  
  [1X8 [33X[0;0YStandard examples[133X[101X
  
  [33X[0;0YIn  this  chapter we describe some standard examples of semigroups which are
  available in the [5XSemigroups[105X package.[133X
  
  
  [1X8.1 [33X[0;0YTransformation semigroups[133X[101X
  
  [33X[0;0YIn  this  section, we describe the operations in [5XSemigroups[105X that can be used
  to create transformation semigroups belonging to several standard classes of
  example.   See  [14X'Reference:  Transformations'[114X  for  more  information  about
  transformations.[133X
  
  [1X8.1-1 CatalanMonoid[101X
  
  [33X[1;0Y[29X[2XCatalanMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X  is a positive integer, then this operation returns the Catalan monoid
  of degree [3Xn[103X. The [13XCatalan monoid[113X is the semigroup of the order-preserving and
  order-decreasing transformations of [10X[1 .. n][110X with the usual ordering.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := CatalanMonoid(6);[127X[104X
    [4X[28X<transformation monoid of degree 6 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X132[128X[104X
  [4X[32X[104X
  
  [1X8.1-2 EndomorphismsPartition[101X
  
  [33X[1;0Y[29X[2XEndomorphismsPartition[102X( [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation monoid.[133X
  
  [33X[0;0YIf  [3Xlist[103X is a list of positive integers, then [10XEndomorphismsPartition[110X returns
  a  monoid of endomorphisms preserving a partition of [10X[1 .. Sum([3Xlist[103X[10X)][110X with a
  part  of  length [10X[3Xlist[103X[10X[i][110X for every [10Xi[110X. For example, if [10X[3Xlist[103X[10X = [1, 2, 3][110X, then
  [10XEndomorphismsPartition[110X  returns the monoid of endomorphisms of the partition
  [10X[[1], [2, 3], [4, 5, 6]][110X.[133X
  
  [33X[0;0YIf  [10Xf[110X  is  a  transformation  of  [10X[1  .. n][110X, then it is an [12Xendomorphism[112X of a
  partition [10XP[110X on [10X[1 .. n][110X if [10X(i, j)[110X in [10XP[110X implies that [10X(i ^ f, j ^ f)[110X is in [10XP[110X.[133X
  
  [33X[0;0Y[10XEndomorphismsPartition[110X  returns a monoid with a minimal size generating set,
  as described in [ABMS15].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := EndomorphismsPartition([3, 3, 3]);[127X[104X
    [4X[28X<transformation semigroup of degree 9 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X531441[128X[104X
  [4X[32X[104X
  
  [1X8.1-3 PartialTransformationMonoid[101X
  
  [33X[1;0Y[29X[2XPartialTransformationMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA transformation monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a  positive  integer,  then  this function returns a semigroup of
  transformations  on  [10X[3Xn[103X[10X  +  1[110X  points  which  is  isomorphic to the semigroup
  consisting  of  all partial transformation on [3Xn[103X points. This monoid has [10X([3Xn[103X[10X +
  1) ^ [3Xn[103X[10X[110X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := PartialTransformationMonoid(5);[127X[104X
    [4X[28X<regular transformation monoid of degree 6 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X7776[128X[104X
  [4X[32X[104X
  
  [1X8.1-4 SingularTransformationSemigroup[101X
  
  [33X[1;0Y[29X[2XSingularTransformationSemigroup[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularTransformationMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe semigroup of non-invertible transformations.[133X
  
  [33X[0;0YIf  [3Xn[103X  is a integer greater than 1, then this function returns the semigroup
  of  non-invertible  transformations,  which  is  generated  by  the [10X[3Xn[103X[10X([3Xn[103X[10X - 1)[110X
  idempotents of degree [3Xn[103X and rank [10X[3Xn[103X[10X - 1[110X and has [22Xn ^ n - n![122X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SingularTransformationSemigroup(4);[127X[104X
    [4X[28X<regular transformation semigroup ideal of degree 4 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X232[128X[104X
  [4X[32X[104X
  
  
  [1X8.1-5 [33X[0;0YSemigroups of order-preserving transformations[133X[101X
  
  [33X[1;0Y[29X[2XOrderEndomorphisms[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularOrderEndomorphisms[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOrderAntiEndomorphisms[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPartialOrderEndomorphisms[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPartialOrderAntiEndomorphisms[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA semigroup of transformations related to a linear order.[133X
  
  [8X[10XOrderEndomorphisms([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XOrderEndomorphisms([3Xn[103X[10X)[110X  returns  the  monoid  of  transformations  that
        preserve  the  usual  order  on  [22X{1, 2, ..., n}[122X, where [3Xn[103X is a positive
        integer.[133X
  
  [8X[10XSingularOrderEndomorphisms([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XSingularOrderEndomorphisms([3Xn[103X[10X)[110X      returns      the      ideal      of
        [10XOrderEndomorphisms([3Xn[103X[10X)[110X  consisting of the non-invertible elements, when
        [3Xn[103X  is at least [10X2[110X. The only invertible element in [10XOrderEndomorphisms([3Xn[103X[10X)[110X
        is the identity transformation.[133X
  
  [8X[10XOrderAntiEndomorphisms([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XOrderAntiEndomorphisms([3Xn[103X[10X)[110X  returns  the monoid of transformations that
        preserve  or  reverse  the usual order on [22X{1, 2, ..., n}[122X, where [3Xn[103X is a
        positive   integer.  [10XOrderAntiEndomorphisms([3Xn[103X[10X)[110X  is  generated  by  the
        generators   of   [10XOrderEndomorphisms([3Xn[103X[10X)[110X   along   with  the  bijective
        transformation  that  reverses the order on [22X{1, 2, ..., n}[122X. The monoid
        [10XOrderAntiEndomorphisms([3Xn[103X[10X)[110X has [22X2n-1choose n-1 - n[122X elements.[133X
  
  [8X[10XPartialOrderEndomorphisms([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XPartialOrderEndomorphisms([3Xn[103X[10X)[110X  returns a monoid of transformations on [10X[3Xn[103X[10X
        +  1[110X points that is isomorphic to the monoid consisting of all partial
        transformations that preserve the usual order on [22X{1, 2, ..., n}[122X.[133X
  
