110X,
  and another to remove the original (now redundant) relation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3], [F.2 ^ 2, F.2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 3 relations of length 22>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 3 relations[128X[104X
    [4X[28X with length 22>[128X[104X
    [4X[25Xgap>[125X [27XStzSubstituteRelation(stz, 3, 1);[127X[104X
    [4X[25Xgap>[125X [27XRelationsOfStzPresentation(stz);[127X[104X
    [4X[28X[ [ [ 1 ], [ 2, 2, 2, 3 ] ], [ [ 2, 2, 2 ], [ 2, 2 ] ], [128X[104X
    [4X[28X  [ [ 2, 2 ], [ 2 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XStzSubstituteRelation(stz, 3, 1);[127X[104X
    [4X[25Xgap>[125X [27XRelationsOfStzPresentation(stz);[127X[104X
    [4X[28X[ [ [ 1 ], [ 2, 2, 3 ] ], [ [ 2, 2 ], [ 2 ] ], [ [ 2, 2 ], [ 2 ] ] ][128X[104X
  [4X[32X[104X
  
  
  [1X15.6 [33X[0;0YConverting a Tietze transformation object into a fp semigroup[133X[101X
  
  [33X[0;0YSemigroup Tietze transformation objects ([2XIsStzPresentation[102X ([14X15.3-2[114X)) are not
  actual   fp   semigroups   in   the   sense   of  [2XIsFpSemigroup[102X  ([14XReference:
  IsFpSemigroup[114X).  This  is  because  their  generators  and  relations can be
  modified  (see  section  [14X15.5[114X).  However, an [2XStzPresentation[102X ([14X15.3-1[114X) can be
  converted back into an actual finitely presented semigroup using the methods
  described in this section.[133X
  
  [33X[0;0YThe  intended  use  of  semigroup  Tietze objects is as follows: given an fp
  semigroup    [10XS[110X,    create   a   modifiable   presentation   using   [10Xstz   :=
  StzPresentation(S)[110X,  apply Tietze transformations to it (perhaps in order to
  simplify  the presentation), then generate a new fp semigroup [10XT[110X given by [10Xstz[110X
  which  is  isomorphic  to  [10XS[110X,  but  has  a  simpler  presentation. Once [10XT[110X is
  obtained,  it  may be of interest to map elements of [10XS[110X over to [10XT[110X, where they
  may  be represented by different combinations of generators. The isomorphism
  achieving this is described in this section (see [2XStzIsomorphism[102X ([14X15.6-3[114X)).[133X
  
  [1X15.6-1 TietzeForwardMap[101X
  
  [33X[1;0Y[29X[2XTietzeForwardMap[102X( [3Xstz[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of lists of positive integers.[133X
  
  [33X[0;0YIf  [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object, then [10XTietzeForwardMap[110X returns
  a  list  of lists of positive integers. There is an element of this list for
  every  generator  of  the  unreduced  semigroup  of  the  presentation  (see
  [2XUnreducedFpSemigroup[102X  ([14X15.3-5[114X))  that indicates the word (in [10XLetterRep[110X form)
  in  the  semigroup  object  currently  defined  by the presentation that the
  generator maps to.[133X
  
  [33X[0;0YThis mapping is updated as the presentation object is transformed. It begins
  as  a  list of the form [10X[[1], [2], [3], . . ., [n]][110X where [10Xn[110X is the number of
  generators of the unreduced semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XS := F / [[F.1, F.3 ^ 3], [F.2 ^ 2, F.3 ^ 2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 11>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(S);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 11>[128X[104X
    [4X[25Xgap>[125X [27XStzRemoveGenerator(stz, 1);[127X[104X
    [4X[25Xgap>[125X [27XTietzeForwardMap(stz);[127X[104X
    [4X[28X[ [ 2, 2, 2 ], [ 1 ], [ 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X15.6-2 TietzeBackwardMap[101X
  
  [33X[1;0Y[29X[2XTietzeBackwardMap[102X( [3Xstz[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of lists of positive integers.[133X
  
  [33X[0;0YIf [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object, then [10XTietzeBackwardMap[110X returns
  a  list  of lists of positive integers. There is an element of this list for
  every  generator  of  the  semigroup that the presentation currently defines
  that  indicates  the  word (in [10XLetterRep[110X form) in the unreduced semigroup of
  the presentation (see [2XUnreducedFpSemigroup[102X ([14X15.3-5[114X)) that the generator maps
  to.[133X
  
