  
  [1X6 [33X[0;0YFree inverse semigroups and free bands[133X[101X
  
  [33X[0;0YThis  chapter  describes  the  functions in [5XSemigroups[105X for dealing with free
  inverse semigroups and free bands. This part of the manual and the functions
  described herein were written by Julius Jonušas.[133X
  
  
  [1X6.1 [33X[0;0YFree inverse semigroups[133X[101X
  
  [33X[0;0Y[22XF[122X  is  a  [13Xfree  inverse  semigroup[113X  on  a non-empty set [22XX[122X if [22XF[122X is an inverse
  semigroup  with  a map [22Xf[122X from [22XF[122X to [22XX[122X such that for every inverse semigroup [22XS[122X
  and  a  map  [22Xg[122X from [22XX[122X to [22XS[122X there exists a unique homomorphism [22Xg'[122X from [22XF[122X to [22XS[122X
  such  that  [22Xfg'  =  g[122X.  Moreover,  by  the universal property, every inverse
  semigroup can be expressed as a quotient of a free inverse semigroup.[133X
  
  [33X[0;0YThe internal representation of an element of a free inverse semigroup uses a
  Munn  tree.  A  [13XMunn  tree[113X  is  a directed tree with distinguished start and
  terminal  vertices and where the edges are labeled by generators so that two
  edges  labeled by the same generator are only incident to the same vertex if
  one  of the edges is coming in and the other is leaving the vertex. For more
  information  regarding  free inverse semigroups and the Munn representations
  see  Section  5.10  of  [How95]. See also [14X'Reference: Inverse semigroups and
  monoids'[114X,  [14X'Reference:  Partial  permutations'[114X  and [14X'Reference: Free Groups,
  Monoids and Semigroups'[114X.[133X
  
  [33X[0;0YAn  element  of  a  free inverse semigroup in [5XSemigroups[105X is be displayed, by
  default,  as  a  shortest  word corresponding to the element. However, there
  might  be  more than one word of the minimum length. For example, if [22Xx[122X and [22Xy[122X
  are generators of a free inverse semigroups, then[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yxyy^{-1}xx^{-1}x^{-1} = xxx^{-1}yy^{-1}x^{-1}.[133X [124X[133X
  
  
  [33X[0;0YSee  [2XMinimalWord[102X  ([14X6.3-2[114X) Therefore we provide a another method for printing
  elements  of  a  free inverse semigroup: a unique canonical form. Suppose an
  element  of  a  free inverse semigroup is given as a Munn tree. Let [22XL[122X be the
  set  of  words  corresponding to the shortest paths from the start vertex to
  the  leaves  of the tree. Also let [22Xw[122X be a word corresponding to the shortest
  path  from  start  to terminal vertices. The word [22Xvv^-1[122X is an idempotent for
  every  [22Xv[122X in [22XL[122X. The canonical form is given by multiplying these idempotents,
  in  shortlex order, and then postmultiplying by [22Xw[122X. For example, consider the
  word  [22Xxyy^-1xx^-1x^-1[122X  again.  The  words  corresponding to the paths to the
  leaves  are  in  this case [22Xxx[122X and [22Xxy[122X. And [22Xw[122X is an empty word since start and
  terminal vertices are the same. Therefore, the canonical form is[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yxxx^{-1}x^{-1}xyy^{-1}x^{-1}.[133X [124X[133X
  
  
  [33X[0;0YSee [2XCanonicalForm[102X ([14X6.3-1[114X).[133X
  
  [1X6.1-1 FreeInverseSemigroup[101X
  
  [29X[2XFreeInverseSemigroup[102X( [3Xrank[103X[, [3Xname[103X] ) [32X function
  [29X[2XFreeInverseSemigroup[102X( [3Xname1[103X, [3Xname2[103X, [3X...[103X ) [32X function
  [29X[2XFreeInverseSemigroup[102X( [3Xnames[103X ) [32X function
  [6XReturns:[106X  [33X[0;10YA free inverse semigroup.[133X
  