  [8X[10XPartialOrderAntiEndomorphisms([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XPartialAntiOrderEndomorphisms([3Xn[103X[10X)[110X  returns  a monoid of transformations
        on  [10X[3Xn[103X[10X  +  1[110X  points that is isomorphic to the monoid consisting of all
        partial  transformations  that  preserve or reverse the usual order on
        [22X{1, 2, ..., n}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := OrderEndomorphisms(5);[127X[104X
    [4X[28X<regular transformation monoid of degree 5 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsIdempotentGenerated(S);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(S) = Binomial(2 * 5 - 1, 5 - 1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDifference(S, SingularOrderEndomorphisms(5));[127X[104X
    [4X[28X[ IdentityTransformation ][128X[104X
    [4X[25Xgap>[125X [27XSingularOrderEndomorphisms(10);[127X[104X
    [4X[28X<regular transformation semigroup ideal of degree 10 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XT := OrderAntiEndomorphisms(4);[127X[104X
    [4X[28X<regular transformation monoid of degree 4 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XTransformation([4, 2, 2, 1]) in T;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XU := PartialOrderEndomorphisms(6);[127X[104X
    [4X[28X<regular transformation monoid of degree 7 with 12 generators>[128X[104X
    [4X[25Xgap>[125X [27XV := PartialOrderAntiEndomorphisms(6);[127X[104X
    [4X[28X<regular transformation monoid of degree 7 with 13 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroup(V, U);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X8.2 [33X[0;0YSemigroups of partial permutations[133X[101X
  
  [33X[0;0YIn  this  section, we describe the operations in [5XSemigroups[105X that can be used
  to  create  semigroups of partial permutations belonging to several standard
  classes   of   example.  See  [14X'Reference:  Partial  permutations'[114X  for  more
  information about partial permutations.[133X
  
  [1X8.2-1 MunnSemigroup[101X
  
  [33X[1;0Y[29X[2XMunnSemigroup[102X( [3XS[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YThe Munn semigroup of a semilattice.[133X
  
  [33X[0;0YIf  [3XS[103X  is a semilattice, then [10XMunnSemigroup[110X returns the inverse semigroup of
  partial  permutations  of  isomorphisms of principal ideals of [3XS[103X; called the
  [13XMunn semigroup[113X of [3XS[103X.[133X
  
  [33X[0;0YThis  function  was  written  jointly  by  J.  D. Mitchell, Yann Péresse (St
  Andrews), Yanhui Wang (York).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := InverseSemigroup([[127X[104X
    [4X[25X>[125X [27XPartialPerm([1, 2, 3, 4, 5, 6, 7, 10], [4, 6, 7, 3, 8, 2, 9, 5]),[127X[104X
    [4X[25X>[125X [27XPartialPerm([1, 2, 7, 9], [5, 6, 4, 3])]);[127X[104X
    [4X[28X<inverse partial perm semigroup of rank 10 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XT := IdempotentGeneratedSubsemigroup(S);;[127X[104X
    [4X[25Xgap>[125X [27XM := MunnSemigroup(T);[127X[104X
    [4X[28X<inverse partial perm semigroup of rank 60 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XNrIdempotents(M);[127X[104X
    [4X[28X60[128X[104X
    [4X[25Xgap>[125X [27XNrIdempotents(S);[127X[104X
    [4X[28X60[128X[104X
  [4X[32X[104X
  
  [1X8.2-2 RookMonoid[101X
  
  [33X[1;0Y[29X[2XRookMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn inverse monoid of partial permutations.[133X
  
  [33X[0;0Y[10XRookMonoid[110X    is    a   synonym   for   [2XSymmetricInverseMonoid[102X   ([14XReference:
  SymmetricInverseMonoid[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := RookMonoid(4);[127X[104X
    [4X[28X<symmetric inverse monoid of degree 4>[128X[104X
    [4X[25Xgap>[125X [27XS = SymmetricInverseMonoid(4);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X8.2-3 [33X[0;0YInverse monoids of order-preserving partial permutations[133X[101X
  
  [33X[1;0Y[29X[2XPOI[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPODI[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPOPI[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPORI[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn  inverse  monoid  of  partial  permutations related to a linear
            order.[133X
  
  [8X[10XPOI([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XPOI([3Xn[103X[10X)[110X  returns  the  inverse  monoid  of  partial  permutations  that
        preserve  the  usual  order  on  [22X{1, 2, ..., n}[122X, where [3Xn[103X is a positive
        integer.[133X
  
  [8X[10XPODI([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XPODI([3Xn[103X[10X)[110X  returns  the  inverse  monoid  of  partial  permutations that
        preserve  or  reverse  the usual order on [22X{1, 2, ..., n}[122X, where [3Xn[103X is a
        positive  integer.  [10XPODI([3Xn[103X[10X)[110X  is generated by the generators of [10XPOI([3Xn[103X[10X)[110X,
        along  with  the  permutation  that reverses the usual order on [22X{1, 2,
        ..., n}[122X.[133X
  
  [8X[10XPOPI([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XPOPI([3Xn[103X[10X)[110X  returns  the  inverse  monoid  of  partial  permutations that
        preserve  the  orientation  of  [22X{1,2,...,  n}[122X,  where  [22Xn[122X is a positive
        integer.[133X
  
