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  51 Semigroups and Monoids
  
  This  chapter  describes  functions  for creating semigroups and monoids and
  determining information about them.
  
  
  51.1 Semigroups
  
  51.1-1 IsSemigroup
  
  IsSemigroup( D )  Synonym
  
  returns true if the object D is a semigroup. A semigroup is a magma (see 35)
  with associative multiplication.
  
  
  51.1-2 Semigroup
  
  Semigroup( gen1, gen2, ... )  function
  Semigroup( gens )  function
  
  In  the  first  form,  Semigroup  returns  the  semigroup  generated  by the
  arguments  gen1,  gen2,  ...,  that  is, the closure of these elements under
  multiplication.   In  the  second  form,  Semigroup  returns  the  semigroup
  generated  by  the elements in the homogeneous list gens; a square matrix as
  only argument is treated as one generator, not as a list of generators.
  
  It  is not checked whether the underlying multiplication is associative, use
  Magma  (35.2-1)  and  IsAssociative  (35.4-7) if you want to check whether a
  magma is in fact a semigroup.
  
    Example  
    gap> a:= Transformation( [ 2, 3, 4, 1 ] );
    Transformation( [ 2, 3, 4, 1 ] )
    gap> b:= Transformation( [ 2, 2, 3, 4 ] );
    Transformation( [ 2, 2 ] )
    gap> s:= Semigroup(a, b);
    <transformation semigroup of degree 4 with 2 generators>
  
  
  51.1-3 Subsemigroup
  
  Subsemigroup( S, gens )  function
  SubsemigroupNC( S, gens )  function
  
  are   just   synonyms   of   Submagma   (35.2-7)  and  SubmagmaNC  (35.2-7),
  respectively.
  
    Example  
    gap> a:=GeneratorsOfSemigroup(s)[1];
    Transformation( [ 2, 3, 4, 1 ] )
    gap> t:=Subsemigroup(s,[a]);
    <commutative transformation semigroup of degree 4 with 1 generator>
  
  
  51.1-4 IsSubsemigroup
  
  IsSubsemigroup( S, T )  operation
  Returns:  true or false.
  
  This  operation  returns  true  if  the semigroup T is a subsemigroup of the
  semigroup S and false if it is not.
  
    Example  
    gap> f := Transformation([5, 6, 7, 1, 4, 3, 2, 7]);
    Transformation( [ 5, 6, 7, 1, 4, 3, 2, 7 ] )
    gap> T := Semigroup(f);;
    gap> IsSubsemigroup(FullTransformationSemigroup(4), T);
    false
    gap> S := Semigroup(f);;
    gap> T := Semigroup(f ^ 2);;
    gap> IsSubsemigroup(S, T);
    true
  
  
  51.1-5 SemigroupByGenerators
  
  SemigroupByGenerators( gens )  operation
  
  is the underlying operation of Semigroup (51.1-2).
  
  51.1-6 AsSemigroup
  
  AsSemigroup( C )  operation
  
  If  C  is  a  collection  whose  elements  form a semigroup (see IsSemigroup
  (51.1-1))  then  AsSemigroup  returns  this  semigroup.  Otherwise  fail  is
  returned.
  
  51.1-7 AsSubsemigroup
  
  AsSubsemigroup( D, C )  operation
  
  Let  D  be  a  domain and C a collection. If C is a subset of D that forms a
  semigroup  then  AsSubsemigroup  returns  this  semigroup,  with  parent  D.
  Otherwise fail is returned.
  
  51.1-8 GeneratorsOfSemigroup
  
  GeneratorsOfSemigroup( S )  attribute
  
  Semigroup  generators  of  a  semigroup  D are the same as magma generators,
  see GeneratorsOfMagma (35.4-1).
  
    Example  
    gap> GeneratorsOfSemigroup(s);
    [ Transformation( [ 2, 3, 4, 1 ] ), Transformation( [ 2, 2 ] ) ]
    gap> GeneratorsOfSemigroup(t);
    [ Transformation( [ 2, 3, 4, 1 ] ) ]
  
  
  51.1-9 IsGeneratorsOfSemigroup
  
  IsGeneratorsOfSemigroup( C )  property
  
  This  property  reflects  whether  the  list  or  collection  C  generates a
  semigroup.       IsAssociativeElementCollection       (31.15-1)      implies
   IsGeneratorsOfSemigroup,  but  is  not  used  directly  in  semigroup code,
  because of conflicts with matrices.
  
    Example  
    gap> IsGeneratorsOfSemigroup([Transformation([2,3,1])]);
    true
  
  
  
  51.1-10 FreeSemigroup
  
  FreeSemigroup( [wfilt, ]rank[, name] )  function
  FreeSemigroup( [wfilt, ]name1, name2, ... )  function
  FreeSemigroup( [wfilt, ]names )  function
  FreeSemigroup( [wfilt, ]infinity, name, init )  function
  
  Called  with a positive integer rank, FreeSemigroup returns a free semigroup
  on  rank  generators.  If  the  optional  argument  name  is  given then the
  generators  are  printed  as  name1,  name2  etc., that is, each name is the
  concatenation of the string name and an integer from 1 to range. The default
  for name is the string "s".
  
  Called in the second form, FreeSemigroup returns a free semigroup on as many
  generators as arguments, printed as name1, name2 etc.
  
  Called  in the third form, FreeSemigroup returns a free semigroup on as many
  generators as the length of the list names, the i-th generator being printed
  as names[i].
  
  Called  in  the  fourth  form,  FreeSemigroup  returns  a  free semigroup on
  infinitely  many  generators,  where the first generators are printed by the
  names  in  the  list  init, and the other generators by name and an appended
  number.
  
  If    the   extra   argument   wfilt   is   given,   it   must   be   either
  IsSyllableWordsFamily    (37.6-6)   or   IsLetterWordsFamily   (37.6-2)   or
  IsWLetterWordsFamily  (37.6-4) or IsBLetterWordsFamily (37.6-4). This filter
  then  specifies  the  representation  used  for  the  elements  of  the free
  semigroup (see 37.6). If no such filter is given, a letter representation is
  used.
  
    Example  
    gap> f1 := FreeSemigroup( 3 );
    <free semigroup on the generators [ s1, s2, s3 ]>
    gap> f2 := FreeSemigroup( 3 , "generator" );
    <free semigroup on the generators 
    [ generator1, generator2, generator3 ]>
    gap> f3 := FreeSemigroup( "gen1" , "gen2" );
    <free semigroup on the generators [ gen1, gen2 ]>
    gap> f4 := FreeSemigroup( ["gen1" , "gen2"] );
    <free semigroup on the generators [ gen1, gen2 ]>
  
  
  For more on associative words see Chapter 37.
  
  Each  free  object defines a unique alphabet (and a unique family of words).
  Its generators are the letters of this alphabet, thus words of length one.
  
    Example  
    gap> FreeGroup( 5 );
    <free group on the generators [ f1, f2, f3, f4, f5 ]>
    gap> FreeGroup( "a", "b" );
    <free group on the generators [ a, b ]>
    gap> FreeGroup( infinity );
    <free group with infinity generators>
    gap> FreeSemigroup( "x", "y" );
    <free semigroup on the generators [ x, y ]>
    gap> FreeMonoid( 7 );
    <free monoid on the generators [ m1, m2, m3, m4, m5, m6, m7 ]>
  
  
  Remember  that  names  are  just  a help for printing and do not necessarily
  distinguish  letters.  It is possible to create arbitrarily weird situations
  by choosing strange names for the letters.
  