  [33X[0;0YThis mapping is updated as the presentation object is transformed. It begins
  as  a  list of the form [10X[[1], [2], [3], . . ., [n]][110X where [10Xn[110X is the number of
  generators of the unreduced semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XS := F / [[F.1, F.3 ^ 3], [F.2 ^ 2, F.3 ^ 2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 11>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(S);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 11>[128X[104X
    [4X[25Xgap>[125X [27XStzRemoveGenerator(stz, 1);[127X[104X
    [4X[25Xgap>[125X [27XTietzeBackwardMap(stz);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ] ][128X[104X
  [4X[32X[104X
  
  [1X15.6-3 StzIsomorphism[101X
  
  [33X[1;0Y[29X[2XStzIsomorphism[102X( [3Xstz[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA mapping object.[133X
  
  [33X[0;0YIf  [3Xstz[103X is an [2XStzPresentation[102X ([14X15.3-1[114X) object, then [10XStzIsomorphism[110X returns a
  mapping  object  that  maps the unreduced semigroup of the presentation (see
  [2XUnreducedFpSemigroup[102X  ([14X15.3-5[114X))  to an FpSemigroup object that is defined by
  the  generators  and  relations  of the semigroup presentation at the moment
  this function is ran.[133X
  
  [33X[0;0YIf  a  [10XStzIsomorphism[110X  is  generated  from  [3Xstz[103X, and the presentation [3Xstz[103X is
  further   modified   afterwards   (for   example  by  applying  more  Tietze
  transformations   or   [2XStzSimplifyOnce[102X   ([14X15.7-1[114X)   to  [3Xstz[103X),  then  running
  [10XStzIsomorphism([3Xstz[103X[10X)[110X a second time will produce a different result consistent
  with the [13Xnew[113X generators and relations of [3Xstz[103X.[133X
  
  [33X[0;0YThis   mapping   is   built   from   the   [2XTietzeForwardMap[102X   ([14X15.6-1[114X)   and
  [2XTietzeBackwardMap[102X ([14X15.6-2[114X) attributes from the presentation object, since if
  we know how to map the generators of the respective semigroups, then we know
  how to map any element of that semigroup.[133X
  
  [33X[0;0YThis function is the primary way to obtain the simplified semigroup from the
  presentation  object,  by  applying  [10XRange[110X to the mapping that this function
  returns.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XS := F / [[F.1, F.3 ^ 3], [F.2 ^ 2, F.3 ^ 2]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 11>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(S);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 11>[128X[104X
    [4X[25Xgap>[125X [27XStzRemoveGenerator(stz, "a");[127X[104X
    [4X[25Xgap>[125X [27Xmap := StzIsomorphism(stz);[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 11> -> [128X[104X
    [4X[28X<fp semigroup with 2 generators and 1 relation of length 6>[128X[104X
    [4X[25Xgap>[125X [27XS.1 ^ map;[127X[104X
    [4X[28Xc^3[128X[104X
  [4X[32X[104X
  
  
  [1X15.7 [33X[0;0YAutomatically simplifying a Tietze transformation object[133X[101X
  
  [33X[0;0YIt  is  possible to create a presentation object from an fp semigroup object
  and  attempt  to  manually  reduce  it  by  applying Tietze transformations.
  However,  there  may  be  many  different reductions that can be applied, so
  [10XStzSimplifyOnce[110X  can  be  used  to automatically check a number of different
  possible  reductions  and apply the best one. Then, [10XStzSimplifyPresentation [110X
  repeatedly applies StzSimplifyOnce to the presentation object until it fails
  to reduce the presentation any further. The metric with respect to which the
  [10XIsStzPresentation[110X object is reduced is [2XLength[102X ([14X15.3-6[114X).[133X
  
  [1X15.7-1 StzSimplifyOnce[101X
  
  [33X[1;0Y[29X[2XStzSimplifyOnce[102X( [3Xstz[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YIf  [3Xstz[103X  is  an  [2XStzPresentation[102X  ([14X15.3-1[114X) object, then [10XStzSimplifyOnce[110X will
  check  the  possible reductions in length for a number of different possible
  Tietze  transformations,  and apply the choice which gives the least length.
  If  a  valid transformation was found then the function returns [9Xtrue[109X, and if
  no  transformation  was  performed because none would lead to a reduction in
  length, then the function returns [9Xfalse[109X.[133X
  