  [8X[10XPORI([3Xn[103X[8X[10X)[110X[8X[108X
        [33X[0;6Y[10XPORI([3Xn[103X[10X)[110X  returns  the  inverse  monoid  of  partial  permutations that
        preserve  or  reverse  the orientation of [22X{1, 2, ..., n}[122X, where [22Xn[122X is a
        positive  integer.  [10XPORI([3Xn[103X[10X)[110X is generated by the generators of [10XPOPI([3Xn[103X[10X)[110X,
        along  with  the  permutation  that reverses the usual order on [22X{1, 2,
        ..., n}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := PORI(10);[127X[104X
    [4X[28X<inverse partial perm monoid of rank 10 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XS := POPI(10);[127X[104X
    [4X[28X<inverse partial perm monoid of rank 10 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S) = 1 + 5 * Binomial(20, 10);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XS := PODI(10);[127X[104X
    [4X[28X<inverse partial perm monoid of rank 10 with 11 generators>[128X[104X
    [4X[25Xgap>[125X [27XS := POI(10);[127X[104X
    [4X[28X<inverse partial perm monoid of rank 10 with 10 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S) = Binomial(20, 10);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroup(PORI(10), PODI(10))[127X[104X
    [4X[25X>[125X [27Xand IsSubsemigroup(PORI(10), POPI(10))[127X[104X
    [4X[25X>[125X [27Xand IsSubsemigroup(PODI(10), POI(10))[127X[104X
    [4X[25X>[125X [27Xand IsSubsemigroup(POPI(10), POI(10));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X8.3 [33X[0;0YSemigroups of bipartitions[133X[101X
  
  [33X[0;0YIn  this  section, we describe the operations in [5XSemigroups[105X that can be used
  to  create  bipartition  semigroups belonging to several standard classes of
  example. See Chapter [14X3[114X for more information about bipartitions.[133X
  
  [1X8.3-1 PartitionMonoid[101X
  
  [33X[1;0Y[29X[2XPartitionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRookPartitionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularPartitionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a non-negative integer, then this operation returns the partition
  monoid  of  degree  [3Xn[103X.  The  [13Xpartition  monoid  of  degree  [3Xn[103X[113X  is the monoid
  consisting of all the bipartitions of degree [3Xn[103X.[133X
  
  [33X[0;0Y[10XSingularPartitionMonoid[110X returns the ideal of the partition monoid consisting
  of  the non-invertible elements (i.e. those not in the group of units), when
  [3Xn[103X is positive.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  positive,  then  [10XRookPartitionMonoid[110X  returns  submonoid  of  the
  partition monoid of degree [10X[3Xn[103X[10X + 1[110X consisting of those bipartitions with [10X[3Xn[103X[10X + 1[110X
  and [10X-[3Xn[103X[10X - 1[110X in the same block; see [HR05], [Gro06], and [Eas19].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := PartitionMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of size 4140, degree 4 with 4 [128X[104X
    [4X[28X generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X4140[128X[104X
    [4X[25Xgap>[125X [27XT := SingularPartitionMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-semigroup ideal of degree 4 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XSize(S) - Size(T) = Factorial(4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XS := RookPartitionMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 5 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X21147[128X[104X
  [4X[32X[104X
  
  [1X8.3-2 BrauerMonoid[101X
  
  [33X[1;0Y[29X[2XBrauerMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPartialBrauerMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularBrauerMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a  non-negative  integer,  then this operation returns the Brauer
  monoid  of  degree  [3Xn[103X.  The  [13XBrauer monoid[113X is the submonoid of the partition
  monoid consisiting of those bipartitions where the size of every block is 2.[133X
  
  [33X[0;0Y[10XPartialBrauerMonoid[110X   returns  the  partial  Brauer  monoid,  which  is  the
  submonoid of the partition monoid consisting of those bipartitions where the
  size  of  every  block  is [13Xat most[113X 2. The partial Brauer monoid contains the
  Brauer monoid as a submonoid.[133X
  
  [33X[0;0Y[10XSingularBrauerMonoid[110X  returns  the  ideal of the Brauer monoid consisting of
  the  non-invertible  elements (i.e. those not in the group of units), when [3Xn[103X
  is at least 2.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := BrauerMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroup(S, JonesMonoid(4));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X105[128X[104X
    [4X[25Xgap>[125X [27XSingularBrauerMonoid(8);[127X[104X
    [4X[28X<regular bipartition *-semigroup ideal of degree 8 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XS := PartialBrauerMonoid(3);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 3 with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroup(S, BrauerMonoid(3));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X76[128X[104X
  [4X[32X[104X
  
  [1X8.3-3 JonesMonoid[101X
  
  [33X[1;0Y[29X[2XJonesMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XTemperleyLiebMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularJonesMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf [3Xn[103X is a non-negative integer, then this operation returns the Jones monoid
  of  degree  [3Xn[103X.  The  [13XJones  monoid[113X  is the subsemigroup of the Brauer monoid
  consisting  of those bipartitions that are planar; see [2XPlanarPartitionMonoid[102X
  ([14X8.3-9[114X).  The  Jones  monoid  is sometimes referred to as the [12XTemperley-Lieb
  monoid[112X.[133X
  
  [33X[0;0Y[10XSingularJonesMonoid[110X  returns the ideal of the Jones monoid consisting of the
  non-invertible elements (i.e. those not in the group of units), when [3Xn[103X is at
  least 2.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := JonesMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XS = TemperleyLiebMonoid(4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSingularJonesMonoid(8);[127X[104X
    [4X[28X<regular bipartition *-semigroup ideal of degree 8 with 1 generator>[128X[104X
  [4X[32X[104X
  
  [1X8.3-4 PartialJonesMonoid[101X
  
  [33X[1;0Y[29X[2XPartialJonesMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X is a non-negative integer, then [10XPartialJonesMonoid[110X returns the partial
  Jones  monoid of degree [3Xn[103X. The [13Xpartial Jones monoid[113X is a subsemigroup of the
  partial  Brauer monoid. An element of the partial Brauer monoid is contained
  in  the  partial Jones monoid if the partition that it defines is finer than
  the  partition  defined by some element of the Jones monoid. In other words,
  an  element of the partial Jones monoid can be formed from some element [10Xx[110X of
  the  Jones  monoid  by replacing some blocks [10X[a, b][110X of [10Xx[110X by singleton blocks
  [10X[a], [b][110X.[133X
  
  [33X[0;0YNote  that,  in  general,  the  partial Jones monoid of degree [3Xn[103X is strictly
  contained in the Motzkin monoid of the same degree.[133X
  
