  
  [1X16 [33X[0;0YAttributes of inverse semigroups[133X[101X
  
  [33X[0;0YIn  this  chapter  we  describe the attributes which are specific to inverse
  semigroups that can be determined using [5XSemigroups[105X.[133X
  
  [33X[0;0YThe functions[133X
  
  [30X    [33X[0;6Y[2XIsJoinIrreducible[102X ([14X17.2-5[114X)[133X
  
  [30X    [33X[0;6Y[2XIsMajorantlyClosed[102X ([14X17.2-6[114X)[133X
  
  [30X    [33X[0;6Y[2XJoinIrreducibleDClasses[102X ([14X16.1-2[114X)[133X
  
  [30X    [33X[0;6Y[2XMajorantClosure[102X ([14X16.1-3[114X)[133X
  
  [30X    [33X[0;6Y[2XMinorants[102X ([14X16.1-4[114X)[133X
  
  [30X    [33X[0;6Y[2XRightCosetsOfInverseSemigroup[102X ([14X16.1-6[114X)[133X
  
  [30X    [33X[0;6Y[2XSmallerDegreePartialPermRepresentation[102X ([14X16.1-8[114X)[133X
  
  [30X    [33X[0;6Y[2XVagnerPrestonRepresentation[102X ([14X16.1-9[114X)[133X
  
  [33X[0;0Ywere written by Wilf Wilson and Robert Hancock.[133X
  
  [33X[0;0YThe  function  [2XCharacterTableOfInverseSemigroup[102X  ([14X16.1-10[114X)  was  written  by
  Jhevon Smith and Ben Steinberg.[133X
  
  
  [1X16.1 [33X[0;0YAttributes of inverse semigroups[133X[101X
  
  [1X16.1-1 NaturalLeqInverseSemigroup[101X
  
  [29X[2XNaturalLeqInverseSemigroup[102X( [3XS[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YAn function.[133X
  
  [33X[0;0Y[10XNaturalLeqInverseSemigroup[110X  returns a function that, when given two elements
  [10Xx,  y[110X of the inverse semigroup [3XS[103X, returns [9Xtrue[109X if [10Xx[110X is less than or equal to
  [10Xy[110X in the natural partial order on [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := Monoid(Transformation([1, 3, 4, 4]), [127X[104X
    [4X[25X>[125X [27X               Transformation([1, 4, 2, 4]));[127X[104X
    [4X[28X<transformation monoid of degree 4 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsInverseSemigroup(S);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XNaturalPartialOrder(S);[127X[104X
    [4X[28X[ [ 2, 5, 6 ], [ 6 ], [ 6 ], [ 6 ], [ 6 ], [  ] ][128X[104X
  [4X[32X[104X
  
  [1X16.1-2 JoinIrreducibleDClasses[101X
  
  [29X[2XJoinIrreducibleDClasses[102X( [3XS[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA list of [13XD[113X-classes.[133X
  
  [33X[0;0Y[10XJoinIrreducibleDClasses[110X  returns a list of the join irreducible [13XD[113X-classes of
  the  inverse  semigroup of partial permutations, block bijections or partial
  permutation bipartitions [3XS[103X.[133X
  
