GAP (Semigroups) - Chapter 11: Attributes and operations for semigroups

Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Bib Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

[MathJax off]

11 Attributes and operations for semigroups
11.1 Accessing the elements of a semigroup
11.1-1 AsListCanonical
11.1-2 PositionCanonical
11.1-3 Enumerate
11.1-4 IsEnumerated
11.2 Cayley graphs
11.2-1 RightCayleyDigraph
11.3 Random elements of a semigroup
11.3-1 Random
11.4 Properties of elements in a semigroup
11.4-1 IndexPeriodOfSemigroupElement
11.4-2 SmallestIdempotentPower
11.5 Expressing semigroup elements as words in generators
11.5-1 EvaluateWord
11.5-2 Factorization
11.5-3 MinimalFactorization
11.5-4 NonTrivialFactorization
11.6 Generating sets
11.6-1 Generators
11.6-2 SmallGeneratingSet
11.6-3 IrredundantGeneratingSubset
11.6-4 MinimalSemigroupGeneratingSet
11.6-5 GeneratorsSmallest
11.6-6 IndecomposableElements
11.7 Minimal ideals and multiplicative zeros
11.7-1 MinimalIdeal
11.7-2 RepresentativeOfMinimalIdeal
11.7-3 MultiplicativeZero
11.7-4 UnderlyingSemigroupOfSemigroupWithAdjoinedZero
11.8 Group of units and identity elements
11.8-1 GroupOfUnits
11.9 Idempotents
11.9-1 Idempotents
11.9-2 NrIdempotents
11.9-3 IdempotentGeneratedSubsemigroup
11.10 Maximal subsemigroups
11.10-1 MaximalSubsemigroups
11.10-2 NrMaximalSubsemigroups
11.10-3 IsMaximalSubsemigroup
11.11 Attributes of transformations and transformation semigroups
11.11-1 ComponentRepsOfTransformationSemigroup
11.11-2 ComponentsOfTransformationSemigroup
11.11-3 CyclesOfTransformationSemigroup
11.11-4 DigraphOfAction
11.11-5 DigraphOfActionOnPoints
11.11-6 FixedPointsOfTransformationSemigroup
11.11-7 IsTransitive
11.11-8 SmallestElementSemigroup
11.11-9 CanonicalTransformation
11.11-10 IsConnectedTransformationSemigroup
11.12 Attributes of partial perm semigroups
11.12-1 ComponentRepsOfPartialPermSemigroup
11.12-2 ComponentsOfPartialPermSemigroup
11.12-3 CyclesOfPartialPerm
11.12-4 CyclesOfPartialPermSemigroup
11.13 Attributes of Rees (0-)matrix semigroups
11.13-1 RZMSDigraph
11.13-2 RZMSConnectedComponents
11.14 Attributes of inverse semigroups
11.14-1 NaturalLeqInverseSemigroup
11.14-2 JoinIrreducibleDClasses
11.14-3 MajorantClosure
11.14-4 Minorants
11.14-5 PrimitiveIdempotents
11.14-6 RightCosetsOfInverseSemigroup
11.14-7 SameMinorantsSubgroup
11.14-8 SmallerDegreePartialPermRepresentation
11.14-9 VagnerPrestonRepresentation
11.14-10 CharacterTableOfInverseSemigroup
11.14-11 EUnitaryInverseCover
11.15 Nambooripad partial order
11.15-1 NambooripadLeqRegularSemigroup
11.15-2 NambooripadPartialOrder

### 11 Attributes and operations for semigroups

In this chapter we decribe the methods that are available in Semigroups for determining the attributes of a semigroup, and the operations which can be applied to a semigroup.

#### 11.1 Accessing the elements of a semigroup

##### 11.1-1 AsListCanonical

`‣ AsListCanonical`( S ) | ( attribute )

`‣ EnumeratorCanonical`( S ) | ( attribute )

`‣ IteratorCanonical`( S ) | ( operation )

Returns: A list, enumerator, or iterator.

When the argument S is a semigroup satisfying `CanUseFroidurePin` (6.1-4), `AsListCanonical` returns a list of the elements of S in the order they are enumerated by the Froidure-Pin Algorithm. This is the same as the order used to index the elements of S in `RightCayleyDigraph` (11.2-1) and `LeftCayleyDigraph` (11.2-1).

`EnumeratorCanonical` and `IteratorCanonical` return an enumerator and an iterator where the elements are ordered in the same way as `AsListCanonical`. Using `EnumeratorCanonical` or `IteratorCanonical` will often use less memory than `AsListCanonical`, but may have slightly worse performance if the elements of the semigroup are looped over repeatedly. `EnumeratorCanonical` returns the same list as `AsListCanonical` if `AsListCanonical` has ever been called for S.