  [33X[0;0YSee  [2XPartialBrauerMonoid[102X  ([14X8.3-2[114X),  [2XJonesMonoid[102X  ([14X8.3-3[114X),  and [2XMotzkinMonoid[102X
  ([14X8.3-6[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := PartialJonesMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 4 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XT := JonesMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XU := MotzkinMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 4 with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroup(U, S);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroup(S, T);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(U);[127X[104X
    [4X[28X323[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X143[128X[104X
    [4X[25Xgap>[125X [27XSize(T);[127X[104X
    [4X[28X14[128X[104X
  [4X[32X[104X
  
  [1X8.3-5 AnnularJonesMonoid[101X
  
  [33X[1;0Y[29X[2XAnnularJonesMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X is a non-negative integer, then [10XAnnularJonesMonoid[110X returns the annular
  Jones  monoid  of  degree [3Xn[103X. The [13Xannular Jones monoid[113X is the subsemigroup of
  the  partition  monoid  consisting  of all annular bipartitions whose blocks
  have   size   2  (annular  bipartitions  are  defined  in  Chapter  [14X3[114X).  See
  [2XBrauerMonoid[102X ([14X8.3-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := AnnularJonesMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 4 with 2 generators>[128X[104X
  [4X[32X[104X
  
  [1X8.3-6 MotzkinMonoid[101X
  
  [33X[1;0Y[29X[2XMotzkinMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a  non-negative  integer, then this operation returns the Motzkin
  monoid  of  degree  [3Xn[103X. The [13XMotzkin monoid[113X is the subsemigroup of the partial
  Brauer  monoid  consisting  of  those  bipartitions  that are planar (planar
  bipartitions are defined in Chapter [14X3[114X).[133X
  
  [33X[0;0YNote  that  the Motzkin monoid of degree [3Xn[103X contains the partial Jones monoid
  of   degree   [3Xn[103X,   but   in  general,  these  monoids  are  not  equal;  see
  [2XPartialJonesMonoid[102X ([14X8.3-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := MotzkinMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 4 with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27XT := PartialJonesMonoid(4);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 4 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroup(S, T);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X323[128X[104X
    [4X[25Xgap>[125X [27XSize(T);[127X[104X
    [4X[28X143[128X[104X
  [4X[32X[104X
  
  [1X8.3-7 DualSymmetricInverseSemigroup[101X
  
  [33X[1;0Y[29X[2XDualSymmetricInverseSemigroup[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDualSymmetricInverseMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularDualSymmetricInverseMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPartialDualSymmetricInverseMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn inverse bipartition monoid.[133X
  
  [33X[0;0YIf     [3Xn[103X     is     a     positive     integer,    then    the    operations
  [10XDualSymmetricInverseSemigroup[110X and [10XDualSymmetricInverseMonoid[110X return the dual
  symmetric  inverse  monoid  of  degree  [3Xn[103X,  which is the subsemigroup of the
  partition monoid consisting of the block bijections of degree [3Xn[103X.[133X
  
  [33X[0;0Y[10XSingularDualSymmetricInverseMonoid[110X  returns  the ideal of the dual symmetric
  inverse  monoid consisting of the non-invertible elements (i.e. those not in
  the group of units), when [3Xn[103X is at least 2.[133X
  
  [33X[0;0Y[10XPartialDualSymmetricInverseMonoid[110X   returns   the   submonoid  of  the  dual
  symmetric  inverse  monoid  of  degree  [10X[3Xn[103X[10X  +  1[110X  consisting  of  those block
  bijections with [10X[3Xn[103X[10X + 1[110X and [10X-[3Xn[103X[10X - 1[110X in the same block; see [KM11] and [KMU15].[133X
  
  [33X[0;0YSee [2XIsBlockBijection[102X ([14X3.5-16[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumber(PartitionMonoid(3), IsBlockBijection);[127X[104X
    [4X[28X25[128X[104X
    [4X[25Xgap>[125X [27XS := DualSymmetricInverseSemigroup(3);[127X[104X
    [4X[28X<inverse block bijection monoid of degree 3 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X25[128X[104X
    [4X[25Xgap>[125X [27XS := PartialDualSymmetricInverseMonoid(5);[127X[104X
    [4X[28X<inverse block bijection monoid of degree 6 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X29072[128X[104X
  [4X[32X[104X
  
  [1X8.3-8 UniformBlockBijectionMonoid[101X
  
  [33X[1;0Y[29X[2XUniformBlockBijectionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XFactorisableDualSymmetricInverseMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularUniformBlockBijectionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPartialUniformBlockBijectionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularFactorisableDualSymmetricInverseMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPlanarUniformBlockBijectionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularPlanarUniformBlockBijectionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn inverse bipartition monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a positive integer, then this operation returns the uniform block
  bijection  monoid  of  degree  [3Xn[103X.  The [13Xuniform block bijection monoid[113X is the
  submonoid  of  the  partition  monoid  consisting of the block bijections of
  degree  [10Xn[110X where the number of positive integers in a block equals the number
  of  negative  integers  in that block. The uniform block bijection monoid is
  also referred to as the [13Xfactorisable dual symmetric inverse monoid[113X.[133X
  
  [33X[0;0Y[10XSingularUniformBlockBijectionMonoid[110X  returns  the ideal of the uniform block
  bijection  monoid  consisting of the non-invertible elements (i.e. those not
  in the group of units), when [3Xn[103X is at least 2.[133X
  
  [33X[0;0Y[10XPlanarUniformBlockBijectionMonoid[110X returns the submonoid of the uniform block
  bijection monoid consisting of the planar elements (i.e. those in the planar
  partition monoid, see [2XPlanarPartitionMonoid[102X ([14X8.3-9[114X)).[133X
  
  [33X[0;0Y[10XSingularPlanarUniformBlockBijectionMonoid[110X  returns  the  ideal of the planar
  uniform  block  bijection  monoid  consisting of the non-invertible elements
  (i.e. those not in the group of units), when [3Xn[103X is at least 2.[133X
  
