  
  [1X4 [33X[0;0YFunctionality[133X[101X
  
  
  [1X4.1 [33X[0;0YIndividual Semigroups[133X[101X
  
  [33X[0;0YThe  semigroups  of  sizes  1  to  8  are  available  up  to isomorphism and
  anti-isomorphism  in [5XSmallsemi[105X. Every semigroup in the library is identified
  by its size [22Xm[122X and a number [22Xn[122X lying between 1 and the number of semigroups of
  size [22Xm[122X (see Table [14X1.[114X). We call the pair [22X(m,n)[122X the [13XID[113X of the semigroup.[133X
  
  [33X[0;0YIn  this  section we give details about the functions relating to individual
  semigroups  in  [5XSmallsemi[105X.  This  includes  how  to access semigroups in the
  library  and  how  to identify the semigroup in the library equivalent to an
  arbitrary semigroup (of size 1 to 8).[133X
  
  [33X[0;0YIf  you  are  interested  in the properties of a semigroup in the library or
  would  like  to find all the semigroups satisfying a given set of properties
  please see Section [14X4.2[114X or Section [14X4.5[114X respectively.[133X
  
  [1X4.1-1 SmallSemigroup[101X
  
  [29X[2XSmallSemigroup[102X( [3Xm[103X, [3Xn[103X ) [32X function
  [29X[2XSmallSemigroupNC[102X( [3Xm[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Yreturns  the  semigroup  with  ID  [22X([3Xm,n[103X)[122X  from  the library, that is the [3Xn[103Xth
  semigroup with [3Xm[103X elements.[133X
  
  [33X[0;0YIn  [10XSmallSemigroupNC[110X  no check is performed to verify that [3Xm[103X and [3Xn[103X are valid
  arguments.[133X
  
  [33X[0;0YIn  [10XSmallSemigroup[110X an error is signalled if the semigroups of size [3Xm[103X are not
  classified or if [3Xn[103X is greater than the number of semigroups with [3Xm[103X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSmallSemigroup(8,1353452);[127X[104X
    [4X[28X<small semigroup of size 8>[128X[104X
    [4X[25Xgap>[125X [27XSmallSemigroupNC(5,1);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XSmallSemigroupNC(5,1)=SmallSemigroup(5,1);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.1-2 IsSmallSemigroup[101X
  
  [29X[2XIsSmallSemigroup[102X( [3Xsgrp[103X ) [32X filter
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  [3Xsgrp[103X  is  a semigroup from the library, that is if it was
  created using [2XSmallSemigroup[102X ([14X4.1-1[114X). Otherwise [10Xfalse[110X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgrp:=RandomSmallSemigroup(5);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsSmallSemigroup(sgrp);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xsgrp:=Semigroup(Transformation([1]));;[127X[104X
    [4X[25Xgap>[125X [27XIsSmallSemigroup(sgrp);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X4.1-3 IsSmallSemigroupElt[101X
  
  [29X[2XIsSmallSemigroupElt[102X( [3Xx[103X ) [32X filter
  
  [33X[0;0Yreturns  [10Xtrue[110X  if [3Xx[103X is an element of a semigroup from the library, and [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0Y[3XIsSmallSemigroupElt[103X is a representation satisfying [3XIsPositionalObjectRep[103X and
  [2XIsMultiplicativeElement[102X     ([14XReference:     IsMultiplicativeElement[114X)     and
  [3XIsAttributeStoringRep[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsSmallSemigroupElt(Transformation([1]));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xsgrp:=RandomSmallSemigroup(5);;[127X[104X
    [4X[25Xgap>[125X [27XIsSmallSemigroupElt(Random(sgrp));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.1-4 RecoverMultiplicationTable[101X
  
  [29X[2XRecoverMultiplicationTable[102X( [3Xm[103X, [3Xn[103X ) [32X function
  [29X[2XRecoverMultiplicationTableNC[102X( [3Xm[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Yreturn  the  multiplication table of the [3Xn[103X-th semigroup with [3Xm[103X elements from
  the library.[133X
  
  [33X[0;0YIf [3Xm[103X is greater than 8 or [3Xn[103X greater than the number of semigroups of size [3Xm[103X,
  then  [10Xfail[110X  is  returned.  The  NC version does not perform any tests on the
  input and will most likely run into an error in such a case.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRecoverMultiplicationTable(10,2);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XRecoverMultiplicationTable(1,2); [127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XRecoverMultiplicationTable(2,1);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XRecoverMultiplicationTable(8,11111111);[127X[104X
    [4X[28X[ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 3 ], [128X[104X
    [4X[28X  [ 3, 3, 3, 3, 3, 3, 3, 3 ], [ 1, 1, 1, 4, 4, 4, 4, 1 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 4, 5, 6, 7, 1 ], [ 1, 2, 3, 4, 5, 6, 7, 1 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 4, 5, 6, 7, 1 ], [ 8, 8, 8, 8, 8, 8, 8, 8 ] ][128X[104X
    [4X[25Xgap>[125X [27XRecoverMultiplicationTable(2,11111111);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  no semigroup is created calling this function but just the table
  is  created.  This makes it useful if one wants to perform very simple (i.e.
  quick  in  [5XGAP[105X) tests on a large number of semigroups which can be performed
  on the multiplication table.[133X
  
  [33X[0;0YTo   create   a   semigroup   with  the  multiplication  table  obtained  by
  [10XRecoverMultiplicationTable([3Xm,n[103X[10X)[110X use the function [2XSmallSemigroup[102X ([14X4.1-1[114X) with
  arguments [3Xm,n[103X.[133X
  
  [1X4.1-5 SemigroupByMultiplicationTableNC[101X
  
  [29X[2XSemigroupByMultiplicationTableNC[102X( [3Xtable[103X ) [32X function
  
  [33X[0;0Yreturns   an   object   with   [2XIsSemigroup[102X   ([14XReference:   IsSemigroup[114X)  and
  multiplication table [3Xtable[103X without checking if the multiplication defined by
  the table is associative.[133X
  
  [33X[0;0YIf  [3Xtable[103X  is  not associative, this can lead to errors and wrong results or
  might even crash [5XGAP[105X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SemigroupByMultiplicationTableNC([[1,2],[2,1]]);[127X[104X
    [4X[28X<semigroup of size 2, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSmallSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote that this function is [13Xnot[113X used to create semigroups when [2XSmallSemigroup[102X
  ([14X4.1-1[114X)    is    called.    It   can   be   useful   in   combination   with
  [2XRecoverMultiplicationTable[102X  ([14X4.1-4[114X)  if  one wants to avoid that a semigroup
  knows it comes from the library.[133X
  
  [1X4.1-6 IdSmallSemigroup[101X
  
  [29X[2XIdSmallSemigroup[102X( [3Xsgrp[103X ) [32X attribute
  
  [33X[0;0Yreturns  a pair [10X[m, n][110X such that [22X(m,n)[122X is the ID of a semigroup in [5XSmallsemi[105X
  equivalent  to  [3Xsgrp[103X.  The  argument [3Xsgrp[103X has to be a semigroup of size 8 or
  less, otherwise an error is signalled.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgrp:=Semigroup(Transformation( [ 1, 2, 2 ] ), Transformation( [ 1, 2, 3 ] ));;[127X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(sgrp);[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
  [4X[32X[104X
  
  [1X4.1-7 EquivalenceSmallSemigroup[101X
  
  [29X[2XEquivalenceSmallSemigroup[102X( [3Xsgrp[103X ) [32X attribute
  
  [33X[0;0Yreturns  a mapping [10Xmap[110X from [3Xsgrp[103X to the semigroup in [5XSmallsemi[105X equivalent to
  [3Xsgrp[103X.  The  mapping is an isomorphism if such exists and an anti-isomorphism
  otherwise.  The  argument  [3Xsgrp[103X  has  to  be  a semigroup of size 8 or less,
  otherwise an error is signalled.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgrp:=Semigroup(Transformation( [ 1, 2, 2 ] ), [127X[104X
    [4X[25X>[125X [27XTransformation( [ 1, 2, 3 ] ));;[127X[104X
    [4X[25Xgap>[125X [27XEquivalenceSmallSemigroup(sgrp);[127X[104X
    [4X[28XSemigroupHomomorphismByImages ( Monoid( [128X[104X
    [4X[28X[ Transformation( [ 1, 2, 2 ] ) ] )-><small semigroup of size 2>)[128X[104X
  [4X[32X[104X
  
  [1X4.1-8 InfoSmallsemi[101X
  
  [29X[2XInfoSmallsemi[102X[32X info class
  
  [33X[0;0Yis  the  info class (see [14X'Reference: Info Functions'[114X) of [5XSmallsemi[105X. The info
  level is initially set to 1 which triggers a message whenever data is loaded
  into [5XGAP[105X.[133X
  
  [1X4.1-9 UnloadSmallsemiData[101X
  
  [29X[2XUnloadSmallsemiData[102X( [3Xuse_later[103X ) [32X function
  
  [33X[0;0Ydeletes  most  or  all of the data from the [5XGAP[105X workspace that was loaded by
  [5XSmallsemi[105X.[133X
  
  [33X[0;0YIf  the  boolean  [3Xuse_later[103X is [10Xfalse[110X all data loaded by [5XSmallsemi[105X is deleted
  from  the  workspace,  in  which  case  [5XSmallsemi[105X  is not guaranteed to work
  properly without restarting your [5XGAP[105X session.[133X
  
  [33X[0;0YIf  the boolean [3Xuse_later[103X is [10Xtrue[110X only the recoverable data is deleted. This
  leaves roughly 10 MB of data in the workspace.[133X
  
  
  [1X4.2 [33X[0;0YProperties of Semigroups[133X[101X
  
  [33X[0;0YIn  this  section  we detail the [5XGAP[105X functions that can be used to determine
  whether  a  small  semigroup  satisfies  a  certain  property.  Let  [3XS[103X  be a
  semigroup. Then[133X
  
  [30X    [33X[0;6Y[3XS[103X is a [13Xleft zero semigroup[113X if [3Xxy=x[103X for all [3Xx,y[103X in [3XS[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X is a [13Xright zero semigroup[113X if [3Xxy=y[103X for all [3Xx,y[103X in [3XS[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X is [13Xcommutative[113X if [3Xxy=yx[103X for all [3Xx,y[103X in [3XS[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X is [13Xsimple[113X if it has no proper two-sided ideals.[133X
  
  [30X    [33X[0;6Y[3XS[103X is [13Xzero simple[113X if the only 2-sided ideals are [3X{0}[103X and [3XS[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X is [13Xregular[113X if for all [3Xx[103X in [3XS[103X there exists [3Xy[103X in [3XS[103X such that [3Xxyx=x[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X is [13Xcompletely regular[113X if every element of [3XS[103X lies in a subgroup.[133X
  
  [30X    [33X[0;6Y[3XS[103X  is  an  [13Xinverse  semigroup[113X  if  every  element  [3Xx[103X in [3XS[103X has a unique
        semigroup  inverse,  that  is,  a unique element [3Xy[103X such that [3Xxyx=x[103X and
        [3Xyxy=y[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X  is  a  [13XClifford  semigroup[113X  if  it  is  a  regular  semigroup whose
        idempotents are central, that is, for all [3Xe,x[103X in [3XS[103X where [3Xe^2=e[103X we have
        that [3Xex=xe[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X  is a [13Xband[113X if every element is an idempotent, that is, [3Xx^2=x[103X for all
        [3Xx[103X in [3XS[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X is a [13XBrandt semigroup[113X if it is inverse and zero simple.[133X
  