If S is an acting semigroup, then the value returned by `AsList` may not equal the value returned by `AsListCanonical`. `AsListCanonical` exists so that there is a method for obtaining the elements of S in the particular order used by `RightCayleyDigraph` (11.2-1) and `LeftCayleyDigraph` (11.2-1).

See also `PositionCanonical` (11.1-2).

    gap> S := Semigroup(Transformation([1, 3, 2]));;
    gap> AsListCanonical(S);
    [ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ]
    gap> IteratorCanonical(S);
    <iterator>
    gap> EnumeratorCanonical(S);
    [ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ]
    gap> S := Monoid([Matrix(IsBooleanMat, [[1, 0, 0],
    >                                       [0, 1, 0],
    >                                       [0, 1, 0]])]);
    <commutative monoid of 3x3 boolean matrices with 1 generator>
    gap> it := IteratorCanonical(S);
    <iterator>
    gap> NextIterator(it);
    Matrix(IsBooleanMat, [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
    gap> en := EnumeratorCanonical(S);
    <enumerator of <commutative monoid of size 2, 3x3 boolean matrices
     with 1 generator>>
    gap> en[1];
    Matrix(IsBooleanMat, [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
    gap> Position(en, en[1]);
    1
    gap> Length(en);
    2

##### 11.1-2 PositionCanonical

`‣ PositionCanonical`( S, x ) | ( operation )

Returns: A positive integer or `fail`.

When the argument S is a semigroup satisfying `CanUseFroidurePin` (6.1-4) and x is an element of S, `PositionCanonical` returns the position of x in `AsListCanonical(S)` or equivalently the position of x in `EnumeratorCanonical(S)`.

See also `AsListCanonical` (11.1-1) and `EnumeratorCanonical` (11.1-1).

    gap> S := FullTropicalMaxPlusMonoid(2, 3);
    <monoid of 2x2 tropical max-plus matrices with 13 generators>
    gap> x := Matrix(IsTropicalMaxPlusMatrix, [[1, 3], [2, 1]], 3);
    Matrix(IsTropicalMaxPlusMatrix, [[1, 3], [2, 1]], 3)
    gap> PositionCanonical(S, x);
    234
    gap> EnumeratorCanonical(S)[234] = x;
    true

##### 11.1-3 Enumerate

`‣ Enumerate`( S[, limit] ) | ( operation )

Returns: A semigroup (the argument).

If S is a semigroup with representation `CanUseFroidurePin` (6.1-4) and limit is a positive integer, then this operation can be used to enumerate at least limit elements of S, or `Size(S)` elements if this is less than limit, using the Froidure-Pin Algorithm.

If the optional second argument limit is not given, then the semigroup is enumerated until all of its elements have been found.

    gap> S := FullTransformationMonoid(7);
    <full transformation monoid of degree 7>
    gap> Enumerate(S, 1000);
    <full transformation monoid of degree 7>

##### 11.1-4 IsEnumerated

`‣ IsEnumerated`( S ) | ( operation )

Returns: `true` or `false`.

If S is a semigroup with representation `CanUseFroidurePin` (6.1-4), then this operation returns `true` if the Froidure-Pin Algorithm has been run to completion (i.e. all of the elements of S have been found) and `false` if S has not been fully enumerated.

#### 11.2 Cayley graphs

##### 11.2-1 RightCayleyDigraph

`‣ RightCayleyDigraph`( S ) | ( attribute )

`‣ LeftCayleyDigraph`( S ) | ( attribute )

Returns: A digraph.

When the argument S is a semigroup satisfying `CanUseFroidurePin` (6.1-4), `RightCayleyDigraph` returns the right Cayley graph of S, as a `Digraph` (Digraphs: Digraph) `digraph` where vertex `OutNeighbours(digraph)[i][j]` is `PositionCanonical(S, AsListCanonical(S)[i] * GeneratorsOfSemigroup(S)[j])`. The digraph returned by `LeftCayleyDigraph` is defined analogously.

The digraph returned by this attribute belongs to the category `IsCayleyDigraph` (Digraphs: IsCayleyDigraph), the semigroup S and the generators used to create the digraph can be recovered from the digraph using `SemigroupOfCayleyDigraph` (Digraphs: SemigroupOfCayleyDigraph) and `GeneratorsOfCayleyDigraph` (Digraphs: GeneratorsOfCayleyDigraph).

    gap> S := FullTransformationMonoid(2);
    <full transformation monoid of degree 2>
    gap> RightCayleyDigraph(S);
    <immutable multidigraph with 4 vertices, 12 edges>
    gap> LeftCayleyDigraph(S);
    <immutable multidigraph with 4 vertices, 12 edges>

#### 11.3 Random elements of a semigroup

##### 11.3-1 Random

`‣ Random`( S ) | ( method )

Returns: A random element.