  [33X[0;0Y[10XPartialUniformBlockBijectionMonoid[110X  returns  the  submonoid  of  the uniform
  block  bijection  monoid  of  degree [10X[3Xn[103X[10X + 1[110X consisting of those uniform block
  bijection with [10X[3Xn[103X[10X + 1[110X and [10X-[3Xn[103X[10X - 1[110X in the same block.[133X
  
  [33X[0;0YSee [2XIsUniformBlockBijection[102X ([14X3.5-17[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := UniformBlockBijectionMonoid(4);[127X[104X
    [4X[28X<inverse block bijection monoid of degree 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(PlanarUniformBlockBijectionMonoid(8));[127X[104X
    [4X[28X128[128X[104X
    [4X[25Xgap>[125X [27XS := DualSymmetricInverseMonoid(4);[127X[104X
    [4X[28X<inverse block bijection monoid of degree 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsFactorisableInverseMonoid(S);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XS := UniformBlockBijectionMonoid(4);[127X[104X
    [4X[28X<inverse block bijection monoid of degree 4 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsFactorisableInverseMonoid(S);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XS := AsSemigroup(IsBipartitionSemigroup,[127X[104X
    [4X[25X>[125X [27X                    SymmetricInverseMonoid(5));[127X[104X
    [4X[28X<inverse bipartition monoid of degree 5 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsFactorisableInverseMonoid(S);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XS := PartialUniformBlockBijectionMonoid(5);[127X[104X
    [4X[28X<inverse block bijection monoid of degree 6 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XNrIdempotents(S);[127X[104X
    [4X[28X203[128X[104X
    [4X[25Xgap>[125X [27XIsFactorisableInverseMonoid(S);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X8.3-9 PlanarPartitionMonoid[101X
  
  [33X[1;0Y[29X[2XPlanarPartitionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularPlanarPartitionMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf [3Xn[103X is a positive integer, then this operation returns the planar partition
  monoid  of  degree  [3Xn[103X  which  is  the  monoid  consisting  of all the planar
  bipartitions of degree [3Xn[103X (planar bipartitions are defined in Chapter [14X3[114X).[133X
  
  [33X[0;0Y[10XSingularPlanarPartitionMonoid[110X  returns  the  ideal  of  the planar partition
  monoid  consisting  of  the  non-invertible  elements (i.e. those not in the
  group of units).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := PlanarPartitionMonoid(3);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 3 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X132[128X[104X
    [4X[25Xgap>[125X [27XT := SingularPlanarPartitionMonoid(3);[127X[104X
    [4X[28X<regular bipartition *-semigroup ideal of degree 3 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XSize(T);[127X[104X
    [4X[28X131[128X[104X
    [4X[25Xgap>[125X [27XDifference(S, T);[127X[104X
    [4X[28X[ <block bijection: [ 1, -1 ], [ 2, -2 ], [ 3, -3 ]> ][128X[104X
  [4X[32X[104X
  
  [1X8.3-10 ModularPartitionMonoid[101X
  
  [33X[1;0Y[29X[2XModularPartitionMonoid[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularModularPartitionMonoid[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPlanarModularPartitionMonoid[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularPlanarModularPartitionMonoid[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf  [3Xm[103X and [3Xn[103X are positive integers, then this operation returns the modular-[3Xm[103X
  partition  monoid  of  degree  [3Xn[103X.  The  [13Xmodular-[113X[3Xm[103X  [13Xpartition  monoid[113X  is the
  submonoid  of  the  partition  monoid  such that the numbers of positive and
  negative integers contained in each block are congruent mod [3Xm[103X.[133X
  
  [33X[0;0Y[10XSingularModularPartitionMonoid[110X  returns  the  ideal of the modular partition
  monoid  consisting  of  the  non-invertible  elements (i.e. those not in the
  group of units), when either [3Xm = n = 1[103X or [3Xm, n > 1[103X.[133X
  
  [33X[0;0Y[10XPlanarModularPartitionMonoid[110X   returns   the   submonoid  of  the  modular-[3Xm[103X
  partition monoid consisting of the planar elements (i.e. those in the planar
  partition monoid, see [2XPlanarPartitionMonoid[102X ([14X8.3-9[114X)).[133X
  
  [33X[0;0Y[10XSingularPlanarModularPartitionMonoid[110X returns the ideal of the planar modular
  partition  monoid  consisting of the non-invertible elements (i.e. those not
  in the group of units), when either [3Xm = n = 1[103X or [3Xm, n > 1[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := ModularPartitionMonoid(3, 6);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 6 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X36243[128X[104X
    [4X[25Xgap>[125X [27XS := SingularModularPartitionMonoid(1, 1);[127X[104X
    [4X[28X<commutative inverse bipartition semigroup ideal of degree 1 with[128X[104X
    [4X[28X  1 generator>[128X[104X
    [4X[25Xgap>[125X [27XSize(SingularModularPartitionMonoid(2, 4));[127X[104X
    [4X[28X355[128X[104X
    [4X[25Xgap>[125X [27XS := PlanarModularPartitionMonoid(4, 9);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 9 with 14 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X1795[128X[104X
    [4X[25Xgap>[125X [27XS := SingularPlanarModularPartitionMonoid(3, 5);[127X[104X
    [4X[28X<regular bipartition *-semigroup ideal of degree 5 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XSize(SingularPlanarModularPartitionMonoid(1, 2));[127X[104X
    [4X[28X13[128X[104X
  [4X[32X[104X
  
  [1X8.3-11 ApsisMonoid[101X
  
  [33X[1;0Y[29X[2XApsisMonoid[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularApsisMonoid[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCrossedApsisMonoid[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSingularCrossedApsisMonoid[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA bipartition monoid.[133X
  
  [33X[0;0YIf  [3Xm[103X  and  [3Xn[103X are positive integers, then this operation returns the [3Xm[103X-apsis
  monoid  of  degree  [3Xn[103X.  The  [3Xm[103X[13X-apsis  monoid[113X  is  the monoid of bipartitions
  generated  when  the  diapses in generators of the Jones monoid are replaced
  with  [3Xm[103X-apses.  Note  that  an  [3Xm[103X[13X-apsis[113X is a block that contains precisely [3Xm[103X
  consecutive integers.[133X
  