  [30X    [33X[0;6Y[3XS[103X  is  a [13Xrectangular band[113X if for all [3Xx,y,z[103X in [3XS[103X we have that [3Xx^2=x[103X and
        [3Xxyz=xz[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X  is  a  [13Xsemiband[113X if it is generated by its idempotent elements, that
        is, elements satisfying [3Xx^2=x[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X  is  an  [13Xorthodox  semigroup[113X  if  it  is regular and its idempotents
        (elements satisfying [3Xx^2=x[103X) form a subsemigroup.[133X
  
  [30X    [33X[0;6Y[3XS[103X is a [13Xzero semigroup[113X if there exists an element [3X0[103X in [3XS[103X such that [3Xxy=0[103X
        for all [3Xx,y[103X in [3XS[103X.[133X
  
  [30X    [33X[0;6Y[3XS[103X  is  a  [13Xzero  group[113X  if  there  exists an element [3X0[103X in [3XS[103X such that [3XS[103X
        without [3X0[103X is a group and for all [3Xx[103X in [3XS[103X we have that [3Xx0=0x=0[103X.[133X
  
  [33X[0;0YThe  [5XMONOID[105X  package  was used to determined which semigroups in the library
  satisfy  the properties above. All of the resulting information is stored in
  the library.[133X
  
  [1X4.2-1 Annihilators[101X
  
  [29X[2XAnnihilators[102X( [3Xsgrp[103X ) [32X attribute
  
  [33X[0;0Yreturns  the set of annihilators of [3Xsgrp[103X if [3Xsgrp[103X contains a zero element and
  [10Xfail[110X otherwise.[133X
  
  [33X[0;0YAn  element  [22Xx[122X  in  a semigroup with zero [22Xz[122X is an [13Xannihilator[113X if [22Xxy=yx=z[122X for
  every element [22Xy[122X in the semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := SmallSemigroup(5,6);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XAnnihilators(s);[127X[104X
    [4X[28X[ s1, s2 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-2 DiagonalOfMultiplicationTable[101X
  
  [29X[2XDiagonalOfMultiplicationTable[102X( [3Xsgrp[103X ) [32X attribute
  
  [33X[0;0Yreturns the diagonal of the multiplication table of the semigroup [3Xsgrp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(8,10101);;[127X[104X
    [4X[25Xgap>[125X [27XDiagonalOfMultiplicationTable(s);[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7,10101);;[127X[104X
    [4X[25Xgap>[125X [27XDiagonalOfMultiplicationTable(s);[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-3 DisplaySmallSemigroup[101X
  
  [29X[2XDisplaySmallSemigroup[102X( [3Xsgrp[103X ) [32X function
  
  [33X[0;0Ydisplays  all  the information about the small semigroup [3Xsgrp[103X that is stored
  in the library and its Green's classes and idempotents.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(6, 3838);;[127X[104X
    [4X[25Xgap>[125X [27XDisplaySmallSemigroup(s);[127X[104X
    [4X[28XIsBand:                              false[128X[104X
    [4X[28XIsBrandtSemigroup:                   false[128X[104X
    [4X[28XIsCommutative:                       false[128X[104X
    [4X[28XIsCompletelyRegularSemigroup:        false[128X[104X
    [4X[28XIsFullTransformationSemigroupCopy:   false[128X[104X
    [4X[28XIsGroupAsSemigroup:                  false[128X[104X
    [4X[28XIsIdempotentGenerated:               false[128X[104X
    [4X[28XIsInverseSemigroup:                  false[128X[104X
    [4X[28XIsMonogenicSemigroup:                false[128X[104X
    [4X[28XIsMonoidAsSemigroup:                 false[128X[104X
    [4X[28XIsMultSemigroupOfNearRing:           false[128X[104X
    [4X[28XIsOrthodoxSemigroup:                 false[128X[104X
    [4X[28XIsRectangularBand:                   false[128X[104X
    [4X[28XIsRegularSemigroup:                  false[128X[104X
    [4X[28XIsSelfDualSemigroup:                 false[128X[104X
    [4X[28XIsSemigroupWithClosedIdempotents:    true[128X[104X
    [4X[28XIsSimpleSemigroup:                   false[128X[104X
    [4X[28XIsSingularSemigroupCopy:             false[128X[104X
    [4X[28XIsZeroSemigroup:                     false[128X[104X
    [4X[28XIsZeroSimpleSemigroup:               false[128X[104X
    [4X[28XMinimalGeneratingSet:                [ s3, s4, s5, s6 ][128X[104X
    [4X[28XIdempotents:                         [ s1, s5, s6 ][128X[104X
    [4X[28XGreensRClasses:                      [ {s1}, {s2}, {s3}, {s4}, {s5}, {\[128X[104X
    [4X[28Xs6} ][128X[104X
    [4X[28XGreensLClasses:                      [ {s1}, {s2}, {s3}, {s4}, {s6} ][128X[104X
    [4X[28XGreensHClasses:                      [ {s1}, {s2}, {s3}, {s4}, {s5}, {\[128X[104X
    [4X[28Xs6} ][128X[104X
    [4X[28XGreensDClasses:                      [ {s1}, {s2}, {s3}, {s4}, {s6} ][128X[104X
  [4X[32X[104X
  
  [1X4.2-4 IndexPeriod[101X
  
  [29X[2XIndexPeriod[102X( [3Xx[103X ) [32X attribute
  
  [33X[0;0Yreturns  the  minimum numbers [3Xm, r[103X such that [3Xx^{m+r}=x^m[103X; known as the index
  and period of the small semigroup element [3Xx[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5,116);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27Xx:=Elements(s)[3];[127X[104X
    [4X[28Xs3[128X[104X
    [4X[25Xgap>[125X [27XIndexPeriod(x);[127X[104X
    [4X[28X[ 2, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xx^3=x^2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx^2=x^1;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xx^3=x^1;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X4.2-5 IsBand[101X
  
  [29X[2XIsBand[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns [10Xtrue[110X if the small semigroup [3Xsgrp[103X is a band and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YA semigroup [3Xsgrp[103X is a [13Xband[113X if every element is an idempotent, that is, [3Xx^2=x[103X
  for all [3Xx[103X in [3Xsgrp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5,519);;[127X[104X
    [4X[25Xgap>[125X [27XIsBand(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(5, IsBand, true);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsBand(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);   [127X[104X
    [4X[28X[ 5, 1010 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-6 IsBrandtSemigroup[101X
  
  [29X[2XIsBrandtSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  small semigroup [3Xsgrp[103X is a Brandt semigroup and [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YA  finite  semigroup  [3Xsgrp[103X  is  a [13XBrandt semigroup[113X if it is inverse and zero
  simple.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5,519);;[127X[104X
    [4X[25Xgap>[125X [27XIsBrandtSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(5, IsBrandtSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsBrandtSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);   [127X[104X
    [4X[28X[ 5, 149 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-7 IsCliffordSemigroup[101X
  
  [29X[2XIsCliffordSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if the small semigroup [3Xsgrp[103X is a Clifford semigroup and [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YA  semigroup [3Xsgrp[103X is a [13XClifford semigroup[113X if it is a regular semigroup whose
  idempotents  are  central,  that is, for all [3Xe,x[103X in [3Xsgrp[103X where [3Xe^2=e[103X we have
  that [3Xex=xe[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5,519);; [127X[104X
    [4X[25Xgap>[125X [27XIsBand(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(5, IsBand, true);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsBand(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);   [127X[104X
    [4X[28X[ 5, 1010 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5,519);;[127X[104X
    [4X[25Xgap>[125X [27XIsCliffordSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(5, IsCliffordSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsCliffordSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 5, 148 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-8 IsCommutativeSemigroup[101X
  
  [29X[2XIsCommutativeSemigroup[102X( [3Xsgrp[103X ) [32X property
  [29X[2XIsCommutative[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturn [10Xtrue[110X if the small semigroup [3Xsgrp[103X is commutative and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YA semigroup [3Xsgrp[103X is [13Xcommutative[113X if [3Xxy=yx[103X for all [3Xx,y[103X in [3Xsgrp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(6,871);;[127X[104X
    [4X[25Xgap>[125X [27XIsCommutativeSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(5, IsCommutative, true);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsCommutativeSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCommutative(s);         [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 5, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(5, IsCommutativeSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsCommutativeSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCommutative(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-9 IsCompletelyRegularSemigroup[101X
  
  [29X[2XIsCompletelyRegularSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  semigroup  [3Xsgrp[103X  is  completely  regular  and  [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YA  semigroup  is  [13Xcompletely  regular[113X  if  every  element  is contained in a
  subgroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(6,13131);                           [127X[104X
    [4X[28X<small semigroup of size 6>[128X[104X
    [4X[25Xgap>[125X [27XIsCompletelyRegularSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(6, IsCompletelyRegularSemigroup, true); [127X[104X
    [4X[28X<small semigroup of size 6>[128X[104X
    [4X[25Xgap>[125X [27XIsCompletelyRegularSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 6, 3164 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-10 IsFullTransformationSemigroupCopy[101X
  
  [29X[2XIsFullTransformationSemigroupCopy[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the semigroup [3Xsgrp[103X is isomorphic to a full transformation
  semigroup and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YThe size of the full transformation semigroup on an [3Xn[103X element set is [22Xn^n[122X and
  so there are only two semigroup in the library that have this property.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(1,1);[127X[104X
    [4X[28X<small semigroup of size 1>[128X[104X
    [4X[25Xgap>[125X [27XIsFullTransformationSemigroupCopy(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(4, IsFullTransformationSemigroupCopy, true);[127X[104X
    [4X[28X<small semigroup of size 4>[128X[104X
    [4X[25Xgap>[125X [27XIsFullTransformationSemigroupCopy(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 4, 96 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(6, IsFullTransformationSemigroupCopy, true);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X4.2-11 IsGroupAsSemigroup[101X
  
  [29X[2XIsGroupAsSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  small  semigroup [3Xsgrp[103X is mathematically a group, and
  returns  [10Xfalse[110X  otherwise.  Note  that  no  small  semigroup  can lie in the
  category [2XIsGroup[102X ([14XReference: IsGroup[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7,7);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsGroupAsSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(4,37);;[127X[104X
    [4X[25Xgap>[125X [27XIsGroupAsSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X4.2-12 [33X[0;0YIsIdempotentGenerated[133X[101X
  
  [29X[2XIsIdempotentGenerated[102X( [3Xsgrp[103X ) [32X property
  [29X[2XIsSemiband[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns [10Xtrue[110X if the semigroup [3Xsgrp[103X is a semiband and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YA semigroup [3Xsgrp[103X is [13Xidempotent generated[113X or equivalently a [13Xsemiband[113X if it is
  generated by its idempotent elements, i.e elements satisfying [3Xx^2=x[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(3, 13);[127X[104X
    [4X[28X<small semigroup of size 3>[128X[104X
    [4X[25Xgap>[125X [27XIsIdempotentGenerated(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(3, IsIdempotentGenerated, false);[127X[104X
    [4X[28X<small semigroup of size 3>[128X[104X
    [4X[25Xgap>[125X [27XIsIdempotentGenerated(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 3, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(4, IsIdempotentGenerated, true, [127X[104X
    [4X[25X>[125X [27XIsSingularSemigroupCopy, true);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(2, IsIdempotentGenerated, true, [127X[104X
    [4X[25X>[125X [27XIsSingularSemigroupCopy, true);[127X[104X
    [4X[28X<small semigroup of size 2>[128X[104X
  [4X[32X[104X
  