    Example  
    gap> f:= FreeGroup( "x", "x" );  gens:= GeneratorsOfGroup( f );;
    <free group on the generators [ x, x ]>
    gap> gens[1] = gens[2];
    false
    gap> f:= FreeGroup( "f1*f2", "f2^-1", "Group( [ f1, f2 ] )" );
    <free group on the generators [ f1*f2, f2^-1, Group( [ f1, f2 ] ) ]>
    gap> gens:= GeneratorsOfGroup( f );;
    gap> gens[1]*gens[2];
    f1*f2*f2^-1
    gap> gens[1]/gens[3];
    f1*f2*Group( [ f1, f2 ] )^-1
    gap> gens[3]/gens[1]/gens[2];
    Group( [ f1, f2 ] )*f1*f2^-1*f2^-1^-1
  
  
  51.1-11 SemigroupByMultiplicationTable
  
  SemigroupByMultiplicationTable( A )  function
  
  returns the semigroup whose multiplication is defined by the square matrix A
  (see MagmaByMultiplicationTable   (35.3-1))  if  such  a  semigroup  exists.
  Otherwise fail is returned.
  
    Example  
    gap> SemigroupByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);
    <semigroup of size 3, with 3 generators>
    gap> SemigroupByMultiplicationTable([[1,2,3],[2,3,1],[3,2,1]]);
    fail
  
  
  
  51.2 Monoids
  
  51.2-1 IsMonoid
  
  IsMonoid( D )  Synonym
  
  A monoid is a magma-with-one (see 35) with associative multiplication.
  
  
  51.2-2 Monoid
  
  Monoid( gen1, gen2, ... )  function
  Monoid( gens[, id] )  function
  
  In  the  first  form,  Monoid  returns the monoid generated by the arguments
  gen1, gen2, ..., that is, the closure of these elements under multiplication
  and  taking  the  0-th  power. In the second form, Monoid returns the monoid
  generated  by  the elements in the homogeneous list gens; a square matrix as
  only  argument  is treated as one generator, not as a list of generators. In
  the  second  form,  the  identity  element  id  may  be  given as the second
  argument.
  
  It  is not checked whether the underlying multiplication is associative, use
  MagmaWithOne  (35.2-2)  and  IsAssociative  (35.4-7)  if  you  want to check
  whether a magma-with-one is in fact a monoid.
  
  51.2-3 Submonoid
  
  Submonoid( M, gens )  function
  SubmonoidNC( M, gens )  function
  
  are   just   synonyms  of  SubmagmaWithOne  (35.2-8)  and  SubmagmaWithOneNC
  (35.2-8), respectively.
  
  51.2-4 MonoidByGenerators
  
  MonoidByGenerators( gens[, one] )  operation
  
  is the underlying operation of Monoid (51.2-2).
  
  51.2-5 AsMonoid
  
  AsMonoid( C )  operation
  
  If  C  is  a  collection whose elements form a monoid, then AsMonoid returns
  this monoid. Otherwise fail is returned.
  
  51.2-6 AsSubmonoid
  
  AsSubmonoid( D, C )  operation
  
  Let  D  be  a  domain and C a collection. If C is a subset of D that forms a
  monoid  then  AsSubmonoid returns this monoid, with parent D. Otherwise fail
  is returned.
  
  51.2-7 GeneratorsOfMonoid
  
  GeneratorsOfMonoid( M )  attribute
  
  Monoid  generators  of  a monoid M are the same as magma-with-one generators
  (see GeneratorsOfMagmaWithOne (35.4-2)).
  
  51.2-8 TrivialSubmonoid
  
  TrivialSubmonoid( M )  attribute
  
  is just a synonym for TrivialSubmagmaWithOne (35.4-13).
  
  
  51.2-9 FreeMonoid
  
  FreeMonoid( [wfilt, ]rank[, name] )  function
  FreeMonoid( [wfilt, ]name1, name2, ... )  function
  FreeMonoid( [wfilt, ]names )  function
  FreeMonoid( [wfilt, ]infinity, name, init )  function
  
  Called  with  a  positive  integer rank, FreeMonoid returns a free monoid on
  rank  generators. If the optional argument name is given then the generators
  are printed as name1, name2 etc., that is, each name is the concatenation of
  the  string name and an integer from 1 to range. The default for name is the
  string "m".
  
  Called  in  the  second  form,  FreeMonoid  returns a free monoid on as many
  generators as arguments, printed as name1, name2 etc.
  
  Called  in  the  third  form,  FreeMonoid  returns  a free monoid on as many
  generators as the length of the list names, the i-th generator being printed
  as names[i].
  
  Called  in  the  fourth form, FreeMonoid returns a free monoid on infinitely
  many  generators, where the first generators are printed by the names in the
  list init, and the other generators by name and an appended number.
  
  If    the   extra   argument   wfilt   is   given,   it   must   be   either
  IsSyllableWordsFamily    (37.6-6)   or   IsLetterWordsFamily   (37.6-2)   or
  IsWLetterWordsFamily  (37.6-4) or IsBLetterWordsFamily (37.6-4). This filter
  then  specifies  the representation used for the elements of the free monoid
  (see 37.6). If no such filter is given, a letter representation is used.
  
  For more on associative words see Chapter 37.
  
  51.2-10 MonoidByMultiplicationTable
  
  MonoidByMultiplicationTable( A )  function
  
  returns  the  monoid  whose multiplication is defined by the square matrix A
  (see MagmaByMultiplicationTable (35.3-1)) if such a monoid exists. Otherwise
  fail is returned.
  
    Example  
    gap> MonoidByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);
    <monoid of size 3, with 3 generators>
    gap> MonoidByMultiplicationTable([[1,2,3],[2,3,1],[1,3,2]]);
    fail
  
  
  
  51.3 Inverse semigroups and monoids
  
  51.3-1 InverseSemigroup
  
  InverseSemigroup( obj1, obj2, ... )  function
  Returns:  An inverse semigroup.
  
  If obj1, obj2, ... are (any combination) of associative elements with unique
  semigroup  inverses,  semigroups  of  such  elements, or collections of such
  elements,  then  InverseSemigroup returns the inverse semigroup generated by
  the  union  of  obj1,  obj2, .... This equals the semigroup generated by the
  union of obj1, obj2, ... and their inverses.
  
  For  example  if S and T are inverse semigroups, then InverseSemigroup(S, f,
  Idempotents(T));     is     the     inverse     semigroup    generated    by
  Union(GeneratorsOfInverseSemigroup(S), [f], Idempotents(T)));.
  
  As  present,  the  only associative elements with unique semigroup inverses,
  which  do not always generate a group, are partial permutations; see Chapter
  54.
  