  [33X[0;0Y[10XSingularApsisMonoid[110X  returns the ideal of the apsis monoid consisting of the
  non-invertible elements (i.e. those not in the group of units), when [3Xm[103X [22X≤[122X [3Xn[103X.[133X
  
  [33X[0;0Y[10XCrossedApsisGeneratedMonoid[110X returns the semigroup generated by the symmetric
  group of degree [3Xn[103X and the [3Xm[103X-apsis monoid of degree [3Xn[103X.[133X
  
  [33X[0;0Y[10XSingularCrossedApsisMonoid[110X  returns  the  ideal  of the crossed apsis monoid
  consisting  of  the  non-invertible elements (i.e. those not in the group of
  units), when [3Xm <= n[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := ApsisMonoid(3, 7);[127X[104X
    [4X[28X<regular bipartition *-monoid of degree 7 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X320[128X[104X
    [4X[25Xgap>[125X [27XT := SingularApsisMonoid(3, 7);[127X[104X
    [4X[28X<regular bipartition *-semigroup ideal of degree 7 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XDifference(S, T) = [One(S)];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(CrossedApsisMonoid(2, 5));[127X[104X
    [4X[28X945[128X[104X
    [4X[25Xgap>[125X [27XSingularCrossedApsisMonoid(4, 6);[127X[104X
    [4X[28X<regular bipartition *-semigroup ideal of degree 6 with 1 generator>[128X[104X
  [4X[32X[104X
  
  
  [1X8.4 [33X[0;0YStandard PBR semigroups[133X[101X
  
  [33X[0;0YIn  this  section, we describe the operations in [5XSemigroups[105X that can be used
  to  create  standard  examples of semigroups of partitioned binary relations
  (PBRs). See Chapter [14X4[114X for more information about PBRs.[133X
  
  [1X8.4-1 FullPBRMonoid[101X
  
  [33X[1;0Y[29X[2XFullPBRMonoid[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA PBR monoid.[133X
  
  [33X[0;0YIf  [3Xn[103X  is a positive integer not greater than [10X2[110X, then this operation returns
  the  monoid  consisting of all of the partitioned binary relations (PBRs) of
  degree  [3Xn[103X;  called  the [13Xfull PBR monoid[113X. There are [10X2 ^ ((2 * n) ^ 2)[110X PBRs of
  degree  [3Xn[103X. The full PBR monoid of degree [3Xn[103X is currently too large to compute
  when [22X[3Xn[103X ≥ 3[122X.[133X
  
  [33X[0;0YThe full PBR monoid is not regular in general.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FullPBRMonoid(1);[127X[104X
    [4X[28X<pbr monoid of degree 1 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XS := FullPBRMonoid(2);[127X[104X
    [4X[28X<pbr monoid of degree 2 with 10 generators>[128X[104X
  [4X[32X[104X
  
  
  [1X8.5 [33X[0;0YSemigroups of matrices over a finite field[133X[101X
  
  [33X[0;0YIn  this  section, we describe the operations in [5XSemigroups[105X that can be used
  to  create  semigroups  of  matrices  over  a finite field that belonging to
  several  standard  classes of example. See the section [14X'[33X[0;0YMatrices over finite
  fields[133X'[114X for more information about matrices over a finite field.[133X
  
  [1X8.5-1 FullMatrixMonoid[101X
  
  [33X[1;0Y[29X[2XFullMatrixMonoid[102X( [3Xd[103X, [3Xq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XGeneralLinearMonoid[102X( [3Xd[103X, [3Xq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XGLM[102X( [3Xd[103X, [3Xq[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA matrix monoid.[133X
  
  [33X[0;0YThese  operations  return the full matrix monoid of [3Xd[103X by [3Xd[103X matrices over the
  field  with  [3Xq[103X  elements.  The [13Xfull matrix monoid[113X, also known as the [13Xgeneral
  linear monoid[113X, with these parameters, is the monoid consisting of all [3Xd[103X by [3Xd[103X
  matrices  with  entries  from  the  field [10XGF([3Xq[103X[10X)[110X. This monoid has [10X[3Xq[103X[10X ^ ([3Xd[103X[10X ^ 2)[110X
  elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FullMatrixMonoid(2, 4);[127X[104X
    [4X[28X<general linear monoid 2x2 over GF(2^2)>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X256[128X[104X
    [4X[25Xgap>[125X [27XS = GeneralLinearMonoid(2, 4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XGLM(2, 2);[127X[104X
    [4X[28X<general linear monoid 2x2 over GF(2)>[128X[104X
  [4X[32X[104X
  
  [1X8.5-2 SpecialLinearMonoid[101X
  
  [33X[1;0Y[29X[2XSpecialLinearMonoid[102X( [3Xd[103X, [3Xq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSLM[102X( [3Xd[103X, [3Xq[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA matrix monoid.[133X
  
  [33X[0;0YThese  operations  return  the special linear monoid of [3Xd[103X by [3Xd[103X matrices over
  the  field  with  [3Xq[103X  elements.  The  [13Xspecial  linear  monoid[113X  is  the monoid
  consisting  of  all  [3Xd[103X  by [3Xd[103X matrices with entries from the field [10XGF([3Xq[103X[10X)[110X that
  have determinant [10X0[110X or [10X1[110X. In other words, the special linear monoid is formed
  from the general linear monoid of the same parameters by replacing its group
  of units (the general linear group) by the special linear group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SpecialLinearMonoid(2, 4);[127X[104X
    [4X[28X<regular monoid of 2x2 matrices over GF(2^2) with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XS = SLM(2, 4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X136[128X[104X
  [4X[32X[104X
  
  [1X8.5-3 IsFullMatrixMonoid[101X
  
  [33X[1;0Y[29X[2XIsFullMatrixMonoid[102X( [3XS[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsGeneralLinearMonoid[102X( [3XS[103X ) [32X property[133X
  