  [33X[0;0YThere  are  four different possible transformations that [10XStzSimplifyOnce[110X may
  apply. The function searches for redundant generators and checks if removing
  them  would reduce the overall length of the presentation, it checks whether
  substituting  one side of each relation throughout the rest of the relations
  would  reduce  the length, it checks whether there are any trivial relations
  (of  the  form [3Xw = w[103X for some word [3Xw[103X) or any duplicated relations (relations
  which  are formed from precisely the same words as another relation), and it
  checks  whether  any  frequently  occurring subwords in the relations can be
  replaced  with  a  new generator to reduce the length. For more details, see
  [14X15.5[114X[133X
  
  [33X[0;0YAt  [10XInfoLevel[110X  2  (which  is  the  default value, and which can be set using
  [10XSetInfoLevel(InfoFpSemigroup, 2)[110X), the precise transformations performed are
  printed to the screen.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup with 3 generators and 2 relations of length 19>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 19>[128X[104X
    [4X[25Xgap>[125X [27XStzSimplifyOnce(stz);[127X[104X
    [4X[28X#I  <Removing redundant generator a using relation : 1. a = b^5*c>[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xstz;[127X[104X
    [4X[28X<fp semigroup presentation with 2 generators and 1 relation[128X[104X
    [4X[28X with length 11>[128X[104X
  [4X[32X[104X
  
  [1X15.7-2 StzSimplifyPresentation[101X
  
  [33X[1;0Y[29X[2XStzSimplifyPresentation[102X( [3Xstz[103X ) [32X operation[133X
  
  [33X[0;0YIf  [3Xstz[103X  is  an  [10XStzPresentation[110X  object,  then [10XStzSimplifyPresentation[110X will
  repeatedly  apply  the best of a few possible reductions to [3Xstz[103X until it can
  no longer reduce the length of the presentation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27Xstz := StzPresentation(T);[127X[104X
    [4X[28X<fp semigroup presentation with 3 generators and 2 relations[128X[104X
    [4X[28X with length 19>[128X[104X
    [4X[25Xgap>[125X [27XStzSimplifyPresentation(stz);[127X[104X
    [4X[25Xgap>[125X [27XRelationsOfStzPresentation(stz);[127X[104X
    [4X[28X[ [ [ 1, 1, 1 ], [ 3 ] ], [ [ 3, 3 ], [ 3 ] ] ][128X[104X
  [4X[32X[104X
  
  
  [1X15.8 [33X[0;0YAutomatically simplifying an fp semigroup[133X[101X
  
  [33X[0;0YIt  may  be  the case that, rather than working with a Tietze transformation
  object,  we  want  to  start with an fp semigroup object and obtain the most
  simplified  version  of  that  fp semigroup that [10XStzSimplifyPresentation[110X can
  produce.  In  this  case,  [10XSimplifyFpSemigroup[110X  can  be  applied to obtain a
  mapping  from  its argument to a reduced fp semigroup. If the mapping is not
  of  interest,  [10XSimplifiedFpSemigroup[110X can be used to directly obtain a new fp
  semigroup isomorphic to the first with reduced relations and generators (the
  mapping  is stored as an attribute of the output). With these functions, the
  user never has to consider precisely what Tietze transformations to perform,
  and  never  has  to  worry  about  using  the  [10XStzPresentation[110X object or its
  associated  operations.  They  can  start  with an fp semigroup and obtain a
  simplified fp semigroup.[133X
  
  [1X15.8-1 SimplifyFpSemigroup[101X
  
  [33X[1;0Y[29X[2XSimplifyFpSemigroup[102X( [3XS[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA mapping object.[133X
  
  [33X[0;0YIf [3XS[103X is a finitely presented semigroup, then [10XSimplifyFpSemigroup[110X will return
  a  mapping  object  which will map [3XS[103X to a finitely presented semigroup which
  has   had   its  presentation  simplified.  [10XSimplifyFpSemigroup[110X  creates  an
  [10XStzPresentation[110X  object  [3Xstz[103X  from  [3XS[103X,  which  is  then reduced using Tietze
  transformations  until  the  presentation  cannot  be  reduced in length any
  further.[133X
  
  [33X[0;0Y[10XSimplifyFpSemigroup[110X applies the function [2XStzSimplifyPresentation[102X ([14X15.7-2[114X) to
  [3Xstz[103X,  which  repeatedly  checks  whether  a  number  of  different  possible
  transformations will cause a reduction in length, and if so applies the best
  one.  This  loop continues until no transformations cause any reductions, in
  which  case  the  mapping  is returned. The newly reduced FpSemigroup can be
  accessed   either   by   taking   the   range  of  the  mapping  or  calling
  [10XSimplifiedFpSemigroup[110X, which first runs [10XSimplifyFpSemigroup[110X and then returns
  the range of the mapping with the mapping held as an attribute.[133X
  