  [1X4.2-13 IsInverseSemigroup[101X
  
  [29X[2XIsInverseSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  semigroup  [3Xsgrp[103X  is  an  inverse semigroup and [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YA  semigroup  [3Xsgrp[103X  is an [13Xinverse semigroup[113X if every element [3Xx[103X in [3Xsgrp[103X has a
  unique  semigroup  inverse,  that is, a unique element [3Xy[103X such that [3Xxyx=x[103X and
  [3Xyxy=y[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(7, IsInverseSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsInverseSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7, 101324);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsInverseSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X4.2-14 IsLeftZeroSemigroup[101X
  
  [29X[2XIsLeftZeroSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  semigroup  [3Xsgrp[103X  is  a left zero semigroup and [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YA semigroup [3Xsgrp[103X is a [13Xleft zero semigroup[113X if [3Xxy=x[103X for all [3Xx,y[103X in [3Xsgrp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5, 438);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsLeftZeroSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5, 1141);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsLeftZeroSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-15 IsMonogenicSemigroup[101X
  
  [29X[2XIsMonogenicSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the small semigroup [3Xsgrp[103X is generated by a single element
  and [10Xfalse[110X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=RandomSmallSemigroup(7);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsMonogenicSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(7, IsMonogenicSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsMonogenicSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMinimalGeneratingSet(s);[127X[104X
    [4X[28X[ s7 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup( 7, 406945);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsMonogenicSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X4.2-16 IsMonoidAsSemigroup[101X
  
  [29X[2XIsMonoidAsSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  semigroup  [3Xsgrp[103X  is  a  monoid (i.e. has an identity
  element) and [10Xfalse[110X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(4, 126);[127X[104X
    [4X[28X<small semigroup of size 4>[128X[104X
    [4X[25Xgap>[125X [27XIsMonoidAsSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(4, IsMonoidAsSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 4>[128X[104X
    [4X[25Xgap>[125X [27XIsMonoidAsSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOne(s);[127X[104X
    [4X[28Xs1[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 4, 7 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-17 IsMultSemigroupOfNearRing[101X
  
  [29X[2XIsMultSemigroupOfNearRing[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns   [10Xtrue[110X   if   [3Xsgrp[103X   is  isomorphic  (or  anti-isomorphic?)  to  the
  multiplicative semigroup of a near-ring and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YThose  semigroups  in  the  library  that have this property were identified
  using the [5XSonata[105X package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(7, IsMultSemigroupOfNearRing, true);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 7, 1 ][128X[104X
    [4X[25Xgap>[125X [27XIsMultSemigroupOfNearRing(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(2,2);[127X[104X
    [4X[28X<small semigroup of size 2>[128X[104X
    [4X[25Xgap>[125X [27XIsMultSemigroupOfNearRing(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X4.2-18 IsNGeneratedSemigroup[101X
  
  [29X[2XIsNGeneratedSemigroup[102X( [3Xsgrp[103X, [3Xn[103X ) [32X operation
  
  [33X[0;0Yreturns  [10Xtrue[110X  if the least size of a generating set for the small semigroup
  [3Xsgrp[103X is [3Xn[103X and [10Xfalse[110X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7, 760041);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsNGeneratedSemigroup(s, 4);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsNGeneratedSemigroup(s, 3);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMinimalGeneratingSet(s);[127X[104X
    [4X[28X[ s3, s5, s7 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(4, x-> Length(MinimalGeneratingSet(x)), 4);[127X[104X
    [4X[28X<small semigroup of size 4>[128X[104X
    [4X[25Xgap>[125X [27XIsNGeneratedSemigroup(s, 4);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-19 IsNIdempotentSemigroup[101X
  
  [29X[2XIsNIdempotentSemigroup[102X( [3Xsgrp[103X, [3Xn[103X ) [32X operation
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  small  semigroup  [3Xsgrp[103X  has  [3Xn[103X idempotents and [10Xfalse[110X
  otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(4, 75);;[127X[104X
    [4X[25Xgap>[125X [27XIsNIdempotentSemigroup(s, 1);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsNIdempotentSemigroup(s, 2);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsNIdempotentSemigroup(s, 3);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X4.2-20 [33X[0;0YIsNilpotentSemigroup[133X[101X
  
  [29X[2XIsNilpotentSemigroup[102X( [3Xsgrp[103X ) [32X property
  [29X[2XIsNilpotent[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns [10Xtrue[110X if the small semigroup [3Xsgrp[103X is nilpotent and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YA  semigroup is [13Xnilpotent[113X if it has a zero element and every element to some
  power equals this zero.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5,116);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsNilpotentSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7, 673768);;[127X[104X
    [4X[25Xgap>[125X [27XIsNilpotentSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7, 657867);;[127X[104X
    [4X[25Xgap>[125X [27XIsNilpotent(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-21 IsOrthodoxSemigroup[101X
  
  [29X[2XIsOrthodoxSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns [10Xtrue[110X if the semigroup [3Xsgrp[103X is orthodox and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YA  semigroup  is  [13Xorthodox[113X  if  it  is  regular  and  its idempotents form a
  subsemigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(6, 15858);;[127X[104X
    [4X[25Xgap>[125X [27XIsSemigroupWithClosedIdempotents(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsRegularSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsOrthodoxSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-22 IsRectangularBand[101X
  
  [29X[2XIsRectangularBand[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  small semigroup [3Xsgrp[103X is a rectangular band and [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YA semigroup [3Xsgrp[103X is a [13Xrectangular band[113X if for all [3Xx,y,z[103X in [3Xsgrp[103X we have that
  [3Xx^2=x[103X and [3Xxyz=xz[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5, 216);;[127X[104X
    [4X[25Xgap>[125X [27XIsRectangularBand(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(6, 15854);;[127X[104X
    [4X[25Xgap>[125X [27XIsRectangularBand(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-23 IsRegularSemigroup[101X
  
  [29X[2XIsRegularSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the small semigroup [3Xsgrp[103X is a regular semigroup and [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YA semigroup [3Xsgrp[103X is [13Xregular[113X if for all [3Xx[103X in [3Xsgrp[103X there exists [3Xy[103X in [3Xsgrp[103X such
  that [3Xxyx=x[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(3, 10);;[127X[104X
    [4X[25Xgap>[125X [27XIsRegularSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(3, 1);;[127X[104X
    [4X[25Xgap>[125X [27XIsRegularSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(4, IsFullTransformationSemigroupCopy, true); [127X[104X
    [4X[28X<small semigroup of size 4>[128X[104X
    [4X[25Xgap>[125X [27XIsRegularSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-24 IsRightZeroSemigroup[101X
  
  [29X[2XIsRightZeroSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xfalse[110X  for any small semigroup [3Xsgrp[103X since the library contains only
  left zero semigroups.[133X
  
  [33X[0;0YA semigroup [3Xsgrp[103X is a [13Xright zero semigroup[113X if [3Xxy=y[103X for all [3Xx,y[103X in [3Xsgrp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5, 438);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsRightZeroSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X4.2-25 IsSelfDualSemigroup[101X
  
  [29X[2XIsSelfDualSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns [10Xtrue[110X if the semigroup [3Xsgrp[103X is self dual and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YA  semigroup  is  [13Xself  dual[113X  if  it is isomorphic to its dual, that is, the
  semigroup  [3Xt[103X  with  multiplication  [3X*[103X defined by [3Xx*y=yx[103X where [3Xyx[103X denotes the
  product in [3Xsgrp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5,116);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsSelfDualSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=RandomSmallSemigroup(5, IsSelfDualSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsSelfDualSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-26 IsSemigroupWithClosedIdempotents[101X
  
  [29X[2XIsSemigroupWithClosedIdempotents[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  the  idempotent  elements  of  the  semigroup [3Xsgrp[103X form a
  subsemigroup and [10Xfalse[110X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5, 677);;[127X[104X
    [4X[25Xgap>[125X [27XIsSemigroupWithClosedIdempotents(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5, 659);;[127X[104X
    [4X[25Xgap>[125X [27XIsSemigroupWithClosedIdempotents(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5, 327);;[127X[104X
    [4X[25Xgap>[125X [27XIsSemigroupWithClosedIdempotents(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X4.2-27 IsSemigroupWithZero[101X
  
  [29X[2XIsSemigroupWithZero[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns [10Xtrue[110X if the semigroup [3Xsgrp[103X has a zero element and false otherwise.[133X
  
  [33X[0;0YAn element [22Xz[122X is a [13Xzero[113X if [22Xz*x=x*z=z[122X for all [22Xx[122X in the semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(5,1);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsSemigroupWithZero(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(4,26);[127X[104X
    [4X[28X<small semigroup of size 4>[128X[104X
    [4X[25Xgap>[125X [27XIsSemigroupWithZero(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X4.2-28 [33X[0;0YIsSimpleSemigroup[133X[101X
  
  [29X[2XIsSimpleSemigroup[102X( [3Xsgrp[103X ) [32X property
  [29X[2XIsCompletelySimpleSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturn  [10Xtrue[110X  if the semigroup [3Xsgrp[103X is simple or completely simple and [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YA  semigroup  is  [13Xsimple[113X  if it has no proper 2-sided ideals. A semigroup is
  [13Xcompletely  simple[113X  if  it  is  simple  and possesses minimal left and right
  ideals.[133X
  
  [33X[0;0YA finite semigroup is simple if and only if it is completely simple.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7, 835080);;[127X[104X
    [4X[25Xgap>[125X [27XIsSimpleSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCompletelySimpleSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7, 208242);;[127X[104X
    [4X[25Xgap>[125X [27XIsSimpleSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X4.2-29 IsSingularSemigroupCopy[101X
  
  [29X[2XIsSingularSemigroupCopy[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X if the semigroup [3Xsgrp[103X is isomorphic to a semigroup of singular
  (i.e. non-invertible) mappings on a finite set and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YThe  size  of  this  semigroup on an [22Xn[122X element set is [22Xn^n-n![122X and so there is
  only one semigroup in the library that has this property.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(1,1);[127X[104X
    [4X[28X<small semigroup of size 1>[128X[104X
    [4X[25Xgap>[125X [27XIsSingularSemigroupCopy(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(2, IsSingularSemigroupCopy, true);[127X[104X
    [4X[28X<small semigroup of size 2>[128X[104X
    [4X[25Xgap>[125X [27XIsSingularSemigroupCopy(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 2, 4 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(4, IsSingularSemigroupCopy, true);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X4.2-30 IsZeroGroup[101X
  
  [29X[2XIsZeroGroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns [10Xtrue[110X if the semigroup [3Xsgrp[103X is a zero group and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YThe semigroup [3Xsgrp[103X is a [13Xzero group[113X if there exists an element [22Xz[122X in [3Xsgrp[103X such
  that [3Xsgrp[103X without [22Xz[122X is a group and for all [22Xx[122X in [22Xsgrp[122X we have that [22Xxz=zx=z[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2),(3,4)); [127X[104X
    [4X[28XGroup([ (1,2), (3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(g); [127X[104X
    [4X[28X[ 4, 7 ][128X[104X
    [4X[25Xgap>[125X [27Xs := Range(InjectionZeroMagma(g));[127X[104X
    [4X[28X<Group([ (1,2), (3,4) ]) with 0 adjoined>[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 5, 149 ][128X[104X
    [4X[25Xgap>[125X [27XIsZeroGroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-31 IsZeroSemigroup[101X
  