  [33X[0;0Y[10XIsFullMatrixMonoid[110X  and [10XIsGeneralLinearMonoid[110X return [9Xtrue[109X if the semigroup [10XS[110X
  was  created  using  either  of  the  commands  [2XFullMatrixMonoid[102X  ([14X8.5-1[114X) or
  [2XGeneralLinearMonoid[102X ([14X8.5-1[114X) and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := RandomSemigroup(IsTransformationSemigroup, 4, 4);;[127X[104X
    [4X[25Xgap>[125X [27XIsFullMatrixMonoid(S);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XS := GeneralLinearMonoid(3, 3);[127X[104X
    [4X[28X<general linear monoid 3x3 over GF(3)>[128X[104X
    [4X[25Xgap>[125X [27XIsFullMatrixMonoid(S);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X8.6 [33X[0;0YSemigroups of boolean matrices[133X[101X
  
  [33X[0;0YIn  this  section, we describe the operations in [5XSemigroups[105X that can be used
  to  create  semigroups  of  boolean  matrices  belonging to several standard
  classes  of example. See the section [14X'[33X[0;0YBoolean matrices[133X'[114X for more information
  about boolean matrices.[133X
  
  [1X8.6-1 FullBooleanMatMonoid[101X
  
  [33X[1;0Y[29X[2XFullBooleanMatMonoid[102X( [3Xd[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe monoid of all boolean matrices of dimension [3Xd[103X.[133X
  
  [33X[0;0YIf  [3Xd[103X  is  a  positive  integer less than or equal to [10X5[110X, then this operation
  returns  the  full  boolean  matrix  monoid of dimension [3Xd[103X. The [13Xfull boolean
  matrix  monoid of dimension [3Xd[103X[113X is the monoid consisting of all [3Xd[103X by [3Xd[103X boolean
  matrices, and has [10X2 ^ ([3Xn[103X[10X ^ 2)[110X matrices.[133X
  
  [33X[0;0Y[10XFullBooleanMatMonoid[110X  returns a monoid with a generating set that is minimal
  in size. These generating sets are pre-computed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FullBooleanMatMonoid(3);[127X[104X
    [4X[28X<monoid of 3x3 boolean matrices with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X512[128X[104X
  [4X[32X[104X
  
  [1X8.6-2 RegularBooleanMatMonoid[101X
  
  [33X[1;0Y[29X[2XRegularBooleanMatMonoid[102X( [3Xd[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA monoid of boolean matrices.[133X
  
  [33X[0;0YIf  [3Xd[103X is a positive integer, then [10XRegularBooleanMatMonoid[110X returns the monoid
  generated  by  the regular [3Xd[103X by [3Xd[103X boolean matrices. Note that this monoid is
  [13Xnot[113X  regular in general. [10XRegularBooleanMatMonoid([3Xd[103X[10X)[110X is generated by the four
  boolean matrices [10XA, B, C, D[110X, whose [9Xtrue[109X entries are:[133X
  
  [30X    [33X[0;6Y[10XA[i][i + 1][110X and [10XA[n][1][110X, for [22Xi ∈ {1, ..., n - 1}[122X;[133X
  
  [30X    [33X[0;6Y[10XB[1][2][110X, [10XB[2][1][110X, and [10XB[i][i][110X for [22Xi ∈ {3, ..., n}[122X;[133X
  
  [30X    [33X[0;6Y[10XC[1][2][110X and [10XC[i][i][110X, for [22Xi ∈ {2, ..., n - 1}[122X; and[133X
  
  [30X    [33X[0;6Y[10XD[1][2][110X, [10XD[i][i][110X, for [22Xi ∈ {2, ..., n}[122X, and [10XD[n][1][110X.[133X
  
  [33X[0;0YThis monoid has nearly [10X2 ^ (n ^ 2)[110X elements.[133X
  
  [1X8.6-3 ReflexiveBooleanMatMonoid[101X
  
  [33X[1;0Y[29X[2XReflexiveBooleanMatMonoid[102X( [3Xd[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA monoid of boolean matrices.[133X
  
  [33X[0;0YIf  [3Xd[103X  is  a  positive  integer less than or equal to [10X5[110X, then this operation
  returns  the  monoid  consisting of all reflexive [3Xd[103X by [3Xd[103X boolean matrices. A
  boolean  matrix  [10Xmat[110X  is  [13Xreflexive[113X if each entry of its leading diagonal is
  [9Xtrue[109X, i.e. if [10Xmat[i][i][110X is [9Xtrue[109X for all [22Xi ∈ {1, ..., d}[122X.[133X
  
  [33X[0;0YThe  generating  sets  for the monoids returned by [10XReflexiveBooleanMatMonoid[110X
  are  pre-computed, and read from a file. Small generating sets are not known
  for [22X[3Xd[103X ≥ 6[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := ReflexiveBooleanMatMonoid(3);[127X[104X
    [4X[28X<monoid of 3x3 boolean matrices with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X64[128X[104X
  [4X[32X[104X
  
  [1X8.6-4 HallMonoid[101X
  
  [33X[1;0Y[29X[2XHallMonoid[102X( [3Xd[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA monoid of boolean matrices.[133X
  
  [33X[0;0YIf  [3Xd[103X  is  a  positive  integer less than or equal to [10X5[110X, then this operation
  returns  the monoid consisting Hall matrices of degree [3Xd[103X. A [13XHall matrix[113X is a
  boolean  matrix  in  which  every column and every row contains at least one
  [9Xtrue[109X  entry. Equivalently, a Hall matrix is a boolean matrix than contains a
  permutation.[133X
  
  [33X[0;0YA  Hall  matrix  of dimension [3Xd[103X corresponds to a solution to Hall's Marriage
  Problem,  when  there  are  two  collection  of [3Xd[103X people. Thus the number of
  solutions  to  Hall's  Marriage  Problem  in  this instance is the number of
  elements of [10XHallMonoid([3Xd[103X[10X)[110X.[133X
  
  [33X[0;0YThe  operation  [10XHallMonoid[110X  returns  a  monoid with a generating set that is
  minimal  in  size.  These  generating  sets  are pre-computed, and a minimal
  generating set is not known for larger dimensions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := HallMonoid(3);[127X[104X
    [4X[28X<monoid of 3x3 boolean matrices with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X247[128X[104X
  [4X[32X[104X
  