    Example  
    gap> S := InverseSemigroup(
    > PartialPerm( [ 1, 2, 3, 6, 8, 10 ], [ 2, 6, 7, 9, 1, 5 ] ) );;
    gap> f := PartialPerm( [ 1, 2, 3, 4, 5, 8, 10 ], 
    > [ 7, 1, 4, 3, 2, 6, 5 ] );;
    gap> S := InverseSemigroup(S, f, Idempotents(SymmetricInverseSemigroup(5)));
    <inverse partial perm semigroup of rank 10 with 34 generators>
    gap> Size(S);
    1233
  
  
  51.3-2 InverseMonoid
  
  InverseMonoid( obj1, obj2, ... )  function
  Returns:  An inverse monoid.
  
  If obj1, obj2, ... are (any combination) of associative elements with unique
  semigroup  inverses,  semigroups  of  such  elements, or collections of such
  elements,  then  InverseMonoid  returns  the inverse monoid generated by the
  union  of  obj1, obj2, .... This equals the monoid generated by the union of
  obj1, obj2, ... and their inverses.
  
  As present, the only associative elements with unique semigroup inverses are
  partial permutations; see Chapter 54.
  
  For  example  if  S  and  T  are  inverse  monoids, then InverseMonoid(S, f,
  Idempotents(T));      is     the     inverse     monoid     generated     by
  Union(GeneratorsOfInverseMonoid(S), [f], Idempotents(T)));.
  
    Example  
    gap> S := InverseMonoid(
    > PartialPerm( [ 1, 2, 3, 6, 8, 10 ], [ 2, 6, 7, 9, 1, 5 ] ) );;
    gap> f := PartialPerm( [ 1, 2, 3, 4, 5, 8, 10 ], 
    > [ 7, 1, 4, 3, 2, 6, 5 ] );;
    gap> S := InverseMonoid(S, f, Idempotents(SymmetricInverseSemigroup(5)));
    <inverse partial perm monoid of rank 10 with 35 generators>
    gap> Size(S);
    1243
  
  
  51.3-3 GeneratorsOfInverseSemigroup
  
  GeneratorsOfInverseSemigroup( S )  attribute
  Returns:  The generators of an inverse semigroup.
  
  If  S is an inverse semigroup, then GeneratorsOfInverseSemigroup returns the
  generators used to define S, i.e. an inverse semigroup generating set for S.
  
  The  value  of  GeneratorsOfSemigroup(S), for an inverse semigroup S, is the
  union  of  inverse  semigroup  generator  and  their  inverses. So, S is the
  semigroup,  as  opposed  to  inverse semigroup, generated by the elements of
  GeneratorsOfInverseSemigroup(S) and their inverses.
  
  If  S  is  an  inverse monoid, then GeneratorsOfInverseSemigroup returns the
  generators used to define S, as described above, and the identity of S.
  
    Example  
    gap> S:=InverseMonoid(
    >  PartialPerm( [ 1, 2 ], [ 1, 4 ] ),
    >  PartialPerm( [ 1, 2, 4 ], [ 3, 4, 1 ] ) );;
    gap> GeneratorsOfSemigroup(S);
    [ <identity partial perm on [ 1, 2, 3, 4 ]>, [2,4](1), [2,4,1,3], 
      [4,2](1), [3,1,4,2] ]
    gap> GeneratorsOfInverseSemigroup(S);
    [ [2,4](1), [2,4,1,3], <identity partial perm on [ 1, 2, 3, 4 ]> ]
    gap> GeneratorsOfMonoid(S);
    [ [2,4](1), [2,4,1,3], [4,2](1), [3,1,4,2] ]
  
  
  51.3-4 GeneratorsOfInverseMonoid
  
  GeneratorsOfInverseMonoid( S )  attribute
  Returns:  The generators of an inverse monoid.
  
  If  S  is  an  inverse  monoid,  then  GeneratorsOfInverseMonoid returns the
  generators used to define S, i.e. an inverse monoid generating set for S.
  
  There  are  four  different possible generating sets which define an inverse
  monoid.  More  precisely,  an  inverse monoid can be generated as an inverse
  monoid,  inverse  semigroup,  monoid, or semigroup. The different generating
  sets   in   each  case  can  be  obtained  using  GeneratorsOfInverseMonoid,
  GeneratorsOfInverseSemigroup   (51.3-3),  GeneratorsOfMonoid  (51.2-7),  and
  GeneratorsOfSemigroup (51.1-8), respectively.
  
    Example  
    gap> S:=InverseMonoid(
    >  PartialPerm( [ 1, 2 ], [ 1, 4 ] ),
    >  PartialPerm( [ 1, 2, 4 ], [ 3, 4, 1 ] ) );;
    gap> GeneratorsOfInverseMonoid(S);
    [ [2,4](1), [2,4,1,3] ]
  
  
  51.3-5 IsInverseSubsemigroup
  
  IsInverseSubsemigroup( S, T )  operation
  Returns:  true or false.
  
  If  the semigroup T is an inverse subsemigroup of the semigroup S, then this
  operation returns true.
  
    Example  
    gap> T:=InverseSemigroup(RandomPartialPerm(4));;
    gap> IsInverseSubsemigroup(SymmetricInverseSemigroup(4), T); 
    true
    gap> T:=Semigroup(Transformation( [ 1, 2, 4, 5, 6, 3, 7, 8 ] ),
    > Transformation( [ 3, 3, 4, 5, 6, 2, 7, 8 ] ),
    > Transformation([ 1, 2, 5, 3, 6, 8, 4, 4 ] ));;
    gap> IsInverseSubsemigroup(FullTransformationSemigroup(8), T);
    true
  
  
  
  51.4 Properties of Semigroups
  
  The following functions determine information about semigroups.
  
  51.4-1 IsRegularSemigroup
  
  IsRegularSemigroup( S )  property
  
  returns true if S is regular, i.e., if every D-class of S is regular.
  
  51.4-2 IsRegularSemigroupElement
  
  IsRegularSemigroupElement( S, x )  operation
  
  returns  true if x has a general inverse in S, i.e., there is an element y ∈
  S such that x y x = x and y x y = y.
  
  51.4-3 InversesOfSemigroupElement
  
  InversesOfSemigroupElement( S, x )  operation
  Returns:  A list of the inverses of an element of a semigroup.
  
  InversesOfSemigroupElement  returns  a list of the inverses of the element x
  in the semigroup S.
  
  An  element  y in S is an inverse of x if x*y*x=x and y*x*y=y. The element x
  has an inverse if and only if x is a regular element of S.
  
    Example  
    gap> S := Semigroup([
    >  Transformation([3, 1, 4, 2, 5, 2, 1, 6, 1]), 
    >  Transformation([5, 7, 8, 8, 7, 5, 9, 1, 9]), 
    >  Transformation([7, 6, 2, 8, 4, 7, 5, 8, 3])]);
    <transformation semigroup of degree 9 with 3 generators>
    gap> x := Transformation([3, 1, 4, 2, 5, 2, 1, 6, 1]);;
    gap> InversesOfSemigroupElement(S, x);
    [  ]
    gap> IsRegularSemigroupElement(S, x);
    false
    gap> x := Transformation([1, 9, 7, 5, 5, 1, 9, 5, 1]);;
    gap> Set(InversesOfSemigroupElement(S, x));
    [ Transformation( [ 1, 2, 3, 5, 5, 1, 3, 5, 2 ] ), 
      Transformation( [ 1, 5, 1, 1, 5, 1, 3, 1, 2 ] ), 
      Transformation( [ 1, 5, 1, 2, 5, 1, 3, 2, 2 ] ) ]
    gap> IsRegularSemigroupElement(S, x);
    true
    gap> S := ReesZeroMatrixSemigroup(Group((1,2,3)),
    >  [[(), ()], [(), 0], [(), (1,2,3)]]);;
    gap> x := ReesZeroMatrixSemigroupElement(S, 2, (1,2,3), 3);;
    gap> InversesOfSemigroupElement(S, x);
    [ (1,(1,2,3),3), (1,(1,3,2),1), (2,(),3), (2,(1,2,3),1) ]
  
  
  51.4-4 IsSimpleSemigroup
  
  IsSimpleSemigroup( S )  property
  
  is true if and only if the semigroup S has no proper ideals.
  