  [29X[2XIsZeroSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturns [10Xtrue[110X if the semigroup [3Xsgrp[103X is a zero semigroup and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YThe  semigroup [3Xsgrp[103X is a [13Xzero semigroup[113X if there exists an element [22Xz[122X in [3Xsgrp[103X
  such that [22Xxy=z[122X for all [22Xx,y[122X in [3Xsgrp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(5, IsZeroSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIsZeroSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 5, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=OneSmallSemigroup(5, IsZeroSemigroup, false);[127X[104X
    [4X[28X<small semigroup of size 5>[128X[104X
    [4X[25Xgap>[125X [27XIdSmallSemigroup(s);[127X[104X
    [4X[28X[ 5, 2 ][128X[104X
    [4X[25Xgap>[125X [27XIsZeroSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  for  each  size  the  unique  zero semigroup is always the first
  semigroup of this size in the library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsZeroSemigroup(SmallSemigroup(6,1));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsZeroSemigroup(SmallSemigroup(7,1));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsZeroSemigroup(SmallSemigroup(8,1));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-32 IsZeroSimpleSemigroup[101X
  
  [29X[2XIsZeroSimpleSemigroup[102X( [3Xsgrp[103X ) [32X property
  
  [33X[0;0Yreturn [10Xtrue[110X if the semigroup [3Xsgrp[103X is zero simple and [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YA  semigroup [3Xsgrp[103X is [13Xzero simple[113X if the only 2-sided ideals are the zero [3X{0}[103X
  and [3Xsgrp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7, 519799);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsZeroSimpleSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=RandomSmallSemigroup(7, IsZeroSimpleSemigroup, true); [127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27XIsZeroSimpleSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.2-33 MinimalGeneratingSet[101X
  
  [29X[2XMinimalGeneratingSet[102X( [3Xsgrp[103X ) [32X attribute
  
  [33X[0;0Yreturns a set of generators for [3Xsgrp[103X with minimal size.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(8, 1478885610);;[127X[104X
    [4X[25Xgap>[125X [27XMinimalGeneratingSet(s);[127X[104X
    [4X[28X[ s4, s5, s6, s7, s8 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(7, 673768);;[127X[104X
    [4X[25Xgap>[125X [27XMinimalGeneratingSet(s);[127X[104X
    [4X[28X[ s4, s5, s6, s7 ][128X[104X
    [4X[25Xgap>[125X [27Xs:=SmallSemigroup(4, 4);;[127X[104X
    [4X[25Xgap>[125X [27XMinimalGeneratingSet(s);[127X[104X
    [4X[28X[ s2, s3, s4 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-34 NilpotencyDegree[101X
  
  [29X[2XNilpotencyDegree[102X( [3Xsgrp[103X ) [32X attribute
  
  [33X[0;0Yreturns  the  least [22Xn[122X such that every product of [22Xn[122X elements in the nilpotent
  semigroup  [3Xsgrp[103X  equals  the  zero element and returns [10Xfail[110X if the semigroup
  [3Xsgrp[103X is not nilpotent.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := SmallSemigroup(5, 1121);;[127X[104X
    [4X[25Xgap>[125X [27XNilpotencyDegree(s);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xs := SmallSemigroup(7, 393450);;[127X[104X
    [4X[25Xgap>[125X [27XNilpotencyDegree(s);[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  for size 8 a semigroup in the library with ID [22X(8,n)[122X is nilpotent
  of degree 3 if and only if [22Xn[122X is greater than 11433106.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := SmallSemigroup(8, 11433106+1231);;[127X[104X
    [4X[25Xgap>[125X [27XNilpotencyDegree(s);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27Xs := SmallSemigroup(8,NrSmallSemigroups(8));;[127X[104X
    [4X[25Xgap>[125X [27XNilpotencyDegree(s);[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  
  [1X4.3 [33X[0;0YNilpotent semigroups by coclass[133X[101X
  
  [33X[0;0YA  useful  parameter  in the classification of nilpotent semigroups is their
  coclass. For a finite nilpotent semigroup of order [22Xn[122X and nilpotency degree [22Xd[122X
  the coclass is defined as [22Xn-d[122X. In [Dis14] lists up to (anti-)isomorphism are
  provided  for nilpotent semigroups of coclass 0, 1, and 2. The semigroups in
  the  lists  are given by finite presentations. In this section we describe a
  function that allows to access such lists in [5XGAP[105X.[133X
  
  [33X[0;0YA  further  invariant  of  a nilpotent semigroup [22XS[122X is the size of its unique
  minimal  generating  set [22XSbackslash S^2[122X. The possible sizes for a particular
  coclass  are  restricted. Monogenic nilpotent semigroups are precisely those
  of coclass 0. For coclass [22Xd ≥ 1[122X the size of the minimal generating set is at
  least 2 and at most [22Xd+1[122X.[133X
  
  [1X4.3-1 NilpotentSemigroupsByCoclass[101X
  
  [29X[2XNilpotentSemigroupsByCoclass[102X( [3Xn[103X, [3Xd[103X[, [3Xr[103X] ) [32X function
  
  [33X[0;0Yreturns  for  a  positive integer [3Xn[103X and an integer [3Xd[103X with value 0, 1, or 2 a
  list   of   nilpotent   semigroups   of   order   [3Xn[103X  and  coclass  [3Xd[103X  up  to
  (anti-)isomorphism.  If  the  optional  third  argument [3Xr[103X is given then only
  semigroups  of  rank [3Xr[103X are returned. The semigroups in the list are given by
  finite presentations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNilpotentSemigroupsByCoclass(5,1);[127X[104X
    [4X[28X[ <fp semigroup on the generators [ s1, s2 ]>, [128X[104X
    [4X[28X  <fp semigroup on the generators [ s1, s2 ]>, [128X[104X
    [4X[28X  <fp semigroup on the generators [ s1, s2 ]>, [128X[104X
    [4X[28X  <fp semigroup on the generators [ s1, s2 ]>, [128X[104X
    [4X[28X  <fp semigroup on the generators [ s1, s2 ]>, [128X[104X
    [4X[28X  <fp semigroup on the generators [ s1, s2 ]>, [128X[104X
    [4X[28X  <fp semigroup on the generators [ s1, s2 ]> ][128X[104X
    [4X[25Xgap>[125X [27XNilpotentSemigroupsByCoclass(7,0);[127X[104X
    [4X[28X[ <fp semigroup on the generators [ s1 ]> ][128X[104X
    [4X[25Xgap>[125X [27XNilpotentSemigroupsByCoclass(4,2,3);[127X[104X
    [4X[28X[ <fp semigroup on the generators [ s1, s2, s3 ]> ][128X[104X
  [4X[32X[104X
  
  
  [1X4.4 [33X[0;0YStarred Green's relations[133X[101X
  
  [33X[0;0YIn  this  section  functionality  around  the  starred  Green's relations is
  described.  The  five  starred Green's relations are [22XR^*[122X, [22XL^*[122X, [22XJ^*[122X, [22XH^*[122X, and
  [22XD^*[122X;  two elements [22Xa[122X, [22Xb[122X from a semigroup [22XS[122X are [22XR^*[122X-related if for all [22Xx, y ∈
  S^1: xa=ya[122X if and only if [22Xxb = yb[122X; and [22Xa[122X and [22Xb[122X are [22XL^*[122X-related if for all [22Xx,
  y  ∈ S^1: ax=ay[122X if and only if [22Xbx = by[122X. In parallel to the classical Green's
  relations  [14X'Reference:  Green's  Relations'[114X [22XH^*=R^* ∧ L^*[122X and [22XD^*= R^* ∨ L^*[122X
  (but  [22XR^*  ∘ L^* = L^* ∘ R^*[122X does [13Xnot[113X hold in general). To describe [22XJ^*[122X is a
  bit  more  technical.  For  [22Xa,b  ∈ S[122X one can show that [22Xb[122X lies in [22XJ^*(a)[122X, the
  principal  *-ideal  of  [22Xa[122X,  if and only if there exist [22Xc_0,c_1...,c_n∈ S[122X and
  [22Xx_1,...,x_n,y_1,...,  y_n∈  S^1[122X such that [22Xa=c_0, b=c_n[122X and [22Xc_iD^*x_ic_i-1y_i[122X
  for [22X1≤ i≤ n[122X. Then [22XaJ^*b[122X if and only if both [22Xa∈ J^*(b)[122X and [22Xb∈ J^*(a)[122X[133X
  
  [33X[0;0YNote  that  even  for  finite  semigroups  [22XJ^*[122X does not always equal [22XD^*[122X (in
  contrast  to the situation for classical Green's relations). Using [5XSmallsemi[105X
  it was shown that there exist semigroups of order 8 with [22XJ^*≠ D^*[122X [DMU13].[133X
  
  [1X4.4-1 IsStarRelation[101X
  
  [29X[2XIsStarRelation[102X( [3Xbin-relation[103X ) [32X property
  [29X[2XIsRStarRelation[102X( [3Xequiv-relation[103X ) [32X property
  [29X[2XIsLStarRelation[102X( [3Xequiv-relation[103X ) [32X property
  [29X[2XIsJStarRelation[102X( [3Xequiv-relation[103X ) [32X property
  [29X[2XIsHStarRelation[102X( [3Xequiv-relation[103X ) [32X property
  [29X[2XIsDStarRelation[102X( [3Xequiv-relation[103X ) [32X property
  
  [33X[0;0YThese  functions  return  [10Xtrue[110X  if  the  argument  is the respective type of
  relation and [10Xfalse[110X otherwise.[133X
  
  [1X4.4-2 RStarRelation[101X
  
  [29X[2XRStarRelation[102X( [3Xsemigroup[103X ) [32X attribute
  [29X[2XLStarRelation[102X( [3Xsemigroup[103X ) [32X attribute
  [29X[2XJStarRelation[102X( [3Xsemigroup[103X ) [32X attribute
  [29X[2XDStarRelation[102X( [3Xsemigroup[103X ) [32X attribute
  [29X[2XHStarRelation[102X( [3Xsemigroup[103X ) [32X attribute
  
  [33X[0;0YThe   starred  Green's  relations  (which  are  equivalence  relations)  are
  attributes of the semigroup [3Xsemigroup[103X.[133X
  
  [1X4.4-3 RStarClass[101X
  
  [29X[2XRStarClass[102X( [3XS[103X, [3Xa[103X ) [32X operation
  [29X[2XLStarClass[102X( [3XS[103X, [3Xa[103X ) [32X operation
  [29X[2XDStarClass[102X( [3XS[103X, [3Xa[103X ) [32X operation
  [29X[2XJStarClass[102X( [3XS[103X, [3Xa[103X ) [32X operation
  [29X[2XHStarClass[102X( [3XS[103X, [3Xa[103X ) [32X operation
  
  [33X[0;0YCreates  the  [22XX*[122X-class of the element [3Xa[103X in the semigroup [3XS[103X where [22XX[122X is one of
  [22XL[122X, [22XR[122X, [22XD[122X, [22XJ[122X, or [22XH[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := SmallSemigroup(7, 280142);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27Xelm := AsList(s)[5];;[127X[104X
    [4X[25Xgap>[125X [27Xjclass := JStarClass(s, elm);[127X[104X
    [4X[28X{s5}[128X[104X
    [4X[25Xgap>[125X [27XAsList(jclass);[127X[104X
    [4X[28X[ s2, s3, s4, s5 ][128X[104X
  [4X[32X[104X
  