  [33X[0;0YFor more information on how the mapping is created and used, go to [14X15.6[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27Xmap := SimplifyFpSemigroup(T);;[127X[104X
    [4X[28X#I  Applying StzSimplifyPresentation...[128X[104X
    [4X[28X#I  StzSimplifyPresentation is verbose by default. Use SetInfoLevel(InfoFpSemigroup, 1) to hide[128X[104X
    [4X[28X#I  output while maintaining ability to use StzPrintRelations, StzPrintGenerators, etc.[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 2 relations with length 19>[128X[104X
    [4X[28X#I  <Removing redundant generator a using relation : 1. a = b^5*c>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 2 generators and 1 relation with length 11>[128X[104X
    [4X[28X#I  <Creating new generator to replace instances of word: b^3>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 2 relations with length 10>[128X[104X
    [4X[25Xgap>[125X [27XIsMapping(map);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XT.1;[127X[104X
    [4X[28Xa[128X[104X
    [4X[25Xgap>[125X [27XT.1 ^ map;[127X[104X
    [4X[28Xb^5*c[128X[104X
    [4X[25Xgap>[125X [27XRelationsOfFpSemigroup(Range(map));[127X[104X
    [4X[28X[ [ b^3, d ], [ d^2, d ] ][128X[104X
  [4X[32X[104X
  
  [1X15.8-2 SimplifiedFpSemigroup[101X
  
  [33X[1;0Y[29X[2XSimplifiedFpSemigroup[102X( [3XS[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA finitely presented semigroup.[133X
  
  [33X[0;0YIf  [3XS[103X  is an fp semigroup object ([2XIsFpSemigroup[102X ([14XReference: IsFpSemigroup[114X)),
  then  [10XSimplifiedFpSemigroup[110X  will  return  an  FpSemigroup object [3XT[103X which is
  isomorphic   to   [3XS[103X   which   has  been  reduced  to  minimise  its  length.
  [10XSimplifiedFpSemigroup[110X  applies  [2XSimplifyFpSemigroup[102X ([14X15.8-1[114X) and assigns the
  [10XRange[110X  of  the  isomorphism  object  which  is  returned  to  [3XT[103X,  adding the
  isomorphism  to  [3XT[103X as an attribute. In this way, while [3XT[103X is a completely new
  FpSemigroup  object,  words  in  [3XS[103X can be mapped to [3XT[103X using the map obtained
  from the attribute [2XFpTietzeIsomorphism[102X ([14X15.8-4[114X).[133X
  
  [33X[0;0YFor  more  information  on  the mapping between the semigroups and how it is
  created, see [14X15.6[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");;[127X[104X
    [4X[25Xgap>[125X [27XS := F / [[F.1 ^ 4, F.1], [F.1, F.1 ^ 44], [F.1 ^ 8, F.2 * F.3]];;[127X[104X
    [4X[25Xgap>[125X [27XT := SimplifiedFpSemigroup(S);;[127X[104X
    [4X[28X#I  Applying StzSimplifyPresentation...[128X[104X
    [4X[28X#I  StzSimplifyPresentation is verbose by default. Use SetInfoLevel(InfoFpSemigroup, 1) to hide[128X[104X
    [4X[28X#I  output while maintaining ability to use StzPrintRelations, StzPrintGenerators, etc.[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 3 relations with length 63>[128X[104X
    [4X[28X#I  <Replacing all instances in other relations of relation: 1. a^4 = a>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 3 relations with length 24>[128X[104X
    [4X[28X#I  <Replacing all instances in other relations of relation: 1. a^4 = a>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 3 relations with length 18>[128X[104X
    [4X[28X#I  <Replacing all instances in other relations of relation: 1. a^4 = a>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 3 relations with length 15>[128X[104X
    [4X[28X#I  <Replacing all instances in other relations of relation: 3. a = a^2>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 3 relations with length 12>[128X[104X
    [4X[28X#I  <Removing duplicate relation: 1. a = a^2>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 2 relations with length 9>[128X[104X
    [4X[28X#I  <Removing redundant generator a using relation : 2. a = b*c>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 2 generators and 1 relation with length 8>[128X[104X
    [4X[25Xgap>[125X [27Xmap := FpTietzeIsomorphism(T);;[127X[104X
    [4X[25Xgap>[125X [27XS.1 ^ map;[127X[104X
    [4X[28Xb*c[128X[104X
    [4X[25Xgap>[125X [27XS.1 ^ map = T.1 * T.2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinvmap := InverseGeneralMapping(map);;[127X[104X
    [4X[25Xgap>[125X [27XT.1 ^ invmap = S.2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XT.1 = S.2;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X15.8-3 UnreducedFpSemigroup[101X
  