  51.4-5 IsZeroSimpleSemigroup
  
  IsZeroSimpleSemigroup( S )  property
  
  is  true  if  and  only  if the semigroup has no proper ideals except for 0,
  where  S  is a semigroup with zero. If the semigroup does not find its zero,
  then a break-loop is entered.
  
  51.4-6 IsZeroGroup
  
  IsZeroGroup( S )  property
  
  is true if and only if the semigroup S is a group with zero adjoined.
  
  51.4-7 IsReesCongruenceSemigroup
  
  IsReesCongruenceSemigroup( S )  property
  
  returns  true  if  S  is  a  Rees  Congruence  semigroup,  that  is,  if all
  congruences of S are Rees Congruences.
  
  51.4-8 IsInverseSemigroup
  
  IsInverseSemigroup( S )  property
  IsInverseMonoid( S )  Category
  Returns:  true or false.
  
  A  semigroup  S is an inverse semigroup if every element x in S has a unique
  semigroup  inverse,  that  is, a unique element y in S such that x*y*x=x and
  y*x*y=y.
  
  A  monoid  that  happens  to  be  an  inverse semigroup is called an inverse
  monoid; see IsMonoid (51.2-1).
  
    Example  
    gap> S := Semigroup([
    >  Transformation([1, 2, 4, 5, 6, 3, 7, 8]),
    >  Transformation([3, 3, 4, 5, 6, 2, 7, 8]),
    >  Transformation([1, 2, 5, 3, 6, 8, 4, 4])]);;
    gap> IsInverseSemigroup(S);
    true
  
  
  
  51.5 Ideals of semigroups
  
  Ideals of semigroups are the same as ideals of the semigroup when considered
  as a magma. For documentation on ideals for magmas, see Magma (35.2-1).
  
  51.5-1 SemigroupIdealByGenerators
  
  SemigroupIdealByGenerators( S, gens )  operation
  
  S  is  a  semigroup,  gens is a list of elements of S. Returns the two-sided
  ideal of S generated by gens.
  
  51.5-2 ReesCongruenceOfSemigroupIdeal
  
  ReesCongruenceOfSemigroupIdeal( I )  attribute
  
  A  two sided ideal I of a semigroup S defines a congruence on S given by ∆ ∪
  I × I.
  
  51.5-3 IsLeftSemigroupIdeal
  
  IsLeftSemigroupIdeal( I )  property
  IsRightSemigroupIdeal( I )  property
  IsSemigroupIdeal( I )  property
  
  Categories of semigroup ideals.
  
  
  51.6 Congruences for semigroups
  
  An  equivalence  or  a  congruence  on  a  semigroup  is  the equivalence or
  congruence  on  the  semigroup  considered  as  a  magma.  So,  to deal with
  equivalences  and  congruences  on semigroups, magma functions are used. For
  documentation   on  equivalences  and  congruences  for  magmas,  see  Magma
  (35.2-1).
  
  51.6-1 IsSemigroupCongruence
  
  IsSemigroupCongruence( c )  property
  
  a magma congruence c on a semigroup.
  
  51.6-2 IsReesCongruence
  
  IsReesCongruence( c )  property
  
  returns  true  if  and only if the congruence c has at most one nonsingleton
  congruence class.
  
  
  51.7 Quotients
  
  Given a semigroup and a congruence on the semigroup, one can construct a new
  semigroup:  the  quotient  semigroup.  The  following  functions  deal  with
  quotient  semigroups  in  GAP.  For  a  semigroup  S, elements of a quotient
  semigroup     are     equivalence     classes    of    elements    of    the
  QuotientSemigroupPreimage  (51.7-3)  value under the congruence given by the
  value of QuotientSemigroupCongruence (51.7-3).
  
  It  is  probably most useful for calculating the elements of the equivalence
  classes  by using Elements (30.3-11) or by looking at the images of elements
  of   QuotientSemigroupPreimage   (51.7-3)   under   the   map   returned  by
  QuotientSemigroupHomomorphism       (51.7-3),       which      maps      the
  QuotientSemigroupPreimage (51.7-3) value to S.
  
  For   intensive  computations  in  a  quotient  semigroup,  it  is  probably
  worthwhile finding another representation as the equality test could involve
  enumeration of the elements of the congruence classes being compared.
  
  51.7-1 IsQuotientSemigroup
  
  IsQuotientSemigroup( S )  Category
  
  is  the  category  of  semigroups  constructed  from another semigroup and a
  congruence on it.
  
  51.7-2 HomomorphismQuotientSemigroup
  
  HomomorphismQuotientSemigroup( cong )  function
  
  for  a congruence cong and a semigroup S. Returns the homomorphism from S to
  the quotient of S by cong.
  
  51.7-3 QuotientSemigroupPreimage
  
  QuotientSemigroupPreimage( S )  attribute
  QuotientSemigroupCongruence( S )  attribute
  QuotientSemigroupHomomorphism( S )  attribute
  
  for a quotient semigroup S.
  
  
  51.8 Green's Relations
  
  Green's  equivalence  relations  play  a  very  important  role in semigroup
  theory. In this section we describe how they can be used in GAP.
  
  The  five  Green's  relations  are  R,  L, J, H, D: two elements x, y from a
  semigroup  S are R-related if and only if xS^1 = yS^1, L-related if and only
  if  S^1 x = S^1 y and J-related if and only if S^1 xS^1 = S^1 yS^1; finally,
  H = R ∧ L, and D = R ∘ L.
  
  Recall  that  relations R, L and J induce a partial order among the elements
  of the semigroup S: for two elements x, y from S, we say that x is less than
  or equal to y in the order on R if xS^1 ⊆ yS^1; similarly, x is less than or
  equal  to  y  under  L  if S^1x ⊆ S^1y; finally x is less than or equal to y
  under  J  if S^1 xS^1 ⊆ S^1 tS^1. We extend this preorder to a partial order
  on equivalence classes in the natural way.
  
  51.8-1 GreensRRelation
  
  GreensRRelation( semigroup )  attribute
  GreensLRelation( semigroup )  attribute
  GreensJRelation( semigroup )  attribute
  GreensDRelation( semigroup )  attribute
  GreensHRelation( semigroup )  attribute
  
  The  Green's  relations  (which are equivalence relations) are attributes of
  the semigroup semigroup.
  