  [33X[0;0YA  [13Xjoin  irreducible  [13XD[113X-class[113X  is a [13XD[113X-class containing only join irreducible
  elements.  See  [2XIsJoinIrreducible[102X  ([14X17.2-5[114X).  If a [13XD[113X-class contains one join
  irreducible  element,  then  all  of  the  elements  in the [13XD[113X-class are join
  irreducible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SymmetricInverseSemigroup(3);[127X[104X
    [4X[28X<symmetric inverse monoid of degree 3>[128X[104X
    [4X[25Xgap>[125X [27XJoinIrreducibleDClasses(S);[127X[104X
    [4X[28X[ <Green's D-class: <identity partial perm on [ 2 ]>> ][128X[104X
    [4X[25Xgap>[125X [27XT := InverseSemigroup([127X[104X
    [4X[25X>[125X [27XPartialPerm([1, 2, 3, 4], [1, 2, 4, 3]), [127X[104X
    [4X[25X>[125X [27XPartialPerm([1], [1]), PartialPerm([2], [2]));[127X[104X
    [4X[28X<inverse partial perm semigroup of rank 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XJoinIrreducibleDClasses(T);[127X[104X
    [4X[28X[ <Green's D-class: <identity partial perm on [ 1, 2, 3, 4 ]>>, [128X[104X
    [4X[28X  <Green's D-class: <identity partial perm on [ 1 ]>>, [128X[104X
    [4X[28X  <Green's D-class: <identity partial perm on [ 2 ]>> ][128X[104X
    [4X[25Xgap>[125X [27XD := DualSymmetricInverseSemigroup(3);[127X[104X
    [4X[28X<inverse bipartition monoid of degree 3 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XJoinIrreducibleDClasses(D);[127X[104X
    [4X[28X[ <Green's D-class: <block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ]>> ][128X[104X
  [4X[32X[104X
  
  [1X16.1-3 MajorantClosure[101X
  
  [29X[2XMajorantClosure[102X( [3XS[103X, [3XT[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA majorantly closed list of elements.[133X
  
  [33X[0;0Y[10XMajorantClosure[110X  returns  a majorantly closed subset of an inverse semigroup
  of   partial   permutations,   block   bijections   or  partial  permutation
  bipartitions, [3XS[103X, as a list. See [2XIsMajorantlyClosed[102X ([14X17.2-6[114X).[133X
  
  [33X[0;0YThe result contains all elements of [3XS[103X which are greater than or equal to any
  element    of    [3XT[103X    (with   respect   to   the   natural   partial   order
  [2XNaturalLeqPartialPerm[102X  ([14XReference:  NaturalLeqPartialPerm[114X)).  In particular,
  the result is a superset of [3XT[103X.[133X
  
  [33X[0;0YNote that [3XT[103X can be a subset of [3XS[103X or a subsemigroup of [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SymmetricInverseSemigroup(4);[127X[104X
    [4X[28X<symmetric inverse monoid of degree 4>[128X[104X
    [4X[25Xgap>[125X [27XT := [PartialPerm([1, 0, 3, 0])];[127X[104X
    [4X[28X[ <identity partial perm on [ 1, 3 ]> ][128X[104X
    [4X[25Xgap>[125X [27XU := MajorantClosure(S,T);[127X[104X
    [4X[28X[ <identity partial perm on [ 1, 3 ]>, [128X[104X
    [4X[28X  <identity partial perm on [ 1, 2, 3 ]>, [2,4](1)(3), [4,2](1)(3), [128X[104X
    [4X[28X  <identity partial perm on [ 1, 3, 4 ]>, [128X[104X
    [4X[28X  <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2,4)(3) ][128X[104X
    [4X[25Xgap>[125X [27XB := InverseSemigroup([[127X[104X
    [4X[25X>[125X [27X Bipartition([[1, -2], [2, -1], [3, -3], [4, 5, -4, -5]]),[127X[104X
    [4X[25X>[125X [27X Bipartition([[1, -3], [2, -4], [3, -2], [4, -1], [5, -5]])]);;[127X[104X
    [4X[25Xgap>[125X [27XT := [Bipartition([[1, -2], [2, 3, 5, -1, -3, -5], [4, -4]]),[127X[104X
    [4X[25X>[125X [27X Bipartition([[1, -4], [2, 3, 5, -1, -3, -5], [4, -2]])];;[127X[104X
    [4X[25Xgap>[125X [27XIsMajorantlyClosed(B,T);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XMajorantClosure(B,T);[127X[104X
    [4X[28X[ <block bijection: [ 1, -2 ], [ 2, 3, 5, -1, -3, -5 ], [ 4, -4 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, -4 ], [ 2, 3, 5, -1, -3, -5 ], [ 4, -2 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, -2 ], [ 2, 5, -1, -5 ], [ 3, -3 ], [ 4, -4 ]>[128X[104X
    [4X[28X    , <block bijection: [ 1, -2 ], [ 2, -1 ], [ 3, 5, -3, -5 ], [128X[104X
    [4X[28X     [ 4, -4 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, -4 ], [ 2, 5, -3, -5 ], [ 3, -1 ], [ 4, -2 ]>[128X[104X
    [4X[28X    , <block bijection: [ 1, -4 ], [ 2, -3 ], [ 3, 5, -1, -5 ], [128X[104X
    [4X[28X     [ 4, -2 ]>, <block bijection: [ 1, -4 ], [ 2, -3 ], [ 3, -1 ], [128X[104X
    [4X[28X     [ 4, -2 ], [ 5, -5 ]> ][128X[104X
    [4X[25Xgap>[125X [27XIsMajorantlyClosed(B, last);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X16.1-4 Minorants[101X
  