  [1X4.4-4 RStarClass[101X
  
  [29X[2XRStarClass[102X( [3XC[103X ) [32X attribute
  [29X[2XLStarClass[102X( [3XC[103X ) [32X attribute
  [29X[2XDStarClass[102X( [3XC[103X ) [32X attribute
  [29X[2XJStarClass[102X( [3XC[103X ) [32X attribute
  
  [33X[0;0Yare  attributes  reflecting  the  natural  ordering over the various starred
  Green's classes. They return the respective class in which the given class [22XC[122X
  is contained, where [22XC[122X must be a class from a strictly finer relation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := SmallSemigroup(7, 280142);[127X[104X
    [4X[28X<small semigroup of size 7>[128X[104X
    [4X[25Xgap>[125X [27Xelm := AsList(s)[5];;[127X[104X
    [4X[25Xgap>[125X [27Xhclass := HStarClass(s, elm);[127X[104X
    [4X[28X{s5}[128X[104X
    [4X[25Xgap>[125X [27XAsList(LStarClass(hclass));[127X[104X
    [4X[28X[ s5 ][128X[104X
    [4X[25Xgap>[125X [27XAsList(RStarClass(hclass));[127X[104X
    [4X[28X[ s2, s5 ][128X[104X
    [4X[25Xgap>[125X [27XAsList(DStarClass(hclass));[127X[104X
    [4X[28X[ s2, s3, s4, s5 ][128X[104X
  [4X[32X[104X
  
  [1X4.4-5 IsStarClass[101X
  
  [29X[2XIsStarClass[102X( [3Xequiv-class[103X ) [32X property
  [29X[2XIsRStarClass[102X( [3Xequiv-class[103X ) [32X property
  [29X[2XIsLStarClass[102X( [3Xequiv-class[103X ) [32X property
  [29X[2XIsJStarClass[102X( [3Xequiv-class[103X ) [32X property
  [29X[2XIsHStarClass[102X( [3Xequiv-class[103X ) [32X property
  [29X[2XIsDStarClass[102X( [3Xequiv-class[103X ) [32X property
  
  [33X[0;0Yreturn  [9Xtrue[109X if the equivalence class [3Xequiv-class[103X is a starred Green's class
  of any type, or of [22XR[122X, [22XL[122X, [22XJ[122X, [22XH[122X, [22XD[122X type, respectively, or [9Xfalse[109X otherwise.[133X
  
  [1X4.4-6 RStarClasses[101X
  
  [29X[2XRStarClasses[102X( [3Xsemigroup[103X ) [32X attribute
  [29X[2XLStarClasses[102X( [3Xsemigroup[103X ) [32X attribute
  [29X[2XJStarClasses[102X( [3Xsemigroup[103X ) [32X attribute
  [29X[2XDStarClasses[102X( [3Xsemigroup[103X ) [32X attribute
  [29X[2XHStarClasses[102X( [3Xsemigroup[103X ) [32X attribute
  
  [33X[0;0Yreturn  the  [22XR[122X,  [22XL[122X,  [22XJ[122X,  [22XH[122X,  or  [22XD[122X starred Green's classes, respectively for
  semigroup  [3Xsemigroup[103X.  [10XEquivalenceClasses[110X for a Green's relation lead to one
  of these functions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := SmallSemigroup(6, 54);[127X[104X
    [4X[28X<small semigroup of size 6>[128X[104X
    [4X[25Xgap>[125X [27XJStarClasses(s);[127X[104X
    [4X[28X[ {s1}, {s2}, {s4}, {s5}, {s6} ][128X[104X
  [4X[32X[104X
  
  
  [1X4.5 [33X[0;0YFamilies of Semigroups[133X[101X
  
  [33X[0;0YIn this section we describe how to find semigroups in the library satisfying
  a given set of parameters.[133X
  
  [33X[0;0YThe following functions have the same usage but may return different values:
  [2XEnumeratorOfSmallSemigroups[102X     ([14X4.5-2[114X),     [2XAllSmallSemigroups[102X     ([14X4.5-1[114X),
  [2XEnumeratorSortedOfSmallSemigroups[102X   ([14X4.5-4[114X),  [2XIdsOfSmallSemigroups[102X  ([14X4.5-7[114X),
  [2XIteratorOfSmallSemigroups[102X      ([14X4.5-11[114X),     [2XNrSmallSemigroups[102X     ([14X4.5-15[114X),
  [2XOneSmallSemigroup[102X     ([14X4.5-16[114X),     [2XPositionsOfSmallSemigroups[102X     ([14X4.5-17[114X),
  [2XRandomSmallSemigroup[102X ([14X4.5-20[114X).[133X
  
  [33X[0;0YThe number of arguments should be odd:[133X
  
  [30X    [33X[0;6Ythe  first  argument  [10Xarg[1][110X  should  be a positive integer, a list of
        positive  integers,  or  an enumerator or iterator of small semigroups
        satisfying        [2XIsEnumeratorOfSmallSemigroups[102X       ([14X4.5-8[114X)       or
        [2XIsIteratorOfSmallSemigroups[102X ([14X4.5-10[114X)[133X
  
  [30X    [33X[0;6Ythe even arguments [10Xarg[2i][110X, if present, should be a function[133X
  
  [30X    [33X[0;6Ythe  odd  arguments [10Xarg[2i+1][110X argument should be a possible value that
        can be returned by the function [10Xarg[2i][110X.[133X
  
  [33X[0;0YIn  the case that the function is [2XAllSmallSemigroups[102X ([14X4.5-1[114X) and [10Xarg[1][110X is a
  positive integer, then the returned value is a list of all semigroups [22XS[122X with
  [10Xarg[1][110X elements such that [10Xarg[2i](S)=arg[2i+1][110X.[133X
  
  [33X[0;0YFor  example, to obtain all the commutative semigroups with [3X3[103X idempotents of
  sizes   [3X2[103X   to   [3X5[103X   use   one   of   [2XEnumeratorOfSmallSemigroups[102X   ([14X4.5-2[114X),
  [2XAllSmallSemigroups[102X   ([14X4.5-1[114X),   [2XEnumeratorSortedOfSmallSemigroups[102X   ([14X4.5-4[114X),
  [2XIteratorOfSmallSemigroups[102X ([14X4.5-11[114X) with argument[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[2..5], IsCommutative, true, Is3IdempotentGenerated, true[128X[104X
    [4X[28X	[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[2XAllSmallSemigroups[102X   ([14X4.5-1[114X)   returns   a  list  of  all  such  semigroups,
  [2XEnumeratorOfSmallSemigroups[102X    ([14X4.5-2[114X),    [2XEnumeratorSortedOfSmallSemigroups[102X
  ([14X4.5-4[114X),  and [2XIteratorOfSmallSemigroups[102X ([14X4.5-11[114X) return an enumerator and an
  iterator  of  all  such  semigroups,  respectively.  For more information on
  enumerators   and   iterators   see   [2XEnumerator[102X   ([14XReference:  Enumerator[114X),
  [2XEnumeratorSorted[102X  ([14XReference:  EnumeratorSorted[114X),  or  [2XIterator[102X  ([14XReference:
  Iterator[114X).  The  following  are  rules  of  thumb  regarding  the  different
  situations  when  these  functions  should  be  used  in order of slowest to
  fastest and greatest memory use to least:[133X
  
  [30X    [33X[0;6Y[2XAllSmallSemigroups[102X  ([14X4.5-1[114X) should be used if the number of semigroups
        is  not  too  large  and  you want to keep the created semigroups in a
        list.[133X
  
  [30X    [33X[0;6Y[2XEnumeratorOfSmallSemigroups[102X                 ([14X4.5-2[114X)                 or
        [2XEnumeratorSortedOfSmallSemigroups[102X  ([14X4.5-4[114X)  should  be  used  when the
        functions  in  even  indexed positions are those stored in the library
        (see [2XPrecomputedSmallSemisInfo[102X ([14X4.5-19[114X)) or you want repeatedly search
        the  same set of semigroups and there are too many to store in a list.
        Note that the enumerator stores the id numbers of its elements but not
        the  semigroups  themselves.  Hence  every  time  an  element  of  the
        enumerator  is  required  it must be recreated from the multiplication
        table data.[133X
  
  [30X    [33X[0;6Y[2XIteratorOfSmallSemigroups[102X  ([14X4.5-11[114X) should be used if the functions in
        even   indexed   positions   are   not  stored  in  the  library  (see
        [2XPrecomputedSmallSemisInfo[102X ([14X4.5-19[114X)) or if you just want to run through
        all the semigroups satisfying the specified parameters once only. Note
        that  each new call of [2XIteratorOfSmallSemigroups[102X ([14X4.5-11[114X) requires [5XGAP[105X
        to  recompute  its  elements  which  may  be slow if the functions are
        user-defined or not stored in the library.[133X
  
  [33X[0;0YFurther  information on the relative virtues of these different commands can
  be found in Chapter [14X3[114X.[133X
  
  [33X[0;0YAs  a  further  example,  if we want to obtain a single non-simple semigroup
  with [22X7[122X elements and trivial automorphism group, then we would use one of the
  functions  [2XOneSmallSemigroup[102X  ([14X4.5-16[114X) or [2XRandomSmallSemigroup[102X ([14X4.5-20[114X) with
  argument[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X7, IsSimpleSemigroup, false, x-> IsTrivial(AutomorphismGroup(x)), true[128X[104X
    [4X[28X	[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[2XOneSmallSemigroup[102X  ([14X4.5-16[114X)  should  return  an  answer  more  quickly  than
  [2XRandomSmallSemigroup[102X  ([14X4.5-20[114X).  Also  note  that [2XOneSmallSemigroup[102X ([14X4.5-16[114X)
  will  always  return  the  same  semigroup,  i.e. the first semigroup in the
  library with the given parameters.[133X
  
  [1X4.5-1 AllSmallSemigroups[101X
  
  [29X[2XAllSmallSemigroups[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Ythe  number  of  argument of this function should be odd. The first argument
  [10Xarg[1][110X  should be a positive integer, an enumerator of small semigroups with
  [2XIsEnumeratorOfSmallSemigroups[102X  ([14X4.5-8[114X),  or  an  iterator of small semigroup
  with [2XIsIteratorOfSmallSemigroups[102X ([14X4.5-10[114X).[133X
  
  [33X[0;0YThe  even  arguments  [10Xarg[2i][110X,  if present, should be functions, and the odd
  arguments [10Xarg[2i+1][110X should be a value that the preceeding function can have.
  For  example,  a  typical  input  might  be [10X3, IsRegularSemigroup, true[110X. The
  functions [10Xarg[2i][110X can be user defined or existing [5XGAP[105X functions.[133X
  