  51.8-2 IsGreensRelation
  
  IsGreensRelation( bin-relation )  filter
  IsGreensRRelation( equiv-relation )  filter
  IsGreensLRelation( equiv-relation )  filter
  IsGreensJRelation( equiv-relation )  filter
  IsGreensHRelation( equiv-relation )  filter
  IsGreensDRelation( equiv-relation )  filter
  
  Categories for the Green's relations.
  
  51.8-3 IsGreensClass
  
  IsGreensClass( equiv-class )  filter
  IsGreensRClass( equiv-class )  filter
  IsGreensLClass( equiv-class )  filter
  IsGreensJClass( equiv-class )  filter
  IsGreensHClass( equiv-class )  filter
  IsGreensDClass( equiv-class )  filter
  
  return  true  if the equivalence class equiv-class is a Green's class of any
  type, or of R, L, J, H, D type, respectively, or false otherwise.
  
  51.8-4 IsGreensLessThanOrEqual
  
  IsGreensLessThanOrEqual( C1, C2 )  operation
  
  returns  true  if the Green's class C1 is less than or equal to C2 under the
  respective ordering (as defined above), and false otherwise.
  
  Only defined for R, L and J classes.
  
  51.8-5 RClassOfHClass
  
  RClassOfHClass( H )  attribute
  LClassOfHClass( H )  attribute
  
  are  attributes  reflecting  the  natural  ordering over the various Green's
  classes.  RClassOfHClass  and  LClassOfHClass  return  the  R and L classes,
  respectively, in which an H class is contained.
  
  51.8-6 EggBoxOfDClass
  
  EggBoxOfDClass( Dclass )  attribute
  
  returns for a Green's D class Dclass a matrix whose rows represent R classes
  and columns represent L classes. The entries are the H classes.
  
  51.8-7 DisplayEggBoxOfDClass
  
  DisplayEggBoxOfDClass( Dclass )  function
  
  displays  a  picture  of  the  D class Dclass, as an array of 1s and 0s. A 1
  represents a group H class.
  
  51.8-8 GreensRClassOfElement
  
  GreensRClassOfElement( S, a )  operation
  GreensLClassOfElement( S, a )  operation
  GreensDClassOfElement( S, a )  operation
  GreensJClassOfElement( S, a )  operation
  GreensHClassOfElement( S, a )  operation
  
  Creates the X class of the element a in the semigroup S where X is one of L,
  R, D, J, or H.
  
  51.8-9 GreensRClasses
  
  GreensRClasses( semigroup )  attribute
  GreensLClasses( semigroup )  attribute
  GreensJClasses( semigroup )  attribute
  GreensDClasses( semigroup )  attribute
  GreensHClasses( semigroup )  attribute
  
  return  the  R,  L,  J,  H, or D Green's classes, respectively for semigroup
  semigroup. EquivalenceClasses (33.7-3) for a Green's relation lead to one of
  these functions.
  
  51.8-10 GroupHClassOfGreensDClass
  
  GroupHClassOfGreensDClass( Dclass )  attribute
  
  for a D class Dclass of a semigroup, returns a group H class of the D class,
  or fail if there is no group H class.
  
  51.8-11 IsGroupHClass
  
  IsGroupHClass( Hclass )  property
  
  returns true if the Green's H class Hclass is a group, which in turn is true
  if and only if Hclass^2 intersects Hclass.
  
  51.8-12 IsRegularDClass
  
  IsRegularDClass( Dclass )  property
  
  returns  true  if the Greens D class Dclass is regular. A D class is regular
  if and only if each of its elements is regular, which in turn is true if and
  only  if any one element of Dclass is regular. Idempotents are regular since
  eee  =  e  so  it follows that a Green's D class containing an idempotent is
  regular. Conversely, it is true that a regular D class must contain at least
  one idempotent. (See [How76, Prop. 3.2].)
  
  51.8-13 DisplaySemigroup
  
  DisplaySemigroup( S )  operation
  
  Produces  a  convenient  display  of  a  transformation  semigroup's D-Class
  structure. Let S be a transformation semigroup of degree n. Then for each r≤
  n, we show all D-classes of rank r.
  
  A regular D-class with a single H-class of size 120 appears as
  
    Example  
    *[H size = 120, 1 L-class, 1 R-class] 
  
  
  (the * denoting regularity).
  
  
  51.9 Rees Matrix Semigroups
  
  In  this  section,  we  describe  the  functions  in GAP for Rees matrix and
  0-matrix  semigroups  and  their  subsemigroups.  The  importance  of  these
  semigroups  lies  in  the  fact  that Rees matrix semigroups over groups are
  exactly  the completely simple semigroups, and Rees 0-matrix semigroups over
  groups are the completely 0-simple semigroups.
  
  Let  I and J be sets, let S be a semigroup, and let P=(p_ji)_j∈ J, i∈ I be a
  |J|×  |I|  matrix  with  entries  in  S. Then the Rees matrix semigroup with
  underlying semigroup S and matrix P is just the direct product I× S × J with
  multiplication defined by
  
  
  (i, s, j)(k, t, l)=(i,s\cdot p_{j,k}\cdot t, l).
  
  
  
  Rees  0-matrix  semigroups  are defined as follows. If I, J, S, and P are as
  above  and  0  denotes  a new element, then the Rees 0-matrix semigroup with
  underlying  semigroup  S  and matrix P is (I× S× J)∪ {0} with multiplication
  defined by
  
  
  (i, s, j)(k, t, l)=(i, s\cdot p_{j,k}\cdot t, l)
  
  
  
  when p_j,k is not 0 and 0 if p_j,k is 0.
  
  If R is a Rees matrix or 0-matrix semigroup, then the rows of R is the index
  set  I,  the  columns  of  R  is  the  index  set  J, the semigroup S is the
  underlying semigroup of R, and the matrix P is the matrix of S.
  
  Thoroughout  this  section, wherever the distinction is unimportant, we will
  refer  to  Rees  matrix  or  0-matrix semigroups collectively as Rees matrix
  semigroups.
  
  Multiplication  of  elements of a Rees matrix semigroup obviously depends on
  the  matrix  used  to  create the semigroup. Hence elements of a Rees matrix
  semigroup  can only be created with reference to the semigroup to which they
  belong.  More  specifically,  every  collection  or semigroup of Rees matrix
  semigroup  elements  is created from a specific Rees matrix semigroup, which
  contains  the  whole  family  of  its  elements.  So,  it is not possible to
  multiply  or  compare elements belonging to distinct Rees matrix semigroups,
  since  they belong to different families. This situation is similar to, say,
  free  groups,  and different to, say, permutations, which belong to a single
  family,  and  where  arbitrary  permutations  can be compared and multiplied
  without reference to any group containing them.
  
  A  subsemigroup  of a Rees matrix semigroup is not necessarily a Rees matrix
  semigroup. Every semigroup consisting of elements of a Rees matrix semigroup
  satisfies the property IsReesMatrixSubsemigroup (51.9-6) and every semigroup
  of  Rees  0-matrix semigroup elements satisfies IsReesZeroMatrixSubsemigroup
  (51.9-6).
  
  Rees  matrix  and  0-matrix  semigroups  can be created using the operations
  ReesMatrixSemigroup    (51.9-1)    and   ReesZeroMatrixSemigroup   (51.9-1),
  respectively,  from  an  underlying  semigroup  and  a  matrix.  Rees matrix
  semigroups  created  in this way contain the whole family of their elements.
  Every  element  of  a  Rees  matrix  semigroup belongs to a unique semigroup
  created  in  this  way;  every  subsemigroup of a Rees matrix semigroup is a
  subsemigroup of a unique semigroup created in this way.
  