  [29X[2XMinorants[102X( [3XS[103X, [3Xf[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA list of elements.[133X
  
  [33X[0;0Y[10XMinorants[110X   takes  an  element  [3Xf[103X  from  an  inverse  semigroup  of  partial
  permutations,  block  bijections  or partial permutation bipartitions [3XS[103X, and
  returns a list of the minorants of [3Xf[103X in [3XS[103X.[133X
  
  [33X[0;0YA  [13Xminorant[113X  of  [3Xf[103X  is  an element of [3XS[103X which is strictly less than [3Xf[103X in the
  natural   partial   order   of   [3XS[103X.  See  [2XNaturalLeqPartialPerm[102X  ([14XReference:
  NaturalLeqPartialPerm[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SymmetricInverseSemigroup(3);[127X[104X
    [4X[28X<symmetric inverse monoid of degree 3>[128X[104X
    [4X[25Xgap>[125X [27Xx := Elements(S)[13];[127X[104X
    [4X[28X[1,3](2)[128X[104X
    [4X[25Xgap>[125X [27XMinorants(S, x);[127X[104X
    [4X[28X[ <empty partial perm>, [1,3], <identity partial perm on [ 2 ]> ][128X[104X
    [4X[25Xgap>[125X [27Xx := PartialPerm([3, 2, 4, 0]);[127X[104X
    [4X[28X[1,3,4](2)[128X[104X
    [4X[25Xgap>[125X [27XS := InverseSemigroup(x);[127X[104X
    [4X[28X<inverse partial perm semigroup of rank 4 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XMinorants(S, x);[127X[104X
    [4X[28X[ <identity partial perm on [ 2 ]>, [1,3](2), [3,4](2) ][128X[104X
  [4X[32X[104X
  
  [1X16.1-5 PrimitiveIdempotents[101X
  
  [29X[2XPrimitiveIdempotents[102X( [3XS[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA list of idempotent partial permutations.[133X
  
  [33X[0;0YAn  idempotent  in an inverse semigroup [3XS[103X is [13Xprimitive[113X if it is non-zero and
  minimal    with    respect    to    the    [2XNaturalPartialOrder[102X   ([14XReference:
  NaturalPartialOrder[114X)   on   [3XS[103X.  [10XPrimitiveIdempotents[110X  returns  the  list  of
  primitive idempotents in the inverse semigroup of partial permutations [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := InverseMonoid([127X[104X
    [4X[25X>[125X [27XPartialPerm([1], [4]),[127X[104X
    [4X[25X>[125X [27XPartialPerm([1, 2, 3], [2, 1, 3]),[127X[104X
    [4X[25X>[125X [27XPartialPerm([1, 2, 3], [3, 1, 2]));;[127X[104X
    [4X[25Xgap>[125X [27XMultiplicativeZero(S);[127X[104X
    [4X[28X<empty partial perm>[128X[104X
    [4X[25Xgap>[125X [27XSet(PrimitiveIdempotents(S));[127X[104X
    [4X[28X[ <identity partial perm on [ 1 ]>, <identity partial perm on [ 2 ]>, [128X[104X
    [4X[28X  <identity partial perm on [ 3 ]>, <identity partial perm on [ 4 ]> ][128X[104X
    [4X[25Xgap>[125X [27XS := DualSymmetricInverseMonoid(4);[127X[104X
    [4X[28X<inverse bipartition monoid of degree 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSet(PrimitiveIdempotents(S));[127X[104X
    [4X[28X[ <block bijection: [ 1, 2, 3, -1, -2, -3 ], [ 4, -4 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, 2, 4, -1, -2, -4 ], [ 3, -3 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, 2, -1, -2 ], [ 3, 4, -3, -4 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, 3, 4, -1, -3, -4 ], [ 2, -2 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, 3, -1, -3 ], [ 2, 4, -2, -4 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, 4, -1, -4 ], [ 2, 3, -2, -3 ]>, [128X[104X
    [4X[28X  <block bijection: [ 1, -1 ], [ 2, 3, 4, -2, -3, -4 ]> ][128X[104X
  [4X[32X[104X
  