This function returns a random element of the semigroup S. If the elements of S have been calculated, then one of these is chosen randomly. Otherwise, if the data structure for S is known, then a random element of a randomly chosen \(\mathscr{R}\)-class is returned. If the data structure for S has not been calculated, then a short product (at most `2 * Length(GeneratorsOfSemigroup(S))`) of generators is returned.

#### 11.4 Properties of elements in a semigroup

##### 11.4-1 IndexPeriodOfSemigroupElement

`‣ IndexPeriodOfSemigroupElement`( x ) | ( operation )

Returns: A list of two positive integers.

If x is a semigroup element, then `IndexPeriodOfSemigroupElement(x)` returns the pair `[m, r]`, where `m` and `r` are the least positive integers such that `x^(m + r) = x ^ m`. The number `m` is known as the index of x, and the number`r` is known as the period of x.

    gap> x := Transformation([2, 6, 3, 5, 6, 1]);;
    gap> IndexPeriodOfSemigroupElement(x);
    [ 2, 3 ]
    gap> m := IndexPeriodOfSemigroupElement(x)[1];;
    gap> r := IndexPeriodOfSemigroupElement(x)[2];;
    gap> x ^ (m + r) = x ^ m;
    true
    gap> x := PartialPerm([0, 2, 3, 0, 5]);
    <identity partial perm on [ 2, 3, 5 ]>
    gap> IsIdempotent(x);
    true
    gap> IndexPeriodOfSemigroupElement(x);
    [ 1, 1 ]

##### 11.4-2 SmallestIdempotentPower

`‣ SmallestIdempotentPower`( x ) | ( attribute )

Returns: A positive integer.

If x is a semigroup element, then `SmallestIdempotentPower(x)` returns the least positive integer `n` such that `x^n` is an idempotent. The smallest idempotent power of x is the least multiple of the period of x that is greater than or equal to the index of x; see `IndexPeriodOfSemigroupElement` (11.4-1).

    gap> x := Transformation([4, 1, 4, 5, 1]);
    Transformation( [ 4, 1, 4, 5, 1 ] )
    gap> SmallestIdempotentPower(x);
    3
    gap> ForAll([1 .. 2], i -> not IsIdempotent(x ^ i));
    true
    gap> IsIdempotent(x ^ 3);
    true
    gap> x := Bipartition([[1, 2, -3, -4], [3, -5], [4, -1], [5, -2]]);
    <block bijection: [ 1, 2, -3, -4 ], [ 3, -5 ], [ 4, -1 ], [ 5, -2 ]>
    gap> SmallestIdempotentPower(x);
    4
    gap> ForAll([1 .. 3], i -> not IsIdempotent(x ^ i));
    true
    gap> x := PartialPerm([]);
    <empty partial perm>
    gap> SmallestIdempotentPower(x);
    1
    gap> IsIdempotent(x);
    true

#### 11.5 Expressing semigroup elements as words in generators

It is possible to express an element of a semigroup as a word in the generators of that semigroup. This section describes how to accomplish this in Semigroups.

##### 11.5-1 EvaluateWord

`‣ EvaluateWord`( gens, w ) | ( operation )

Returns: A semigroup element.

The argument gens should be a collection of generators of a semigroup and the argument w should be a list of positive integers less than or equal to the length of gens. This operation evaluates the word w in the generators gens. More precisely, `EvaluateWord(gens, w)` returns the equivalent of:

    Product(List(w, i -> gens[i]));

see also `Factorization` (11.5-2).

for elements of a semigroup

  When gens is a list of elements of a semigroup and w is a list of positive integers less than or equal to the length of gens, this operation returns the product `gens[w[1]] * gens[w[2]] * .. . * gens[w[n]]` when the length of w is `n`.

for elements of an inverse semigroup

  When gens is a list of elements with a semigroup inverse and w is a list of non-zero integers whose absolute value does not exceed the length of gens, this operation returns the product `gens[AbsInt(w[1])] ^ SignInt(w[1]) * .. . * gens[AbsInt(w[n])] ^ SignInt(w[n])` where `n` is the length of w.