  Subsemigroups  of  Rees  matrix semigroups can also be created by specifying
  generators.  A  subsemigroup  of  a  Rees matrix semigroup I× U× J satisfies
  IsReesMatrixSemigroup  (51.9-7)  if  and  only  if it is equal to I'× U'× J'
  where  I'⊆ I, J'⊆ J, and U' is a subsemigroup of U. The analogous statements
  holds for Rees 0-matrix semigroups.
  
  It  is  not  necessarily the case that a simple subsemigroups of Rees matrix
  semigroups satisfies IsReesMatrixSemigroup (51.9-7). A Rees matrix semigroup
  is  simple  if  and  only  if  its  underlying semigroup is simple. A finite
  semigroup  is  simple  if  and  only  if  it  is isomorphic to a Rees matrix
  semigroup  over  a  group; this isomorphism can be obtained explicitly using
  IsomorphismReesMatrixSemigroup (51.9-3).
  
  Similarly, 0-simple subsemigroups of Rees 0-matrix semigroups do not have to
  satisfy  IsReesZeroMatrixSemigroup  (51.9-7). A Rees 0-matrix semigroup with
  more  than  2 elements is 0-simple if and only if every row and every column
  of  its  matrix  contains  a non-zero entry, and its underlying semigroup is
  simple.  A finite semigroup is 0-simple if and only if it is isomorphic to a
  Rees 0-matrix semigroup over a group; again this isomorphism can be found by
  using IsomorphismReesZeroMatrixSemigroup (51.9-3).
  
  Elements  of  a  Rees  matrix or 0-matrix semigroup belong to the categories
  IsReesMatrixSemigroupElement  (51.9-4)  and IsReesZeroMatrixSemigroupElement
  (51.9-4),  respectively.  Such  elements  can  be created directly using the
  functions          ReesMatrixSemigroupElement          (51.9-5)          and
  ReesZeroMatrixSemigroupElement (51.9-5).
  
  A  semigroup  in  GAP can either satisfies IsReesMatrixSemigroup (51.9-7) or
  IsReesZeroMatrixSemigroup (51.9-7) but not both.
  
  51.9-1 ReesMatrixSemigroup
  
  ReesMatrixSemigroup( S, mat )  operation
  ReesZeroMatrixSemigroup( S, mat )  operation
  Returns:  A Rees matrix or 0-matrix semigroup.
  
  When  S  is  a  semigroup and mat is an m by n matrix with entries in S, the
  function ReesMatrixSemigroup returns the n by m Rees matrix semigroup over S
  with multiplication defined by mat.
  
  The arguments of ReesZeroMatrixSemigroup should be a semigroup S and an m by
  n   matrix   mat   with   entries   in   S   or  equal  to  the  integer  0.
  ReesZeroMatrixSemigroup  returns  the  n by m Rees 0-matrix semigroup over S
  with  multiplication defined by mat. In GAP a Rees 0-matrix semigroup always
  contains  a multiplicative zero element, regardless of whether there are any
  entries in mat which are equal to 0.
  
    Example  
    gap> G:=Random(AllGroups(Size, 32));;
    gap> mat:=List([1..5], x-> List([1..3], y-> Random(G)));;
    gap> S:=ReesMatrixSemigroup(G, mat);
    <Rees matrix semigroup 3x5 over <pc group of size 32 with 
     5 generators>>
    gap> mat:=[[(), 0, (), ()], [0, 0, 0, 0]];;
    gap> S:=ReesZeroMatrixSemigroup(DihedralGroup(IsPermGroup, 8), mat);
    <Rees 0-matrix semigroup 4x2 over Group([ (1,2,3,4), (2,4) ])>
  
  
  51.9-2 ReesMatrixSubsemigroup
  
  ReesMatrixSubsemigroup( R, I, U, J )  operation
  ReesZeroMatrixSubsemigroup( R, I, U, J )  operation
  Returns:  A Rees matrix or 0-matrix subsemigroup.
  
  The arguments of ReesMatrixSubsemigroup should be a Rees matrix semigroup R,
  subsets  I  and  J  of  the  rows  and  columns  of  R,  respectively, and a
  subsemigroup  S  of  the  underlying  semigroup of R. ReesMatrixSubsemigroup
  returns  the  subsemigroup of R generated by the direct product of I, U, and
  J.
  
  The usage and returned value of ReesZeroMatrixSubsemigroup is analogous when
  R is a Rees 0-matrix semigroup.
  
    Example  
    gap> G:=CyclicGroup(IsPermGroup, 1007);;
    gap> mat:=[[(), 0, 0], [0, (), 0], [0, 0, ()], 
    > [(), (), ()], [0, 0, ()]];;
    gap> R:=ReesZeroMatrixSemigroup(G, mat);
    <Rees 0-matrix semigroup 3x5 over 
      <permutation group of size 1007 with 1 generators>>
    gap> ReesZeroMatrixSubsemigroup(R, [1,3], G, [1..5]);
    <Rees 0-matrix semigroup 2x5 over 
      <permutation group of size 1007 with 1 generators>>
  
  
  51.9-3 IsomorphismReesMatrixSemigroup
  
  IsomorphismReesMatrixSemigroup( S )  attribute
  IsomorphismReesZeroMatrixSemigroup( S )  attribute
  Returns:  An isomorphism.
  
  Every  finite simple semigroup is isomorphic to a Rees matrix semigroup over
  a  group,  and  every  finite  0-simple  semigroup  is  isomorphic to a Rees
  0-matrix semigroup over a group.
  
  If the argument S is a simple semigroup, then IsomorphismReesMatrixSemigroup
  returns  an isomorphism to a Rees matrix semigroup over a group. If S is not
  simple, then IsomorphismReesMatrixSemigroup returns an error.
  
  If     the     argument     S     is     a    0-simple    semigroup,    then
  IsomorphismReesZeroMatrixSemigroup returns an isomorphism to a Rees 0-matrix
  semigroup    over    a    group.    If    S    is    not    0-simple,   then
  IsomorphismReesMatrixSemigroup returns an error.
  
  See IsSimpleSemigroup (51.4-4) and IsZeroSimpleSemigroup (51.4-5).
  