  [1X16.1-6 RightCosetsOfInverseSemigroup[101X
  
  [29X[2XRightCosetsOfInverseSemigroup[102X( [3XS[103X, [3XT[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA list of lists of elements.[133X
  
  [33X[0;0Y[10XRightCosetsOfInverseSemigroup[110X takes a majorantly closed inverse subsemigroup
  [3XT[103X  of  an  inverse  semigroup  of  partial permutations, block bijections or
  partial  permutation  bipartitions  [3XS[103X.  See [2XIsMajorantlyClosed[102X ([14X17.2-6[114X). The
  result is a list of the right cosets of [3XT[103X in [3XS[103X.[133X
  
  [33X[0;0YFor  [22Xs  ∈ S[122X, the right coset [22XoverlineTs[122X is defined if and only if [22Xss^-1 ∈ T[122X,
  in  which  case  it is defined to be the majorant closure of the set [22XTs[122X. See
  [2XMajorantClosure[102X   ([14X16.1-3[114X).   Distinct   cosets  are  disjoint  but  do  not
  necessarily partition [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SymmetricInverseSemigroup(3);[127X[104X
    [4X[28X<symmetric inverse monoid of degree 3>[128X[104X
    [4X[25Xgap>[125X [27XT := InverseSemigroup(MajorantClosure(S, [PartialPerm([1])]));[127X[104X
    [4X[28X<inverse partial perm monoid of rank 3 with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsMajorantlyClosed(S, T);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRC := RightCosetsOfInverseSemigroup(S, T);[127X[104X
    [4X[28X[ [ <identity partial perm on [ 1 ]>, [128X[104X
    [4X[28X      <identity partial perm on [ 1, 2 ]>, [2,3](1), [3,2](1), [128X[104X
    [4X[28X      <identity partial perm on [ 1, 3 ]>, [128X[104X
    [4X[28X      <identity partial perm on [ 1, 2, 3 ]>, (1)(2,3) ], [128X[104X
    [4X[28X  [ [1,3], [2,1,3], [1,3](2), (1,3), [1,3,2], (1,3,2), (1,3)(2) ], [128X[104X
    [4X[28X  [ [1,2], (1,2), [1,2,3], [3,1,2], [1,2](3), (1,2)(3), (1,2,3) ] ][128X[104X
  [4X[32X[104X
  
  [1X16.1-7 SameMinorantsSubgroup[101X
  
  [29X[2XSameMinorantsSubgroup[102X( [3XH[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA list of elements of the group [13XH[113X-class [3XH[103X.[133X
  
  [33X[0;0YGiven  a  group  [13XH[113X-class  [3XH[103X in an inverse semigroup of partial permutations,
  block     bijections     or     partial    permutation    bipartitions    [10XS[110X,
  [10XSameMinorantsSubgroup[110X  returns  a  list  of the elements of [3XH[103X which have the
  same  strict  minorants as the identity element of [3XH[103X. A [13Xstrict minorant[113X of [10Xx[110X
  in  [3XH[103X  is  an element of [10XS[110X which is less than [10Xx[110X (with respect to the natural
  partial order), but is not equal to [10Xx[110X.[133X
  