  [33X[0;0YPlease see Section [14X4.5[114X or Chapter [14X3[114X for more details.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X  is a positive integer, then [10XAllSmallSemigroups[110X returns a list of
  all   the  small  semigroups  [10XS[110X  in  the  library  with  [10XSize(S)=arg[1][110X  and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X is a list of positive integers, then [10XAllSmallSemigroups[110X returns a
  list of all the small semigroups [10XS[110X in the library with [10XSize(S) in arg[1][110X and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf   [10Xarg[1][110X   is  an  enumerator  or  iterator  of  small  semigroups,  then
  [10XAllSmallSemigroups[110X  returns  a  list  of  all  the small semigroups [10XS[110X in the
  library with [10XS in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X  gap> AllSmallSemigroups(2);[128X[104X
    [4X[28X  [ <small semigroup of size 2>, <small semigroup of size 2>,[128X[104X
    [4X[28X    <small semigroup of size 2>, <small semigroup of size 2> ][128X[104X
    [4X[28X  gap> AllSmallSemigroups([2,3], IsRegularSemigroup, true,[128X[104X
    [4X[28X  > x-> Length(GreensRClasses(x)), 1);[128X[104X
    [4X[28X  [ <small semigroup of size 2>, <small semigroup of size 2>,[128X[104X
    [4X[28X    <small semigroup of size 3>, <small semigroup of size 3> ][128X[104X
    [4X[28X  gap> enum:=EnumeratorOfSmallSemigroups(8, IsInverseSemigroup, true,[128X[104X
    [4X[28X  > IsCommutativeSemigroup, true);;[128X[104X
    [4X[28X  gap> AllSmallSemigroups(enum, x-> Length(GreensRClasses(x)), 1);[128X[104X
    [4X[28X  [ <small semigroup of size 8>, <small semigroup of size 8>,[128X[104X
    [4X[28X    <small semigroup of size 8> ][128X[104X
    [4X[28X  gap> iter:=IteratorOfSmallSemigroups(7, x-> Length(GreensRClasses(x)), 1);;[128X[104X
    [4X[28X  gap> AllSmallSemigroups(iter, IsCommutative, true,[128X[104X
    [4X[28X  > IsSimpleSemigroup, true);[128X[104X
    [4X[28X  [ <small semigroup of size 7> ][128X[104X
  [4X[32X[104X
  
  [1X4.5-2 EnumeratorOfSmallSemigroups[101X
  
  [29X[2XEnumeratorOfSmallSemigroups[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Ythe  number  of  argument of this function should be odd. The first argument
  [10Xarg[1][110X  should be a positive integer, an enumerator of small semigroups with
  [2XIsEnumeratorOfSmallSemigroups[102X  ([14X4.5-8[114X),  or  an  iterator of small semigroup
  with [2XIsIteratorOfSmallSemigroups[102X ([14X4.5-10[114X).[133X
  
  [33X[0;0YThe  even  arguments  [10Xarg[2i][110X,  if present, should be functions, and the odd
  arguments [10Xarg[2i+1][110X should be a value that the preceeding function can have.
  For  example,  a  typical  input  might  be [10X3, IsRegularSemigroup, true[110X. The
  functions [10Xarg[2i][110X can be user defined or existing [5XGAP[105X functions.[133X
  
  [33X[0;0YPlease see Section [14X4.5[114X or Chapter [14X3[114X for more details.[133X
  
  [33X[0;0YIf [10Xarg[1][110X is a positive integer, then [10XEnumeratorOfSmallSemigroups[110X returns an
  enumerator  of all the small semigroups [10XS[110X in the library with [10XSize(S)=arg[1][110X
  and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X  is a list of positive integers, then [10XEnumeratorOfSmallSemigroups[110X
  returns  an  enumerator  of  all  the small semigroups [10XS[110X in the library with
  [10XSize(S) in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf   [10Xarg[1][110X   is  an  enumerator  or  iterator  of  small  semigroups,  then
  [10XEnumeratorOfSmallSemigroups[110X   returns   an   enumerator  of  all  the  small
  semigroups  [10XS[110X  in  the library with [10XS in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for
  all [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroups(7);[127X[104X
    [4X[28X<enumerator of semigroups of size 7>[128X[104X
    [4X[25Xgap>[125X [27XEnumeratorOfSmallSemigroups([2,3], IsRegularSemigroup, true);[127X[104X
    [4X[28X<enumerator of semigroups of sizes [ 2, 3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroups(8, IsInverseSemigroup, true,[127X[104X
    [4X[25X>[125X [27XIsCommutativeSemigroup, true);[127X[104X
    [4X[28X<enumerator of semigroups of size 8>[128X[104X
    [4X[25Xgap>[125X [27XEnumeratorOfSmallSemigroups(enum, IsCommutativeSemigroup, true,[127X[104X
    [4X[25X>[125X [27XIsSimpleSemigroup, false);[127X[104X
    [4X[28X<enumerator of semigroups of size 8>[128X[104X
    [4X[25Xgap>[125X [27Xiter:=IteratorOfSmallSemigroups(8);[127X[104X
    [4X[28X<iterator>[128X[104X
    [4X[25Xgap>[125X [27XEnumeratorOfSmallSemigroups(iter, IsCommutativeSemigroup, true,[127X[104X
    [4X[25X>[125X [27XIsSimpleSemigroup, false);[127X[104X
    [4X[28X<enumerator of semigroups of size 8>[128X[104X
  [4X[32X[104X
  
  [1X4.5-3 EnumeratorOfSmallSemigroupsByIds[101X
  
  [29X[2XEnumeratorOfSmallSemigroupsByIds[102X( [3Xarg[103X ) [32X operation
  [29X[2XEnumeratorOfSmallSemigroupsByIdsNC[102X( [3Xarg[103X ) [32X operation
  
  [33X[0;0Ythe argument of this function should be one of the following:[133X
  
  [30X    [33X[0;6Ya  positive  integer  [3Xarg[1][103X  and a set of positive integers less than
        [2XNrSmallSemigroups[102X  ([14X4.5-15[114X)  with  argument  [3Xarg[1][103X.  For example, the
        argument [3X3, [1..10][103X yields the first 10 semigroups with 3 elements.[133X
  
  [30X    [33X[0;6Ya  set  of  positive  integers  [3Xarg[1][103X  and a list of sets of positive
        integers [3Xarg[2][103X such that [3Xx[103X is at most [2XNrSmallSemigroups[102X ([14X4.5-15[114X) with
        argument  [3Xarg[1][i][103X  for  all  [3Xx[103X  in  [3Xarg[2][i][103X.  For  example, [3X[2,3],
        [[1..2],[1..10]][103X  yields  the  first  2  semigroups of size 2, and the
        first 10 semigroups of size 3.[133X
  
  [30X    [33X[0;6Ya list of id numbers, for example, [3X[[7,1], [6,1], [5,1]][103X.[133X
  
  [33X[0;0YThe  no  check  version  does not check that the arguments are valid and may
  return unpredictable results.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroupsByIds([[7,1],[6,1],[5,1]]);[127X[104X
    [4X[28X<enumerator of semigroups of sizes [ 5, 6, 7 ]>[128X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroupsByIds(7, [1..1000]);[127X[104X
    [4X[28X<enumerator of semigroups of size 7>[128X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroupsByIds([2,3], [[1..2],[1..10]]);[127X[104X
    [4X[28X<enumerator of semigroups of sizes [ 2, 3 ]>[128X[104X
  [4X[32X[104X
  
  [1X4.5-4 EnumeratorSortedOfSmallSemigroups[101X
  
  [29X[2XEnumeratorSortedOfSmallSemigroups[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Yaccepts    the   same   arguments   and   returns   the   same   values   as
  [2XEnumeratorOfSmallSemigroups[102X ([14X4.5-2[114X).[133X
  
  [1X4.5-5 FuncsOfSmallSemisInEnum[101X
  
  [29X[2XFuncsOfSmallSemisInEnum[102X( [3Xenum[103X ) [32X function
  
  [33X[0;0Yreturns  a  list  of the functions and their values that were used to create
  the enumerator of small semigroups [3Xenum[103X. If you only want the names of these
  functions use [2XNamesFuncsSmallSemisInEnum[102X ([14X4.5-12[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroups([2..4], IsSimpleSemigroup, false,[127X[104X
    [4X[25X>[125X [27XIsRegularSemigroup, true);;                                         [127X[104X
    [4X[25Xgap>[125X [27XFuncsOfSmallSemisInEnum(enum);[127X[104X
    [4X[28X[ <Property "IsRegularSemigroup">, true, [128X[104X
    [4X[28X  <Property "IsSimpleSemigroup">, false ][128X[104X
  [4X[32X[104X
  
  [1X4.5-6 FuncsOfSmallSemisInIter[101X
  
  [29X[2XFuncsOfSmallSemisInIter[102X( [3Xiter[103X ) [32X function
  
  [33X[0;0Yreturns  a  list  of the functions and their values that were used to create
  the  iterator  of small semigroups [3Xiter[103X. If you only want the names of these
  functions use [2XNamesFuncsSmallSemisInIter[102X ([14X4.5-13[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:=IteratorOfSmallSemigroups([2..4], IsSimpleSemigroup, false,[127X[104X
    [4X[25X>[125X [27XIsRegularSemigroup, true);;                                        [127X[104X
    [4X[25Xgap>[125X [27XFuncsOfSmallSemisInIter(enum);[127X[104X
    [4X[28X[ <Property "IsRegularSemigroup">, true, [128X[104X
    [4X[28X  <Property "IsSimpleSemigroup">, false ][128X[104X
  [4X[32X[104X
  
  [1X4.5-7 IdsOfSmallSemigroups[101X
  
  [29X[2XIdsOfSmallSemigroups[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Ythe  number  of  argument of this function should be odd. The first argument
  [10Xarg[1][110X  should be a positive integer, an enumerator of small semigroups with
  [2XIsEnumeratorOfSmallSemigroups[102X  ([14X4.5-8[114X),  or  an  iterator of small semigroup
  with [2XIsIteratorOfSmallSemigroups[102X ([14X4.5-10[114X).[133X
  
  [33X[0;0YThe  even  arguments  [10Xarg[2i][110X,  if present, should be functions, and the odd
  arguments [10Xarg[2i+1][110X should be a value that the preceeding function can have.
  For  example,  a  typical  input  might  be [10X3, IsRegularSemigroup, true[110X. The
  functions [10Xarg[2i][110X can be user defined or existing [5XGAP[105X functions.[133X
  
  [33X[0;0YPlease see Section [14X4.5[114X or Chapter [14X3[114X for more details.[133X
  
  [33X[0;0YIf [10Xarg[1][110X is a positive integer, then [10XIdsOfSmallSemigroups[110X returns a list of
  the  id  numbers  of  all  the  small  semigroups  [10XS[110X  in  the  library  with
  [10XSize(S)=arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X is a list of positive integers, then [10XIdsOfSmallSemigroups[110X returns
  a  list  of the id numbers of all the small semigroups [10XS[110X in the library with
  [10XSize(S) in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf   [10Xarg[1][110X   is  an  enumerator  or  iterator  of  small  semigroups,  then
  [10XIdsOfSmallSemigroups[110X  returns  a  list  of  the  id numbers of all the small
  semigroups  [10XS[110X  in  the library with [10XS in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for
  all [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X enum:=EnumeratorOfSmallSemigroups(5, x-> Length(GreensRClasses(x)), 1);;[127X[104X
    [4X[25Xgap>[125X [27XIdsOfSmallSemigroups(enum, IsCommutativeSemigroup, true,[127X[104X
    [4X[25X>[125X [27XIsSimpleSemigroup, false);                                              [127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XIdsOfSmallSemigroups([2,3], IsRegularSemigroup, true);  [127X[104X
    [4X[28X[ [ 2, 2 ], [ 2, 3 ], [ 2, 4 ], [ 3, 10 ], [ 3, 11 ], [ 3, 12 ], [128X[104X
    [4X[28X  [ 3, 13 ], [ 3, 14 ], [ 3, 15 ], [ 3, 16 ], [ 3, 17 ], [ 3, 18 ] ][128X[104X
  [4X[32X[104X
  