  [33X[0;0YReturns  a  free  inverse  semigroup  on  [3Xrank[103X  generators,  where [3Xrank[103X is a
  positive  integer. If [3Xrank[103X is not specified, the number of [3Xnames[103X is used. If
  [10XS[110X  is  a free inverse semigroup, then the generators can be accessed by [10XS.1[110X,
  [10XS.2[110X and so on.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FreeInverseSemigroup(7);[127X[104X
    [4X[28X<free inverse semigroup on the generators [128X[104X
    [4X[28X[ x1, x2, x3, x4, x5, x6, x7 ]>[128X[104X
    [4X[25Xgap>[125X [27XS := FreeInverseSemigroup(7,"s");[127X[104X
    [4X[28X<free inverse semigroup on the generators [128X[104X
    [4X[28X[ s1, s2, s3, s4, s5, s6, s7 ]>[128X[104X
    [4X[25Xgap>[125X [27XS := FreeInverseSemigroup("a", "b", "c");[127X[104X
    [4X[28X<free inverse semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XS := FreeInverseSemigroup(["a", "b", "c"]);[127X[104X
    [4X[28X<free inverse semigroup on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XS.1;[127X[104X
    [4X[28Xa[128X[104X
    [4X[25Xgap>[125X [27XS.2;[127X[104X
    [4X[28Xb[128X[104X
  [4X[32X[104X
  
  [1X6.1-2 IsFreeInverseSemigroupCategory[101X
  
  [29X[2XIsFreeInverseSemigroupCategory[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YEvery  free inverse semigroup in [5XGAP[105X created by [2XFreeInverseSemigroup[102X ([14X6.1-1[114X)
  belongs  to the category [10XIsFreeInverseSemigroup[110X. Basic operations for a free
  inverse     semigroup    are:    [2XGeneratorsOfInverseSemigroup[102X    ([14XReference:
  GeneratorsOfInverseSemigroup[114X)    and    [2XGeneratorsOfSemigroup[102X    ([14XReference:
  GeneratorsOfSemigroup[114X).  Elements  of a free inverse semigroup belong to the
  category [2XIsFreeInverseSemigroupElement[102X ([14X6.1-4[114X).[133X
  
  [1X6.1-3 IsFreeInverseSemigroup[101X
  
  [29X[2XIsFreeInverseSemigroup[102X( [3XS[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YAttempts  to  determine  whether  the  given semigroup [3X S [103X is a free inverse
  semigroup.[133X
  
  [1X6.1-4 IsFreeInverseSemigroupElement[101X
  
  [29X[2XIsFreeInverseSemigroupElement[102X[32X Category
  
  [33X[0;0YEvery   element  of  a  free  inverse  semigroup  belongs  to  the  category
  [10XIsFreeInverseSemigroupElement[110X.[133X
  
  
  [1X6.2 [33X[0;0YDisplaying free inverse semigroup elements[133X[101X
  
  [33X[0;0YThere  is  a  way to change how [5XGAP[105X displays free inverse semigroup elements
  using    the   user   preference   [10XFreeInverseSemigroupElementDisplay[110X.   See
  [2XUserPreference[102X  ([14XReference:  UserPreference[114X) for more information about user
  preferences.[133X
  
  [33X[0;0YThere are two possible values for [10XFreeInverseSemigroupElementDisplay[110X:[133X
  
  [8Xminimal [108X
        [33X[0;6YWith   this   option  selected,  [5XGAP[105X  will  display  a  shortest  word
        corresponding  to  the  free  inverse semigroup element. However, this
        shortest word is not unique. This is a default setting.[133X
  
  [8Xcanonical[108X
        [33X[0;6YWith  this  option selected, [5XGAP[105X will display a free inverse semigroup
        element in the canonical form.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetUserPreference("semigroups", "FreeInverseSemigroupElementDisplay", "minimal");[127X[104X
    [4X[25Xgap>[125X [27XS:=FreeInverseSemigroup(2);[127X[104X
    [4X[28X<free inverse semigroup on the generators [ x1, x2 ]>[128X[104X
    [4X[25Xgap>[125X [27XS.1 * S.2;[127X[104X
    [4X[28Xx1*x2[128X[104X
    [4X[25Xgap>[125X [27XSetUserPreference("semigroups", "FreeInverseSemigroupElementDisplay", "canonical");[127X[104X
    [4X[25Xgap>[125X [27XS.1 * S.2;[127X[104X
    [4X[28Xx1x2x2^-1x1^-1x1x2[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YOperators and operations for free inverse semigroup elements[133X[101X
  
  [8X[10X[3Xw[103X[10X ^ -1[110X[108X
        [33X[0;6Yreturns the semigroup inverse of the free inverse semigroup element [3Xw[103X.[133X
  
  [8X[10X[3Xu[103X[10X * [3Xv[103X[10X[110X[108X
        [33X[0;6Yreturns the product of two free inverse semigroup elements [3Xu[103X and [3Xv[103X.[133X
  