  [33X[0;0YThe returned list of elements of [3XH[103X describe a subgroup of [3XH[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SymmetricInverseSemigroup(3);[127X[104X
    [4X[28X<symmetric inverse monoid of degree 3>[128X[104X
    [4X[25Xgap>[125X [27XH := GroupHClass(DClass(S, PartialPerm([1, 2, 3])));[127X[104X
    [4X[28X<Green's H-class: <identity partial perm on [ 1, 2, 3 ]>>[128X[104X
    [4X[25Xgap>[125X [27XElements(H);[127X[104X
    [4X[28X[ <identity partial perm on [ 1, 2, 3 ]>, (1)(2,3), (1,2)(3), [128X[104X
    [4X[28X  (1,2,3), (1,3,2), (1,3)(2) ][128X[104X
    [4X[25Xgap>[125X [27XSameMinorantsSubgroup(H);[127X[104X
    [4X[28X[ <identity partial perm on [ 1, 2, 3 ]> ][128X[104X
    [4X[25Xgap>[125X [27XT := InverseSemigroup( [127X[104X
    [4X[25X>[125X [27XPartialPerm([1, 2, 3, 4], [1, 2, 4, 3]), [127X[104X
    [4X[25X>[125X [27XPartialPerm([1], [1]), PartialPerm([2], [2]));[127X[104X
    [4X[28X<inverse partial perm semigroup of rank 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XElements(T);[127X[104X
    [4X[28X[ <empty partial perm>, <identity partial perm on [ 1 ]>, [128X[104X
    [4X[28X  <identity partial perm on [ 2 ]>, [128X[104X
    [4X[28X  <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4) ][128X[104X
    [4X[25Xgap>[125X [27Xx := GroupHClass(DClass(T, PartialPerm([1, 2, 3, 4])));[127X[104X
    [4X[28X<Green's H-class: <identity partial perm on [ 1, 2, 3, 4 ]>>[128X[104X
    [4X[25Xgap>[125X [27XElements(x);[127X[104X
    [4X[28X[ <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4) ][128X[104X
    [4X[25Xgap>[125X [27XAsSet(SameMinorantsSubgroup(x));[127X[104X
    [4X[28X[ <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4) ][128X[104X
  [4X[32X[104X
  
  [1X16.1-8 SmallerDegreePartialPermRepresentation[101X
  
  [29X[2XSmallerDegreePartialPermRepresentation[102X( [3XS[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YAn isomorphism.[133X
  
  [33X[0;0Y[10XSmallerDegreePartialPermRepresentation[110X  attempts to find an isomorphism from
  the inverse semigroup [3XS[103X of partial permutations to another inverse semigroup
  of  partial  permutations with smaller degree. If the function cannot reduce
  the degree, the identity mapping is returned.[133X
  
  [33X[0;0YThere  is  no  guarantee that the smallest possible degree representation is
  returned. For more information see [Sch92].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := InverseSemigroup(PartialPerm([2, 1, 4, 3, 6, 5, 8, 7]));[127X[104X
    [4X[28X<commutative inverse partial perm semigroup of rank 8 with 1 [128X[104X
    [4X[28X generator>[128X[104X
    [4X[25Xgap>[125X [27XElements(S);[127X[104X
    [4X[28X[ <identity partial perm on [ 1, 2, 3, 4, 5, 6, 7, 8 ]>, [128X[104X
    [4X[28X  (1,2)(3,4)(5,6)(7,8) ][128X[104X
    [4X[25Xgap>[125X [27XT := SmallerDegreePartialPermRepresentation(S);[127X[104X
    [4X[28XMappingByFunction( <partial perm group of size 2, rank 8 with[128X[104X
    [4X[28X  1 generator>, <commutative inverse partial perm semigroup of rank 2 [128X[104X
    [4X[28Xwith 1 generator>, function( x ) ... end, function( x ) ... end )[128X[104X
    [4X[25Xgap>[125X [27XR := Range(T);[127X[104X
    [4X[28X<commutative inverse partial perm semigroup of rank 2 with 1 [128X[104X
    [4X[28X generator>[128X[104X
    [4X[25Xgap>[125X [27XElements(R);[127X[104X
    [4X[28X[ <identity partial perm on [ 1, 2 ]>, (1,2) ][128X[104X
    [4X[25Xgap>[125X [27XS := DualSymmetricInverseMonoid(5);;[127X[104X
    [4X[25Xgap>[125X [27XT := Range(IsomorphismPartialPermSemigroup(S));[127X[104X
    [4X[28X<inverse partial perm monoid of rank 6721 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSmallerDegreePartialPermRepresentation(T);[127X[104X
    [4X[28XMappingByFunction( <inverse partial perm monoid of rank 6721 with 3 [128X[104X
    [4X[28X generators>, <inverse partial perm semigroup of rank 30 with 3 [128X[104X
    [4X[28X generators>, function( x ) ... end, function( x ) ... end )[128X[104X
  [4X[32X[104X
  