  [1X4.5-8 IsEnumeratorOfSmallSemigroups[101X
  
  [29X[2XIsEnumeratorOfSmallSemigroups[102X( [3Xenum[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  [3Xenum[103X  is  an enumerator of small semigroups created using
  [2XEnumeratorOfSmallSemigroups[102X     ([14X4.5-2[114X),    [2XEnumeratorOfSmallSemigroupsByIds[102X
  ([14X4.5-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroupsByIds([[2,1], [3,1], [4,1]]);;[127X[104X
    [4X[25Xgap>[125X [27XIsEnumeratorOfSmallSemigroups(enum);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.5-9 IsIdSmallSemigroup[101X
  
  [29X[2XIsIdSmallSemigroup[102X( [3Xarg[103X ) [32X property
  
  [33X[0;0Yreturn [10Xtrue[110X if the [3Xarg[103X is the id of a small semigroup or [3X[arg[1], arg[2]][103X is
  the id of a small semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsIdSmallSemigroup(8,1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIdSmallSemigroup([1,2]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsIdSmallSemigroup([3,18]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.5-10 IsIteratorOfSmallSemigroups[101X
  
  [29X[2XIsIteratorOfSmallSemigroups[102X( [3Xiter[103X ) [32X property
  
  [33X[0;0Yreturns  [10Xtrue[110X  if  [3Xiter[103X  is  an  iterator  of small semigroups created using
  [2XIteratorOfSmallSemigroups[102X ([14X4.5-11[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xiter:=IteratorOfSmallSemigroups(8);;[127X[104X
    [4X[25Xgap>[125X [27XIsIteratorOfSmallSemigroups(iter);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.5-11 IteratorOfSmallSemigroups[101X
  
  [29X[2XIteratorOfSmallSemigroups[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Ythe  number  of  argument of this function should be odd. The first argument
  [10Xarg[1][110X  should be a positive integer, an enumerator of small semigroups with
  [2XIsEnumeratorOfSmallSemigroups[102X  ([14X4.5-8[114X),  or  an  iterator of small semigroup
  with [2XIsIteratorOfSmallSemigroups[102X ([14X4.5-10[114X).[133X
  
  [33X[0;0YThe  even  arguments  [10Xarg[2i][110X,  if present, should be functions, and the odd
  arguments [10Xarg[2i+1][110X should be a value that the preceeding function can have.
  For  example,  a  typical  input  might  be [10X3, IsRegularSemigroup, true[110X. The
  functions [10Xarg[2i][110X can be user defined or existing [5XGAP[105X functions.[133X
  
  [33X[0;0YPlease see Section [14X4.5[114X or Chapter [14X3[114X for more details.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X  is a positive integer, then [10XIteratorOfSmallSemigroups[110X returns an
  iterator  of  all  the small semigroups [10XS[110X in the library with [10XSize(S)=arg[1][110X
  and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X  is  a  list of positive integers, then [10XIteratorOfSmallSemigroups[110X
  returns  an  iterator  of  all  the  small  semigroups [10XS[110X in the library with
  [10XSize(S) in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf   [10Xarg[1][110X   is  an  enumerator  or  iterator  of  small  semigroups,  then
  [10XIteratorOfSmallSemigroups[110X  returns an iterator of all the small semigroups [10XS[110X
  in the library with [10XS in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xiter:=IteratorOfSmallSemigroups(8);[127X[104X
    [4X[28X<iterator>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator(iter);[127X[104X
    [4X[28X<small semigroup of size 8>[128X[104X
    [4X[25Xgap>[125X [27XIsDoneIterator(iter);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xiter:=IteratorOfSmallSemigroups([2,3], IsRegularSemigroup, true,[127X[104X
    [4X[25X>[125X [27Xx-> Length(Idempotents(x))=1, true);[127X[104X
    [4X[28X<iterator>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator(iter);[127X[104X
    [4X[28X<small semigroup of size 2>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator(iter);[127X[104X
    [4X[28X<small semigroup of size 3>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator(iter);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroups(5, x-> Length(Idempotents(x))=1, true);[127X[104X
    [4X[28X<enumerator of semigroups of size 5>[128X[104X
    [4X[25Xgap>[125X [27Xiter:=IteratorOfSmallSemigroups(enum, x-> Length(GreensRClasses(x))=2, true);[127X[104X
    [4X[28X<iterator>[128X[104X
  [4X[32X[104X
  
  [1X4.5-12 NamesFuncsSmallSemisInEnum[101X
  
  [29X[2XNamesFuncsSmallSemisInEnum[102X( [3Xenum[103X ) [32X function
  
  [33X[0;0Yreturns  a list of the names of functions and their values that were used to
  create  the enumerator of small semigroups [3Xenum[103X. If you only want the actual
  functions themselves then use [2XFuncsOfSmallSemisInEnum[102X ([14X4.5-5[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroups([2..4], IsSimpleSemigroup, false,[127X[104X
    [4X[25X>[125X [27XIsRegularSemigroup, true);;[127X[104X
    [4X[25Xgap>[125X [27XNamesFuncsSmallSemisInEnum(enum);[127X[104X
    [4X[28X[ "IsRegularSemigroup", true, "IsSimpleSemigroup", false ][128X[104X
  [4X[32X[104X
  
  [1X4.5-13 NamesFuncsSmallSemisInIter[101X
  
  [29X[2XNamesFuncsSmallSemisInIter[102X( [3Xiter[103X ) [32X attribute
  
  [33X[0;0Yreturns  a list of the names of functions and their values that were used to
  create  the  iterator  of small semigroups [3Xiter[103X. If you only want the actual
  functions themselves then use [2XFuncsOfSmallSemisInIter[102X ([14X4.5-6[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xiter:=IteratorOfSmallSemigroups([2..4], IsSimpleSemigroup, false,[127X[104X
    [4X[25X>[125X [27XIsRegularSemigroup, true);;[127X[104X
    [4X[25Xgap>[125X [27XNamesFuncsSmallSemisInIter(iter);[127X[104X
    [4X[28X[ "IsRegularSemigroup", true, "IsSimpleSemigroup", false ][128X[104X
  [4X[32X[104X
  
  [1X4.5-14 Nr3NilpotentSemigroups[101X
  
  [29X[2XNr3NilpotentSemigroups[102X( [3Xn[103X[, [3Xtype[103X] ) [32X function
  
  [33X[0;0Yreturns  the  number  of 3-nilpotent semigroups on a set with [3Xn[103X elements. If
  the  optional  argument  [3Xtype[103X  is given it must be one of [10X"UpToEquivalence",
  "UpToIsomorphism",        "SelfDual",       "Commutative",       "Labelled",
  "Labelled-Commutative"[110X.  The number will be returned for the respective type
  of semigroup. By default [3Xtype[103X is [10X"UpToEquivalence"[110X.[133X
  
  [33X[0;0YThe  function  implements the formulae calculating the number of 3-nilpotent
  semigroups developed in [Dis10][133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNr3NilpotentSemigroups( 4 );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XNr3NilpotentSemigroups( 9, "UpToIsomorphism" );[127X[104X
    [4X[28X105931872028455[128X[104X
    [4X[25Xgap>[125X [27XNr3NilpotentSemigroups( 9, "Labelled" );[127X[104X
    [4X[28X38430603831264883632[128X[104X
    [4X[25Xgap>[125X [27XNr3NilpotentSemigroups( 16, "SelfDual" );[127X[104X
    [4X[28X4975000837941847814744710290469890455985530[128X[104X
    [4X[25Xgap>[125X [27XNr3NilpotentSemigroups( 19, "Commutative" );[127X[104X
    [4X[28X12094270656160403920767935604624748908993169949317454767617795[128X[104X
  [4X[32X[104X
  
  [1X4.5-15 NrSmallSemigroups[101X
  
  [29X[2XNrSmallSemigroups[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Ythe  number  of  argument of this function should be odd. The first argument
  [10Xarg[1][110X  should be a positive integer, an enumerator of small semigroups with
  [2XIsEnumeratorOfSmallSemigroups[102X  ([14X4.5-8[114X),  or  an  iterator of small semigroup
  with [2XIsIteratorOfSmallSemigroups[102X ([14X4.5-10[114X).[133X
  
  [33X[0;0YThe  even  arguments  [10Xarg[2i][110X,  if present, should be functions, and the odd
  arguments [10Xarg[2i+1][110X should be a value that the preceeding function can have.
  For  example,  a  typical  input  might  be [10X3, IsRegularSemigroup, true[110X. The
  functions [10Xarg[2i][110X can be user defined or existing [5XGAP[105X functions.[133X
  
  [33X[0;0YPlease see Section [14X4.5[114X or Chapter [14X3[114X for more details.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X  is a positive integer, then [10XNrSmallSemigroups[110X returns the number
  of   small   semigroups   [10XS[110X   in   the   library   with  [10XSize(S)=arg[1][110X  and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf [10Xarg[1][110X is a list of positive integers, then [10XNrSmallSemigroups[110X returns the
  number  of  small  semigroups  [10XS[110X  in  the library with [10XSize(S) in arg[1][110X and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf   [10Xarg[1][110X   is  an  enumerator  or  iterator  of  small  semigroups,  then
  [10XNrSmallSemigroups[110X  returns  the  number of small semigroups [10XS[110X in the library
  with [10XS in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList([1..8], NrSmallSemigroups);[127X[104X
    [4X[28X[ 1, 4, 18, 126, 1160, 15973, 836021, 1843120128 ][128X[104X
    [4X[25Xgap>[125X [27XNrSmallSemigroups(8, IsCommutative, true, IsInverseSemigroup, true);[127X[104X
    [4X[28X4443[128X[104X
    [4X[25Xgap>[125X [27XNrSmallSemigroups([1..8], IsCliffordSemigroup, true);[127X[104X
    [4X[28X5610[128X[104X
    [4X[25Xgap>[125X [27XNrSmallSemigroups(8, IsRegularSemigroup, true,[127X[104X
    [4X[25X>[125X [27XIsCompletelyRegularSemigroup, false);[127X[104X
    [4X[28X1164[128X[104X
    [4X[25Xgap>[125X [27XNrSmallSemigroups(5, NilpotencyDegree, 3);[127X[104X
    [4X[28X84[128X[104X
  [4X[32X[104X
  
  [1X4.5-16 OneSmallSemigroup[101X
  
  [29X[2XOneSmallSemigroup[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Ythe  number  of  argument of this function should be odd. The first argument
  [10Xarg[1][110X  should be a positive integer, an enumerator of small semigroups with
  [2XIsEnumeratorOfSmallSemigroups[102X  ([14X4.5-8[114X),  or  an  iterator of small semigroup
  with [2XIsIteratorOfSmallSemigroups[102X ([14X4.5-10[114X).[133X
  
  [33X[0;0YThe  even  arguments  [10Xarg[2i][110X,  if present, should be functions, and the odd
  arguments [10Xarg[2i+1][110X should be a value that the preceeding function can have.
  For  example,  a  typical  input  might  be [10X3, IsRegularSemigroup, true[110X. The
  functions [10Xarg[2i][110X can be user defined or existing [5XGAP[105X functions.[133X
  