    Example  
    gap> S := Semigroup(Transformation([2, 1, 1, 2, 1]), 
    >                   Transformation([3, 4, 3, 4, 4]), 
    >                   Transformation([3, 4, 3, 4, 3]),  
    >                   Transformation([4, 3, 3, 4, 4]));;
    gap> IsSimpleSemigroup(S);
    true
    gap> Range(IsomorphismReesMatrixSemigroup(S));
    <Rees matrix semigroup 4x2 over Group([ (1,2) ])>
    gap> mat := [[(), 0, 0], 
    >            [0, (), 0], 
    >            [0, 0, ()]];;
    gap> R := ReesZeroMatrixSemigroup(Group((1,2,4,5,6)), mat);
    <Rees 0-matrix semigroup 3x3 over Group([ (1,2,4,5,6) ])>
    gap> U := ReesZeroMatrixSubsemigroup(R, [1, 2], Group(()), [2, 3]);
    <subsemigroup of 3x3 Rees 0-matrix semigroup with 4 generators>
    gap> IsZeroSimpleSemigroup(U);
    false
    gap> U := ReesZeroMatrixSubsemigroup(R, [2, 3], Group(()), [2, 3]);
    <subsemigroup of 3x3 Rees 0-matrix semigroup with 3 generators>
    gap> IsZeroSimpleSemigroup(U);
    true
    gap> Rows(U); Columns(U);
    [ 2, 3 ]
    [ 2, 3 ]
    gap> V := Range(IsomorphismReesZeroMatrixSemigroup(U));
    <Rees 0-matrix semigroup 2x2 over Group(())>
    gap> Rows(V); Columns(V); 
    [ 1, 2 ]
    [ 1, 2 ]
  
  
  51.9-4 IsReesMatrixSemigroupElement
  
  IsReesMatrixSemigroupElement( elt )  Category
  IsReesZeroMatrixSemigroupElement( elt )  Category
  Returns:  true or false.
  
  Every   element   of  a  Rees  matrix  semigroup  belongs  to  the  category
  IsReesMatrixSemigroupElement, and every element of a Rees 0-matrix semigroup
  belongs to the category IsReesZeroMatrixSemigroupElement.
  
    Example  
    gap> G:=Group((1,2,3));;
    gap> mat:=[ [ (), (1,3,2) ], [ (1,3,2), () ] ];;
    gap> R:=ReesMatrixSemigroup(G, mat);
    <Rees matrix semigroup 2x2 over Group([ (1,2,3) ])>
    gap> GeneratorsOfSemigroup(R);
    [ (1,(1,2,3),1), (2,(),2) ]
    gap> IsReesMatrixSemigroupElement(last[1]);
    true
    gap> IsReesZeroMatrixSemigroupElement(last2[1]);
    false
  
  
  51.9-5 ReesMatrixSemigroupElement
  
  ReesMatrixSemigroupElement( R, i, x, j )  function
  ReesZeroMatrixSemigroupElement( R, i, x, j )  function
  Returns:  An element of a Rees matrix or 0-matrix semigroup.
  
  The   arguments  of  ReesMatrixSemigroupElement  should  be  a  Rees  matrix
  subsemigroup  R,  elements  i  and  j  of  the  the  rows  and columns of R,
  respectively,   and   an  element  x  of  the  underlying  semigroup  of  R.
  ReesMatrixSemigroupElement  returns  the  element  of  R  with  row index i,
  underlying  element  x in the underlying semigroup of R, and column index j,
  if such an element exist, if such an element exists.
  
  The   usage  of  ReesZeroMatrixSemigroupElement  is  analogous  to  that  of
  ReesMatrixSemigroupElement, when R is a Rees 0-matrix semigroup.
  
  The  row  i,  underlying  element  x, and column j of an element y of a Rees
  matrix (or 0-matrix) semigroup can be recovered from y using y[1], y[2], and
  y[3], respectively.
  
    Example  
    gap> G:=Group((1,2,3));;
    gap> mat:=[ [ 0, () ], [ (1,3,2), (1,3,2) ] ];;
    gap> R:=ReesZeroMatrixSemigroup(G, mat);
    <Rees 0-matrix semigroup 2x2 over Group([ (1,2,3) ])>
    gap> ReesZeroMatrixSemigroupElement(R, 1, (1,2,3), 2);
    (1,(1,2,3),2)
    gap> MultiplicativeZero(R);
    0
  
  
  51.9-6 IsReesMatrixSubsemigroup
  
  IsReesMatrixSubsemigroup( R )  Synonym
  IsReesZeroMatrixSubsemigroup( R )  Synonym
  Returns:  true or false.
  
  Every  semigroup consisting of elements of a Rees matrix semigroup satisfies
  the  property  IsReesMatrixSubsemigroup and every semigroup of Rees 0-matrix
  semigroup elements satisfies IsReesZeroMatrixSubsemigroup.
  
  Note  that  a  subsemigroup  of a Rees matrix semigroup is not necessarily a
  Rees matrix semigroup.
  
    Example  
    gap> G:=DihedralGroup(32);;
    gap> mat:=List([1..2], x-> List([1..10], x-> Random(G)));;
    gap> R:=ReesMatrixSemigroup(G, mat);
    <Rees matrix semigroup 10x2 over <pc group of size 32 with 
     5 generators>>
    gap> S:=Semigroup(GeneratorsOfSemigroup(R));      
    <subsemigroup of 10x2 Rees matrix semigroup with 14 generators>
    gap> IsReesMatrixSubsemigroup(S); 
    true
    gap> S:=Semigroup(GeneratorsOfSemigroup(R)[1]);
    <subsemigroup of 10x2 Rees matrix semigroup with 1 generator>
    gap> IsReesMatrixSubsemigroup(S);
    true
  
  
  51.9-7 IsReesMatrixSemigroup
  
  IsReesMatrixSemigroup( R )  property
  IsReesZeroMatrixSemigroup( R )  property
  Returns:  true or false.
  
  A   subsemigroup   of   a   Rees   matrix   semigroup   I×  U×  J  satisfies
  IsReesMatrixSemigroup  if and only if it is equal to I'× U'× J' where I'⊆ I,
  J'⊆  J,  and  U'  is  a  subsemigroup of U. It can be costly to check that a
  subsemigroup  defined  by  generators  satisfies  IsReesMatrixSemigroup. The
  analogous statements holds for Rees 0-matrix semigroups.
  
  It  is  not  necessarily the case that a simple subsemigroups of Rees matrix
  semigroups  satisfies  IsReesMatrixSemigroup.  A  Rees  matrix  semigroup is
  simple if and only if its underlying semigroup is simple. A finite semigroup
  is  simple if and only if it is isomorphic to a Rees matrix semigroup over a
  group;    this    isomorphism    can    be    obtained    explicitly   using
  IsomorphismReesMatrixSemigroup (51.9-3).
  
  Similarly, 0-simple subsemigroups of Rees 0-matrix semigroups do not have to
  satisfy  IsReesZeroMatrixSemigroup. A Rees 0-matrix semigroup with more than
  2  elements  is  0-simple  if  and only if every row and every column of its
  matrix  contains a non-zero entry, and its underlying semigroup is simple. A
  finite  semigroup  is  0-simple  if  and  only if it is isomorphic to a Rees
  0-matrix  semigroup  over  a  group;  again this isomorphism can be found by
  using IsomorphismReesMatrixSemigroup (51.9-3).
  