  [8X[10X[3Xu[103X[10X = [3Xv[103X[10X [110X[108X
        [33X[0;6Ychecks  if two free inverse semigroup elements are equal, by comparing
        their canonical forms.[133X
  
  [1X6.3-1 CanonicalForm[101X
  
  [29X[2XCanonicalForm[102X( [3Xw[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA string.[133X
  
  [33X[0;0YEvery  element of a free inverse semigroup has a unique canonical form. If [3Xw[103X
  is  such an element, then [10XCanonicalForm[110X returns the canonical form of [3Xw[103X as a
  string.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FreeInverseSemigroup(3);[127X[104X
    [4X[28X<free inverse semigroup on the generators [ x1, x2, x3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xx := S.1; y := S.2;[127X[104X
    [4X[28Xx1[128X[104X
    [4X[28Xx2[128X[104X
    [4X[25Xgap>[125X [27XCanonicalForm(x^3*y^3);[127X[104X
    [4X[28X"x1x1x1x2x2x2x2^-1x2^-1x2^-1x1^-1x1^-1x1^-1x1x1x1x2x2x2"[128X[104X
  [4X[32X[104X
  
  [1X6.3-2 MinimalWord[101X
  
  [29X[2XMinimalWord[102X( [3Xw[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA string.[133X
  
  [33X[0;0YFor  an  element [3Xw[103X of a free inverse semigroup [10XS[110X, [10XMinimalWord[110X returns a word
  of minimal length equal to [3Xw[103X in [10XS[110X as a string.[133X
  
  [33X[0;0YNote that there maybe more than one word of minimal length which is equal to
  [3Xw[103X in [10XS[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FreeInverseSemigroup(3);[127X[104X
    [4X[28X<free inverse semigroup on the generators [ x1, x2, x3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xx := S.1;[127X[104X
    [4X[28Xx1[128X[104X
    [4X[25Xgap>[125X [27Xy := S.2;[127X[104X
    [4X[28Xx2[128X[104X
    [4X[25Xgap>[125X [27XMinimalWord(x^3 * y^3);[127X[104X
    [4X[28X"x1*x1*x1*x2*x2*x2"[128X[104X
  [4X[32X[104X
  
  
  [1X6.4 [33X[0;0YFree bands[133X[101X
  
  [33X[0;0YA  semigroup [22XB[122X is a [13Xfree band[113X on a non-empty set [22XX[122X if [22XB[122X is a band with a map
  [22Xf[122X  from  [22XB[122X to [22XX[122X such that for every band [22XS[122X and every map [22Xg[122X from [22XX[122X to [22XB[122X there
  exists a unique homomorphism [22Xg'[122X from [22XB[122X to [22XS[122X such that [22Xfg' = g[122X. The free band
  on a set [22XX[122X is unique up to isomorphism. Moreover, by the universal property,
  every band can be expressed as a quotient of a free band.[133X
  
  [33X[0;0YFor an alternative description of a free band. Suppose that [22XX[122X is a non-empty
  set  and  [22XX^+[122X  a  free  semigroup  on [22XX[122X. Also suppose that [22Xb[122X is the smallest
  congurance on [22XX^+[122X containing the set[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\{ (w^2, w) : w \in X^+ \}.[133X [124X[133X
  
  
  [33X[0;0YThen  the  free  band  on  [22XX[122X  is isomorphic to the quotient of [22XX^+[122X by [22Xb[122X. See
  Section 4.5 of [How95] for more information on free bands.[133X
  
  [1X6.4-1 FreeBand[101X
  
  [29X[2XFreeBand[102X( [3Xrank[103X[, [3Xname[103X] ) [32X function
  [29X[2XFreeBand[102X( [3Xname1[103X, [3Xname2[103X, [3X...[103X ) [32X function
  [29X[2XFreeBand[102X( [3Xnames[103X ) [32X function
  [6XReturns:[106X  [33X[0;10YA free band.[133X
  