  [1X16.1-9 VagnerPrestonRepresentation[101X
  
  [29X[2XVagnerPrestonRepresentation[102X( [3XS[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YAn isomorphism to an inverse semigroup of partial permutations.[133X
  
  [33X[0;0Y[10XVagnerPrestonRepresentation[110X returns an isomorphism from an inverse semigroup
  [3XS[103X  where  the  elements  of [3XS[103X have a unique semigroup inverse accessible via
  [2XInverse[102X   ([14XReference:   Inverse[114X),   to  the  inverse  semigroup  of  partial
  permutations [3XT[103X of degree equal to the size of [3XS[103X, which is obtained using the
  Vagner-Preston Representation Theorem.[133X
  
  [33X[0;0YMore    precisely,   if   [22Xf:S->   T[122X   is   the   isomorphism   returned   by
  [10XVagnerPrestonRepresentation([3XS[103X[10X)[110X  and  [22Xx[122X  is  in  [3XS[103X,  then [22Xf(x)[122X is the partial
  permutation  with  domain  [22XSx^-1[122X  and  range  [22XSx^-1x[122X defined by [22Xf(x): sx^-1↦
  sx^-1x[122X.[133X
  
  [33X[0;0YIn  many  cases, it is possible to find a smaller degree representation than
  that        provided        by       [10XVagnerPrestonRepresentation[110X       using
  [2XIsomorphismPartialPermSemigroup[102X ([14XReference: IsomorphismPartialPermSemigroup[114X)
  or [2XSmallerDegreePartialPermRepresentation[102X ([14X16.1-8[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SymmetricInverseSemigroup(2);[127X[104X
    [4X[28X<symmetric inverse monoid of degree 2>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27Xiso := VagnerPrestonRepresentation(S);[127X[104X
    [4X[28XMappingByFunction( <symmetric inverse monoid of degree 2>, [128X[104X
    [4X[28X<inverse partial perm monoid of rank 7 with 2 generators>[128X[104X
    [4X[28X , function( x ) ... end, function( x ) ... end )[128X[104X
    [4X[25Xgap>[125X [27XRespectsMultiplication(iso);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinv := InverseGeneralMapping(iso);;[127X[104X
    [4X[25Xgap>[125X [27XForAll(S, x-> (x ^ iso) ^ inv = x);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XV := InverseSemigroup([127X[104X
    [4X[25X>[125X [27XBipartition([[1, -4], [2, -1], [3, -5], [127X[104X
    [4X[25X>[125X [27X             [4], [5], [-2], [-3]]),[127X[104X
    [4X[25X>[125X [27XBipartition([[1, -5], [2, -1], [3, -3], [127X[104X
    [4X[25X>[125X [27X             [4], [5], [-2], [-4]]),[127X[104X
    [4X[25X>[125X [27XBipartition([[1, -2], [2, -4], [3, -5], [127X[104X
    [4X[25X>[125X [27X             [4, -1], [5, -3]]));[127X[104X
    [4X[28X<inverse bipartition semigroup of degree 5 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsInverseSemigroup(V);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XVagnerPrestonRepresentation(V);[127X[104X
    [4X[28XMappingByFunction( <inverse bipartition semigroup of size 394, [128X[104X
    [4X[28X degree 5 with 3 generators>, <inverse partial perm semigroup of [128X[104X
    [4X[28X rank 394 with 5 generators>[128X[104X
    [4X[28X , function( x ) ... end, function( x ) ... end )[128X[104X
  [4X[32X[104X
  