    Example  
    gap> G:=PSL(2,5);;
    gap> mat:=[ [ 0, (), 0, (2,6,3,5,4) ], 
    > [ (), 0, (), 0 ], [ 0, 0, 0, () ] ];;
    gap> R:=ReesZeroMatrixSemigroup(G, mat);
    <Rees 0-matrix semigroup 4x3 over Group([ (3,5)(4,6), (1,2,5)
    (3,4,6) ])>
    gap> IsReesZeroMatrixSemigroup(R);
    true
    gap> U:=ReesZeroMatrixSubsemigroup(R, [1..3], Group(()), [1..2]);
    <subsemigroup of 4x3 Rees 0-matrix semigroup with 4 generators>
    gap> IsReesZeroMatrixSemigroup(U);
    true
    gap> V:=Semigroup(GeneratorsOfSemigroup(U));
    <subsemigroup of 4x3 Rees 0-matrix semigroup with 4 generators>
    gap> IsReesZeroMatrixSemigroup(V);
    true
    gap> S:=Semigroup(Transformation([1,1]), Transformation([1,2]));
    <commutative transformation monoid of degree 2 with 1 generator>
    gap> IsSimpleSemigroup(S);
    false
    gap> mat:=[[0, One(S), 0, One(S)], [One(S), 0, One(S), 0], 
    > [0, 0, 0, One(S)]];;
    gap> R:=ReesZeroMatrixSemigroup(S, mat);;
    gap> U:=ReesZeroMatrixSubsemigroup(R, [1..3], 
    > Semigroup(Transformation([1,1])), [1..2]);
    <subsemigroup of 4x3 Rees 0-matrix semigroup with 6 generators>
    gap> V:=Semigroup(GeneratorsOfSemigroup(U));
    <subsemigroup of 4x3 Rees 0-matrix semigroup with 6 generators>
    gap> IsReesZeroMatrixSemigroup(V);
    true
    gap> T:=Semigroup(
    > ReesZeroMatrixSemigroupElement(R, 3, Transformation( [ 1, 1 ] ), 3), 
    > ReesZeroMatrixSemigroupElement(R, 2, Transformation( [ 1, 1 ] ), 2));
    <subsemigroup of 4x3 Rees 0-matrix semigroup with 2 generators>
    gap> IsReesZeroMatrixSemigroup(T);
    false
  
  
  51.9-8 Matrix
  
  Matrix( R )  attribute
  Returns:  A matrix.
  
  If  R is a Rees matrix or 0-matrix semigroup, then Matrix returns the matrix
  used to define multiplication in R.
  
  More  specifically,  if R is a Rees matrix or 0-matrix semigroup, which is a
  proper  subsemigroup  of  another  such  semigroup,  then Matrix returns the
  matrix  used to define the Rees matrix (or 0-matrix) semigroup consisting of
  the  whole  family to which the elements of R belong. Thus, for example, a 1
  by 1 Rees matrix semigroup can have a 65 by 15 matrix.
  
  Arbitrary  subsemigroups of Rees matrix or 0-matrix semigroups do not have a
  matrix.  Such  a  subsemigroup  R  has  a matrix if and only if it satisfies
  IsReesMatrixSemigroup (51.9-7) or IsReesZeroMatrixSemigroup (51.9-7).
  
    Example  
    gap> G:=AlternatingGroup(5);;
    gap> mat:=[[(), (), ()], [(), (), ()]];;
    gap> R:=ReesMatrixSemigroup(G, mat);
    <Rees matrix semigroup 3x2 over Alt( [ 1 .. 5 ] )>
    gap> Matrix(R); 
    [ [ (), (), () ], [ (), (), () ] ]
    gap> R:=ReesMatrixSubsemigroup(R, [1,2], Group(()), [2]);
    <subsemigroup of 3x2 Rees matrix semigroup with 2 generators>
    gap> Matrix(R);
    [ [ (), (), () ], [ (), (), () ] ]
  
  
  
  51.9-9 Rows and columns
  
  Rows( R )  attribute
  Columns( R )  attribute
  Returns:  The rows or columns of R.
  
  Rows  returns the rows of the Rees matrix or 0-matrix semigroup R. Note that
  the  rows  of  the semigroup correspond to the columns of the matrix used to
  define multiplication in R.
  
  Columns returns the columns of the Rees matrix or 0-matrix semigroup R. Note
  that  the columns of the semigroup correspond to the rows of the matrix used
  to define multiplication in R.
  
  Arbitrary  subsemigroups  of  Rees matrix or 0-matrix semigroups do not have
  rows  or  columns. Such a subsemigroup R has rows and columns if and only if
  it  satisfies  IsReesMatrixSemigroup  (51.9-7)  or IsReesZeroMatrixSemigroup
  (51.9-7).
  
    Example  
    gap> G:=Group((1,2,3));;                      
    gap> mat:=List([1..100], x-> List([1..200], x->Random(G)));;
    gap> R:=ReesZeroMatrixSemigroup(G, mat); 
    <Rees 0-matrix semigroup 200x100 over Group([ (1,2,3) ])>
    gap> Rows(R);
    [ 1 .. 200 ]
    gap> Columns(R);
    [ 1 .. 100 ]
  
  
  51.9-10 UnderlyingSemigroup
  
  UnderlyingSemigroup( R )  attribute
  UnderlyingSemigroup( R )  attribute
  Returns:  A semigroup.
  
  UnderlyingSemigroup  returns  the underlying semigroup of the Rees matrix or
  0-matrix semigroup R.
  
  Arbitrary subsemigroups of Rees matrix or 0-matrix semigroups do not have an
  underlying  semigroup.  Such a subsemigroup R has an underlying semigroup if
  and    only    if    it    satisfies   IsReesMatrixSemigroup   (51.9-7)   or
  IsReesZeroMatrixSemigroup (51.9-7).
  
    Example  
    gap> S:=Semigroup(Transformation( [ 2, 1, 1, 2, 1 ] ), 
    > Transformation( [ 3, 4, 3, 4, 4 ] ), Transformation([ 3, 4, 3, 4, 3 ] ),
    > Transformation([ 4, 3, 3, 4, 4 ] ) );;
    gap> R:=Range(IsomorphismReesMatrixSemigroup(S));    
    <Rees matrix semigroup 4x2 over Group([ (1,2) ])>
    gap> UnderlyingSemigroup(R);
    Group([ (1,2) ])
  
  
  51.9-11 AssociatedReesMatrixSemigroupOfDClass
  
  AssociatedReesMatrixSemigroupOfDClass( D )  attribute
  Returns:  A Rees matrix or 0-matrix semigroup.
  
  If  D is a regular D-class of a finite semigroup S, then there is a standard
  way  of  associating a Rees matrix semigroup to D. If D is a subsemigroup of
  S,  then  D is simple and hence is isomorphic to a Rees matrix semigroup. In
  this case, the associated Rees matrix semigroup of D is just the Rees matrix
  semigroup isomorphic to D.
  
  If  D is not a subsemigroup of S, then we define a semigroup with elements D
  and a new element 0 with multiplication of x,y∈ D defined by:
  
  xy equals the product of x and y if it belongs to D and 0 if it does not.
  
  The  semigroup  thus  defined  is 0-simple and hence is isomorphic to a Rees
  0-matrix  semigroup.  This  semigroup  can  also  be  described  as the Rees
  quotient  of the ideal generated by D by it maximal subideal. The associated
  Rees matrix semigroup of D is just the Rees 0-matrix semigroup isomorphic to
  the semigroup defined above.
  
    Example  
    gap> S:=FullTransformationSemigroup(5);;
    gap> D:=GreensDClasses(S)[3];
    {Transformation( [ 1, 1, 1, 2, 3 ] )}
    gap> AssociatedReesMatrixSemigroupOfDClass(D);
    <Rees 0-matrix semigroup 25x10 over Group([ (1,2)(3,5)(4,6), (1,3)
    (2,4)(5,6) ])>