  [33X[1;0Y[29X[2XUnreducedFpSemigroup[102X( [3XS[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Y[3XT[103X, an fp semigroup object.[133X
  
  [33X[0;0YIf  [3XS[103X  is  an  fp  semigroup  object  that has been obtained through calling
  [2XSimplifiedFpSemigroup[102X    ([14X15.8-2[114X)    on    some    fp   semigroup   [3XT[103X   then
  [10XUnreducedFpSemigroup[110X   returns   the   original   semigroup   object  before
  simplification.  These  are  unrelated semigroup objects, except that [3XS[103X will
  have  a  [2XFpTietzeIsomorphism[102X  ([14X15.8-4[114X)  attribute that returns an isomorphic
  mapping from [3XT[103X to [3XS[103X.[133X
  
  [33X[0;0YIf  [2XSimplifyFpSemigroup[102X  ([14X15.8-1[114X) has been called on an fp semigroup [3XT[103X, then
  [10XUnreducedFpSemigroup[110X  can  be  used on the [10XRange[110X of the resultant mapping to
  obtain the domain.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XS := SimplifiedFpSemigroup(T);[127X[104X
    [4X[28X#I  Applying StzSimplifyPresentation...[128X[104X
    [4X[28X#I  StzSimplifyPresentation is verbose by default. Use SetInfoLevel(InfoFpSemigroup, 1) to hide[128X[104X
    [4X[28X#I  output while maintaining ability to use StzPrintRelations, StzPrintGenerators, etc.[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 2 relations with length 19>[128X[104X
    [4X[28X#I  <Removing redundant generator a using relation : 1. a = b^5*c>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 2 generators and 1 relation with length 11>[128X[104X
    [4X[28X#I  <Creating new generator to replace instances of word: b^3>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 2 relations with length 10>[128X[104X
    [4X[28X<fp semigroup on the generators [ b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XUnreducedFpSemigroup(S) = T;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X15.8-4 FpTietzeIsomorphism[101X
  
  [33X[1;0Y[29X[2XFpTietzeIsomorphism[102X( [3XS[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA mapping object.[133X
  
  [33X[0;0YIf  [3XS[103X  is  an  fp  semigroup  object  that has been obtained through calling
  [2XSimplifiedFpSemigroup[102X    ([14X15.8-2[114X)    on    some   fp   semigroup   [3XT[103X,   then
  [10XFpTietzeIsomorphism[110X  returns  an  isomorphism  from  [3XT[103X  to [3XS[103X. Simplification
  produces  an fp semigroup isomorphic to the original fp semigroup, and these
  two  fp  semigroup  objects can interact with each other through the mapping
  given by this function.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XT := F / [[F.1, F.2 ^ 5 * F.3],[127X[104X
    [4X[25X>[125X [27X             [F.2 ^ 6, F.2 ^ 3]];[127X[104X
    [4X[28X<fp semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XS := SimplifiedFpSemigroup(T);[127X[104X
    [4X[28X#I  Applying StzSimplifyPresentation...[128X[104X
    [4X[28X#I  StzSimplifyPresentation is verbose by default. Use SetInfoLevel(InfoFpSemigroup, 1) to hide[128X[104X
    [4X[28X#I  output while maintaining ability to use StzPrintRelations, StzPrintGenerators, etc.[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 2 relations with length 19>[128X[104X
    [4X[28X#I  <Removing redundant generator a using relation : 1. a = b^5*c>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 2 generators and 1 relation with length 11>[128X[104X
    [4X[28X#I  <Creating new generator to replace instances of word: b^3>[128X[104X
    [4X[28X#I  Current: <fp semigroup presentation with 3 generators and 2 relations with length 10>[128X[104X
    [4X[28X<fp semigroup on the generators [ b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XT.2;[127X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XS.1;[127X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XT.2 = S.1;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xmap := FpTietzeIsomorphism(S);;[127X[104X
    [4X[25Xgap>[125X [27XT.2 ^ map = S.1;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