  [1X16.1-10 CharacterTableOfInverseSemigroup[101X
  
  [29X[2XCharacterTableOfInverseSemigroup[102X( [3XS[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe  character  table  of  the  inverse  semigroup [3XS[103X and a list of
            conjugacy class representatives of [3XS[103X.[133X
  
  [33X[0;0YReturns  a  list with two entries: the first entry being the character table
  of  the inverse semigroup [3XS[103X as a matrix, while the second entry is a list of
  conjugacy class representatives of [3XS[103X.[133X
  
  [33X[0;0YThe  order of the columns in the character table matrix follows the order of
  the  conjugacy class representatives list. The conjugacy representatives are
  grouped by [13XD[113X-class and then sorted by rank. Also, as is typical of character
  tables,  the rows of the matrix correspond to the irreducible characters and
  the columns correspond to the conjugacy classes.[133X
  
  [33X[0;0YThis function was contributed by Jhevon Smith and Ben Steinberg.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := InverseMonoid( [ PartialPerm( [ 1, 2 ], [ 3, 1 ] ), [127X[104X
    [4X[25X>[125X [27XPartialPerm( [ 1, 2, 3 ], [ 1, 3, 4 ] ), [127X[104X
    [4X[25X>[125X [27XPartialPerm( [ 1, 2, 3 ], [ 2, 4, 1 ] ), [127X[104X
    [4X[25X>[125X [27XPartialPerm( [ 1, 3, 4 ], [ 3, 4, 1 ] ) ] );;[127X[104X
    [4X[25Xgap>[125X [27XCharacterTableOfInverseSemigroup(S);[127X[104X
    [4X[28X[ [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 3, 1, 1, 1, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 3, 1, E(3), E(3)^2, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 3, 1, E(3)^2, E(3), 0, 0, 0, 0 ], [ 6, 3, 0, 0, 1, -1, 0, 0 ],[128X[104X
    [4X[28X      [ 6, 3, 0, 0, 1, 1, 0, 0 ], [ 4, 3, 0, 0, 2, 0, 1, 0 ], [128X[104X
    [4X[28X      [ 1, 1, 1, 1, 1, 1, 1, 1 ] ], [128X[104X
    [4X[28X  [ <identity partial perm on [ 1, 2, 3, 4 ]>, [128X[104X
    [4X[28X      <identity partial perm on [ 1, 3, 4 ]>, (1,3,4), (1,4,3), [128X[104X
    [4X[28X      <identity partial perm on [ 1, 3 ]>, (1,3), [128X[104X
    [4X[28X      <identity partial perm on [ 3 ]>, <empty partial perm> ] ][128X[104X
    [4X[25Xgap>[125X [27XS := SymmetricInverseMonoid(4);;[127X[104X
    [4X[25Xgap>[125X [27XCharacterTableOfInverseSemigroup(S);[127X[104X
    [4X[28X[ [ [ 1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 3, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 2, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 3, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 4, -2, 1, 0, 0, 1, -1, 1, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 8, 0, -1, 0, 0, 2, 0, -1, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 4, 2, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 6, 0, 0, -2, 0, 3, -1, 0, 1, -1, 0, 0 ], [128X[104X
    [4X[28X      [ 6, 2, 0, 2, 0, 3, 1, 0, 1, 1, 0, 0 ], [128X[104X
    [4X[28X      [ 4, 2, 1, 0, 0, 3, 1, 0, 2, 0, 1, 0 ], [128X[104X
    [4X[28X      [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ], [128X[104X
    [4X[28X  [ <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4), [128X[104X
    [4X[28X      (1)(2,3,4), (1,2)(3,4), (1,2,3,4), [128X[104X
    [4X[28X      <identity partial perm on [ 1, 2, 3 ]>, (1)(2,3), (1,2,3), [128X[104X
    [4X[28X      <identity partial perm on [ 2, 3 ]>, (2,3), [128X[104X
    [4X[28X      <identity partial perm on [ 1 ]>, <empty partial perm> ] ][128X[104X
  [4X[32X[104X
  