  [1X8.6-5 GossipMonoid[101X
  
  [33X[1;0Y[29X[2XGossipMonoid[102X( [3Xd[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA monoid of boolean matrices.[133X
  
  [33X[0;0YIf  [3Xd[103X  is  a positive integer, then this operation returns the [3Xd[103X by [3Xd[103X gossip
  monoid.  The  [13Xgossip  monoid[113X  is  defined  to be the monoid generated by the
  collection  of  all  [3Xd[103X  by  [3Xd[103X  boolean  matrices  that define an equivalence
  relation; see [2XIsEquivalenceBooleanMat[102X ([14X5.3-16[114X).[133X
  
  [33X[0;0YFor  [22X[3Xd[103X ≥ 2[122X, [10XGossipMonoid([3Xd[103X[10X)[110X returns a monoid with [22Xd choose 2[122X generators. The
  generating  set  is  the  collection  of  boolean  matrices  that  define an
  equivalence  relation that has one equivalence class of size [10X2[110X, and no other
  non-trivial  equivalence  classes. Note that this generating set is strictly
  contained   within  the  collection  of  all  equivalence  relation  boolean
  matrices.[133X
  
  [33X[0;0YThe  number of elements of [10XGossipMonoid([3Xd[103X[10X)[110X is known for some small values of
  [3Xd[103X  —  see [BDF15] for more information about the gossip monoid, and its size
  for [22X[3Xd[103X ≤ 9[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := GossipMonoid(3);[127X[104X
    [4X[28X<monoid of 3x3 boolean matrices with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X11[128X[104X
  [4X[32X[104X
  
  [1X8.6-6 TriangularBooleanMatMonoid[101X
  
  [33X[1;0Y[29X[2XTriangularBooleanMatMonoid[102X( [3Xd[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XUnitriangularBooleanMatMonoid[102X( [3Xd[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA monoid of boolean matrices.[133X
  
  [33X[0;0YIf  [3Xd[103X  is  a  positive  integer, then [10XTriangularBooleanMatMonoid[110X returns the
  monoid consisting of the upper-triangular [3Xd[103X by [3Xd[103X boolean matrices. A boolean
  matrix is [13Xupper-triangular[113X if the entry in row [10Xi[110X, column [10Xj[110X is [9Xfalse[109X whenever
  [10Xi > j[110X.[133X
  
  [33X[0;0Y[10XUnitriangularBooleanMatMonoid[110X    returns    the    subsemigroup    of    the
  [10XTriangularBooleanMatMonoid[110X   that  consists  of  reflexive  upper-triangular
  boolean matrices; see [2XReflexiveBooleanMatMonoid[102X ([14X8.6-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := TriangularBooleanMatMonoid(3);[127X[104X
    [4X[28X<monoid of 3x3 boolean matrices with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X64[128X[104X
    [4X[25Xgap>[125X [27XT := UnitriangularBooleanMatMonoid(4);[127X[104X
    [4X[28X<monoid of 4x4 boolean matrices with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(T);[127X[104X
    [4X[28X64[128X[104X
  [4X[32X[104X
  
  
  [1X8.7 [33X[0;0YSemigroups of matrices over a semiring[133X[101X
  
  [33X[0;0YIn  this  section, we describe the operations in [5XSemigroups[105X that can be used
  to  create  semigroups  of  matices  over  a semiring that belong to several
  standard  classes  of  example.  See  Chapter  [14X5[114X  for more information about
  matrices over a semiring.[133X
  
  [1X8.7-1 FullTropicalMaxPlusMonoid[101X
  
  [33X[1;0Y[29X[2XFullTropicalMaxPlusMonoid[102X( [3Xd[103X, [3Xt[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA monoid of tropical max plus matrices.[133X
  
  [33X[0;0YIf [10X[3Xd[103X[10X = 2[110X and [3Xt[103X is a positive integer, then [10XFullTropicalMaxPlusMonoid[110X returns
  the  monoid consisting of all [3Xd[103X by [3Xd[103X matrices with entries from the tropical
  max-plus semiring with threshold [3Xt[103X. A small generating set for larger values
  of [3Xd[103X is not currently known.[133X
  
  [33X[0;0YThis monoid contains [10X([3Xt[103X[10X + 2) ^ ([3Xd[103X[10X ^ 2)[110X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FullTropicalMaxPlusMonoid(2, 5);[127X[104X
    [4X[28X<monoid of 2x2 tropical max-plus matrices with 24 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X2401[128X[104X
    [4X[25Xgap>[125X [27X(5 + 2) ^ (2 ^ 2);[127X[104X
    [4X[28X2401[128X[104X
  [4X[32X[104X
  
  [1X8.7-2 FullTropicalMinPlusMonoid[101X
  
  [33X[1;0Y[29X[2XFullTropicalMinPlusMonoid[102X( [3Xd[103X, [3Xt[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA monoid of tropical min plus matrices.[133X
  
  [33X[0;0YIf   [3Xd[103X   is   equal   to  [10X2[110X  or  [10X3[110X,  and  [3Xt[103X  is  a  positive  integer,  then
  [10XFullTropicalMinPlusMonoid[110X  returns  the  monoid  consisting  of  all  [3Xd[103X by [3Xd[103X
  matrices  with entries from the tropical min-plus semiring with threshold [3Xt[103X.
  A small generating set for larger values of [3Xd[103X is not currently known.[133X
  
  [33X[0;0YThis monoid contains [10X([3Xt[103X[10X + 2) ^ ([3Xd[103X[10X ^ 2)[110X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := FullTropicalMinPlusMonoid(2, 3);[127X[104X
    [4X[28X<monoid of 2x2 tropical min-plus matrices with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X625[128X[104X
    [4X[25Xgap>[125X [27X(3 + 2) ^ (2 ^ 2);[127X[104X
    [4X[28X625[128X[104X
  [4X[32X[104X
  