  [33X[0;0YReturns a free band on [3Xrank[103X generators, for a positive integer [3Xrank[103X. If [3Xrank[103X
  is  not  specified,  the number of [3Xnames[103X is used. The resulting semigroup is
  always finite.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFreeBand(6);[127X[104X
    [4X[28X<free band on the generators [ x1, x2, x3, x4, x5, x6 ]>[128X[104X
    [4X[25Xgap>[125X [27XFreeBand(6, "b");[127X[104X
    [4X[28X<free band on the generators [ b1, b2, b3, b4, b5, b6 ]>[128X[104X
    [4X[25Xgap>[125X [27XFreeBand("a", "b", "c");[127X[104X
    [4X[28X<free band on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XFreeBand("a", "b", "c");[127X[104X
    [4X[28X<free band on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27Xs := FreeBand(["a", "b", "c"]);[127X[104X
    [4X[28X<free band on the generators [ a, b, c ]>[128X[104X
    [4X[25Xgap>[125X [27XSize(s);[127X[104X
    [4X[28X159[128X[104X
    [4X[25Xgap>[125X [27Xgens := Generators(s);[127X[104X
    [4X[28X[ a, b, c ][128X[104X
    [4X[25Xgap>[125X [27Xa := gens[1];; b := gens[2];;[127X[104X
    [4X[25Xgap>[125X [27Xa * b;[127X[104X
    [4X[28Xab[128X[104X
  [4X[32X[104X
  
  [1X6.4-2 IsFreeBandCategory[101X
  
  [29X[2XIsFreeBandCategory[102X[32X Category
  
  [33X[0;0Y[10XIsFreeBandCategory[110X  is  the  category  of  semigroups created using [2XFreeBand[102X
  ([14X6.4-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsFreeBandCategory(FreeBand(3));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsFreeBand(SymmetricGroup(6));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.4-3 IsFreeBand[101X
  
  [29X[2XIsFreeBand[102X( [3XS[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0Y[10XIsFreeBand[110X returns [9Xtrue[109X if the given semigroup [3XS[103X is a free band.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsFreeBand(FreeBand(3));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsFreeBand(SymmetricGroup(6));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsFreeBand(FullTransformationMonoid(7));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.4-4 IsFreeBandElement[101X
  
  [29X[2XIsFreeBandElement[102X[32X Category
  
  [33X[0;0Y[10XIsFreeBandElement[110X is a [10XCategory[110X containing the elements of a free band.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsFreeBandElement(Generators(FreeBand(4))[1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsFreeBandElement(Transformation([1,3,4,1]));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsFreeBandElement((1,2,3,4));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.4-5 IsFreeBandSubsemigroup[101X
  
  [29X[2XIsFreeBandSubsemigroup[102X[32X filter
  
  [33X[0;0Y[10XIsFreeBandSubsemigroup[110X     is     a     synonym    for    [10XIsSemigroup[110X    and
  [10XIsFreeBandElementCollection[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FreeBand(2);[127X[104X
    [4X[28X<free band on the generators [ x1, x2 ]>[128X[104X
    [4X[25Xgap>[125X [27Xx := Generators(S)[1];[127X[104X
    [4X[28Xx1[128X[104X
    [4X[25Xgap>[125X [27Xy := Generators(S)[2];[127X[104X
    [4X[28Xx2[128X[104X
    [4X[25Xgap>[125X [27Xnew := Semigroup([x*y, x]);[127X[104X
    [4X[28X<semigroup with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsFreeBand(new);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsFreeBandSubsemigroup(new);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X6.5 [33X[0;0YOperators and operations for free band elements[133X[101X
  
  [8X[10X[3Xu[103X[10X * [3Xv[103X[10X[110X[108X
        [33X[0;6Yreturns the product of two free band elements [3Xu[103X and [3Xv[103X.[133X
  
  [8X[10X[3Xu[103X[10X = [3Xv[103X[10X [110X[108X
        [33X[0;6Ychecks if two free band elements are equal.[133X
  
  [8X[10X[3Xu[103X[10X < [3Xv[103X[10X [110X[108X
        [33X[0;6Ycompares  the  sizes  of the internal representations of two free band
        elements.[133X
  
  [1X6.5-1 GreensDClassOfElement[101X
  
  [29X[2XGreensDClassOfElement[102X( [3Xs[103X, [3Xx[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA Green's D-class[133X
  
  [33X[0;0YLet  [3X  S  [103X  be a free band. Two elements of [3X S [103X are [13XD[113X-related if and only if
  they  have  the  same  content  i.e.  the set of generators appearing in any
  factorization  of the elements. Therefore, a [13XD[113X-class of a free band element [3X
  x [103X is the set of elements of [3X S [103X which have the same content as [3X x [103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FreeBand(3, "b");[127X[104X
    [4X[28X<free band on the generators [ b1, b2, b3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xx := Generators(S)[1] * Generators(S)[2];[127X[104X
    [4X[28Xb1b2[128X[104X
    [4X[25Xgap>[125X [27XD := GreensDClassOfElement(S, x);[127X[104X
    [4X[28X{b1b2}[128X[104X
    [4X[25Xgap>[125X [27XIsGreensDClass(D);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