  [33X[0;0YPlease see Section [14X4.5[114X or Chapter [14X3[114X for more details.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X  is  a positive integer, then [10XOneSmallSemigroup[110X returns the first
  small    semigroup    [10XS[110X    in    the   library   with   [10XSize(S)=arg[1][110X   and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf [10Xarg[1][110X is a list of positive integers, then [10XOneSmallSemigroup[110X returns the
  first  small  semigroup  [10XS[110X  in  the  library  with  [10XSize(S)  in  arg[1][110X  and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf   [10Xarg[1][110X   is  an  enumerator  or  iterator  of  small  semigroups,  then
  [10XOneSmallSemigroup[110X  returns the first small semigroup [10XS[110X in the library with [10XS
  in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOneSmallSemigroup(8, IsCommutative, true, IsInverseSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 8>[128X[104X
    [4X[25Xgap>[125X [27XOneSmallSemigroup([1..8], IsCliffordSemigroup, true);[127X[104X
    [4X[28X<small semigroup of size 1>[128X[104X
    [4X[25Xgap>[125X [27Xiter:=IteratorOfSmallSemigroups(8, IsCommutative, false);[127X[104X
    [4X[28X<iterator>[128X[104X
    [4X[25Xgap>[125X [27XOneSmallSemigroup(iter);[127X[104X
    [4X[28X<small semigroup of size 8>[128X[104X
  [4X[32X[104X
  
  [1X4.5-17 PositionsOfSmallSemigroups[101X
  
  [29X[2XPositionsOfSmallSemigroups[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Ythe  number  of  argument of this function should be odd. The first argument
  [10Xarg[1][110X    should   be   a   positive   integer   or   an   enumerator   with
  [2XIsEnumeratorOfSmallSemigroups[102X   ([14X4.5-8[114X),  the  even  arguments  [10Xarg[2i][110X,  if
  present,  should  be  functions, and the odd arguments [10Xarg[2i+1][110X should be a
  value  that  the  preceeding function can have. For example, a typical input
  might  be  [10X3,  IsRegularSemigroup,  true[110X.  The functions [10Xarg[2i][110X can be user
  defined  or  existing [5XGAP[105X functions. The argument can be a list [10Xarg[110X with the
  same components as given above.[133X
  
  [33X[0;0YThe function returns a list of the second components of the [2XIdSmallSemigroup[102X
  ([14X4.1-6[114X)  of  all the small semigroups [10XS[110X in the library satisfying [10XSize(S)[110X in
  [10Xarg[1][110X     or     [10XSize(S)[110X     in     [10XSizesOfSmallSemisInEnum(arg[1])[110X     and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X partitioned by size of the semigroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X  gap> PositionsOfSmallSemigroups(3);[128X[104X
    [4X[28X  [ [ 1 .. 18 ] ][128X[104X
    [4X[28X  gap> PositionsOfSmallSemigroups(3, IsRegularSemigroup, false);[128X[104X
    [4X[28X  [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ][128X[104X
    [4X[28X  gap> enum:=EnumeratorOfSmallSemigroups(3, IsRegularSemigroup, false);;[128X[104X
    [4X[28X  gap> PositionsOfSmallSemigroups(enum);[128X[104X
    [4X[28X  [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ][128X[104X
    [4X[28X  gap> PositionsOfSmallSemigroups([1..4], IsBand, true);[128X[104X
    [4X[28X  [ [ 1 ], [ 3, 4 ], [ 12 .. 17 ], [ 98 .. 123 ] ][128X[104X
    [4X[28X  gap> PositionsOfSmallSemigroups(enum, Is1IdempotentSemigroup, true,[128X[104X
    [4X[28X  > Is2GeneratedSemigroup, true, IsCliffordSemigroup, false);[128X[104X
    [4X[28X  [ [ 1, 2 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 34, 35, 36, 97 ],[128X[104X
    [4X[28X    [ 5, 20, 21, 22, 23, 26, 29, 32, 35, 54, 55, 56, 60, 61, 62, 63, 64, 65,[128X[104X
    [4X[28X        152, 156, 159, 177, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190,[128X[104X
    [4X[28X        191, 192, 193, 540, 1009, 1157, 1158 ] ][128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X4.5-18 PositionsOfSmallSemisInEnum[101X
  
  [29X[2XPositionsOfSmallSemisInEnum[102X( [3Xenum[103X ) [32X function
  
  [33X[0;0Yreturns  the  second components of the id numbers of the small semigroups in
  the  enumerator of small semigroups [3Xenum[103X in a list partitioned according the
  size   of   the   semigroup.   The   same   value   is   returned  by  using
  [2XPositionsOfSmallSemigroups[102X ([14X4.5-17[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum := EnumeratorOfSmallSemigroups([2..4],IsSimpleSemigroup,true);;[127X[104X
    [4X[25Xgap>[125X [27XPositionsOfSmallSemisInEnum[127X[104X
    [4X[25X>[125X [27X(enum);[127X[104X
    [4X[28X[ [ 2, 4 ], [ 17, 18 ], [ 7, 37, 52, 122, 123 ] ][128X[104X
  [4X[32X[104X
  
  [1X4.5-19 PrecomputedSmallSemisInfo[101X
  
  [29X[2XPrecomputedSmallSemisInfo[102X[32X global variable
  
  [33X[0;0Ythe  global  variable  [10XPrecomputedSmallSemisInfo[110X  contains a list of all the
  names  of  precomputed  properties stored in the library. The [10Xi[110Xth element of
  the  list contains the list of properties that have been precomputed for all
  semigroups in the library of order [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X  gap> PrecomputedSmallSemisInfo[3];[128X[104X
    [4X[28X  [ "Is1GeneratedSemigroup", "Is2GeneratedSemigroup", "Is3GeneratedSemigroup", [128X[104X
    [4X[28X    "IsBand", "IsCliffordSemigroup", "IsCommutative", [128X[104X
    [4X[28X    "IsCompletelyRegularSemigroup", "IsFullTransformationSemigroupCopy", [128X[104X
    [4X[28X    "IsGroupAsSemigroup", "IsIdempotentGenerated", "IsInverseSemigroup", [128X[104X
    [4X[28X    "IsMonoidAsSemigroup", "IsMultSemigroupOfNearRing", "IsRegularSemigroup", [128X[104X
    [4X[28X    "IsSelfDualSemigroup", "IsSemigroupWithoutClosedIdempotents", [128X[104X
    [4X[28X    "IsSimpleSemigroup", "IsSingularSemigroupCopy", "IsZeroSemigroup", [128X[104X
    [4X[28X    "IsZeroSimpleSemigroup" ][128X[104X
  [4X[32X[104X
  
  [1X4.5-20 RandomSmallSemigroup[101X
  
  [29X[2XRandomSmallSemigroup[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0Ythe  number  of  argument of this function should be odd. The first argument
  [10Xarg[1][110X  should be a positive integer, an enumerator of small semigroups with
  [2XIsEnumeratorOfSmallSemigroups[102X  ([14X4.5-8[114X),  or  an  iterator of small semigroup
  with [2XIsIteratorOfSmallSemigroups[102X ([14X4.5-10[114X).[133X
  
  [33X[0;0YThe  even  arguments  [10Xarg[2i][110X,  if present, should be functions, and the odd
  arguments [10Xarg[2i+1][110X should be a value that the preceeding function can have.
  For  example,  a  typical  input  might  be [10X3, IsRegularSemigroup, true[110X. The
  functions [10Xarg[2i][110X can be user defined or existing [5XGAP[105X functions.[133X
  
  [33X[0;0YPlease see Section [14X4.5[114X or Chapter [14X3[114X for more details.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X is a positive integer, then [10XRandomSmallSemigroup[110X returns a random
  small    semigroup    [10XS[110X    in    the   library   with   [10XSize(S)=arg[1][110X   and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf  [10Xarg[1][110X is a list of positive integers, then [10XRandomSmallSemigroup[110X returns
  the  a  random  small  semigroup [10XS[110X in the library with [10XSize(S) in arg[1][110X and
  [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [33X[0;0YIf   [10Xarg[1][110X   is  an  enumerator  or  iterator  of  small  semigroups,  then
  [10XRandomSmallSemigroup[110X  returns  the a random small semigroup [10XS[110X in the library
  with [10XS in arg[1][110X and [10Xarg[2i](S)=arg[2i+1][110X for all [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X  gap> RandomSmallSemigroup(8, IsCommutative, true,[128X[104X
    [4X[28X  > IsInverseSemigroup, true);[128X[104X
    [4X[28X  <small semigroup of size 8>[128X[104X
    [4X[28X  gap> RandomSmallSemigroup([1..8], IsCliffordSemigroup, true);[128X[104X
    [4X[28X  <small semigroup of size 8>[128X[104X
    [4X[28X  gap> iter:=IteratorOfSmallSemigroups([1..7]);[128X[104X
    [4X[28X  <iterator of semigroups of size [ 1 .. 7 ]>[128X[104X
    [4X[28X  gap> RandomSmallSemigroup(iter);[128X[104X
    [4X[28X  <small semigroup of size 7>[128X[104X
  [4X[32X[104X
  
  [1X4.5-21 SizesOfSmallSemisInEnum[101X
  
  [29X[2XSizesOfSmallSemisInEnum[102X( [3Xenum[103X ) [32X function
  
  [33X[0;0Yreturns  the  sizes  of the semigroups in the enumerator of small semigroups
  [3Xenum[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:=EnumeratorOfSmallSemigroups([2..4], IsSimpleSemigroup, false);[127X[104X
    [4X[28X<enumerator of semigroups of sizes [ 2, 3, 4 ]>[128X[104X
    [4X[25Xgap>[125X [27XSizesOfSmallSemisInEnum(enum);[127X[104X
    [4X[28X[ 2, 3, 4 ][128X[104X
  [4X[32X[104X
  
  [1X4.5-22 SizesOfSmallSemisInIter[101X
  
  [29X[2XSizesOfSmallSemisInIter[102X( [3Xiter[103X ) [32X function
  
  [33X[0;0Yreturns  the  sizes  of  the  semigroups  in  the  iterator  [3Xiter[103X  of  small
  semigroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X  gap> iter:=IteratorOfSmallSemigroups(7, IsCommutative, false);[128X[104X
    [4X[28X  <iterator of semigroups of size 7>[128X[104X
    [4X[28X  gap> SizesOfSmallSemisInIter(iter);[128X[104X
    [4X[28X  [ 7 ][128X[104X
  [4X[32X[104X
  
  [1X4.5-23 UpToIsomorphism[101X
  
  [29X[2XUpToIsomorphism[102X( [3Xsgrps[103X ) [32X operation
  
  [33X[0;0Ytakes  a  list  [3Xsgrps[103X of non-equivalent semigroups from the library as input
  and  returns  a  list  of non-isomorphic semigroups containing an isomorphic
  semigroup and an anti-isomorphic semigroup for every semigroup in [3Xsgrps[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XUpToIsomorphism([SmallSemigroup(5,126),SmallSemigroup(6,2)]);[127X[104X
    [4X[28X[ <small semigroup of size 5>, <small semigroup of size 6> ][128X[104X
    [4X[25Xgap>[125X [27XUpToIsomorphism([SmallSemigroup(5,126),SmallSemigroup(5,3)]);[127X[104X
    [4X[28X[ <small semigroup of size 5>, <small semigroup of size 5>, [128X[104X
    [4X[28X  <semigroup of size 5, with 5 generators> ][128X[104X
  [4X[32X[104X
  