Note that `EvaluateWord(gens, [])` returns `One(gens)` if gens belongs to the category `IsMultiplicativeElementWithOne` (Reference: IsMultiplicativeElementWithOne).

    gap> gens := [
    > Transformation([2, 4, 4, 6, 8, 8, 6, 6]),
    > Transformation([2, 7, 4, 1, 4, 6, 5, 2]),
    > Transformation([3, 6, 2, 4, 2, 2, 2, 8]),
    > Transformation([4, 3, 6, 4, 2, 1, 2, 6]),
    > Transformation([4, 5, 1, 3, 8, 5, 8, 2])];;
    gap> S := Semigroup(gens);;
    gap> x := Transformation([1, 4, 6, 1, 7, 2, 7, 6]);;
    gap> word := Factorization(S, x);
    [ 4, 2 ]
    gap> EvaluateWord(gens, word);
    Transformation( [ 1, 4, 6, 1, 7, 2, 7, 6 ] )
    gap> S := SymmetricInverseMonoid(10);;
    gap> x := PartialPerm([2, 6, 7, 0, 0, 9, 0, 1, 0, 5]);
    [3,7][8,1,2,6,9][10,5]
    gap> word := Factorization(S, x);
    [ -2, -2, -2, -2, -3, -2, -2, -2, -2, -2, 5, 2, 5, 5, 2, 5, 2, 2, 2,
      2, -3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2 ]
    gap> EvaluateWord(GeneratorsOfSemigroup(S), word);
    [3,7][8,1,2,6,9][10,5]

##### 11.5-2 Factorization

`‣ Factorization`( S, x ) | ( operation )

Returns: A word in the generators.

for semigroups

  When S is a semigroup and x belongs to S, `Factorization` return a word in the generators of S that is equal to x. In this case, a word is a list of positive integers where an entry `i` corresponds to `GeneratorsOfSemigroups(S)[i]`. More specifically,

      EvaluateWord(GeneratorsOfSemigroup(S), Factorization(S, x)) = x;

for inverse semigroups

  When S is an inverse semigroup and x belongs to S, `Factorization` return a word in the generators of S that is equal to x. In this case, a word is a list of non-zero integers where an entry `i` corresponds to `GeneratorsOfSemigroup(S)[i]` and `-i` corresponds to `GeneratorsOfSemigroup(S)[i] ^ -1`. As in the previous case,

      EvaluateWord(GeneratorsOfSemigroup(S), Factorization(S, x)) = x;

Note that `Factorization` does not always return a word of minimum length; see `MinimalFactorization` (11.5-3).

See also `EvaluateWord` (11.5-1) and `GeneratorsOfSemigroup` (Reference: GeneratorsOfSemigroup).

    gap> gens := [Transformation([2, 2, 9, 7, 4, 9, 5, 5, 4, 8]),
    >             Transformation([4, 10, 5, 6, 4, 1, 2, 7, 1, 2])];;
    gap> S := Semigroup(gens);;
    gap> x := Transformation([1, 10, 2, 10, 1, 2, 7, 10, 2, 7]);;
    gap> word := Factorization(S, x);
    [ 2, 2, 1, 2 ]
    gap> EvaluateWord(gens, word);
    Transformation( [ 1, 10, 2, 10, 1, 2, 7, 10, 2, 7 ] )
    gap> S := SymmetricInverseMonoid(8);
    <symmetric inverse monoid of degree 8>
    gap> x := PartialPerm([1, 2, 3, 4, 5, 8], [7, 1, 4, 3, 2, 6]);
    [5,2,1,7][8,6](3,4)
    gap> word := Factorization(S, x);
    [ -2, -2, -2, -2, -2, -2, 2, 4, 4, 2, 3, 2, -3, -2, -2, 3, 2, -3, -2,
      -2, 4, -3, -4, 2, 2, 3, -2, -3, 4, -3, -4, 2, 2, 3, -2, -3, 2, 2,
      3, -2, -3, 2, 2, 3, -2, -3, 4, -3, -4, 3, 2, -3, -2, -2, 3, 2, -3,
      -2, -2, 4, 3, -4, 3, 2, -3, -2, -2, 3, 2, -3, -2, -2, 3, 2, 2, 3,
      2, 2, 2, 2 ]
    gap> EvaluateWord(GeneratorsOfSemigroup(S), word);
    [5,2,1,7][8,6](3,4)
    gap> S := DualSymmetricInverseMonoid(6);;
    gap> x := S.1 * S.2 * S.3 * S.2 * S.1;
    <block bijection: [ 1, 6, -4 ], [ 2, -2, -3 ], [ 3, -5 ], [ 4, -6 ],
     [ 5, -1 ]>
    gap> word := Factorization(S, x);
    [ -2, -2, -2, -2, -2, 4, 2 ]
    gap> EvaluateWord(GeneratorsOfSemigroup(S), word);
    <block bijection: [ 1, 6, -4 ], [ 2, -2, -3 ], [ 3, -5 ], [ 4, -6 ],
     [ 5, -1 ]>

##### 11.5-3 MinimalFactorization

`‣ MinimalFactorization`( S, x ) | ( operation )

Returns: A minimal word in the generators.

This operation returns a minimal length word in the generators of the semigroup S that equals the element x. In this case, a word is a list of positive integers where an entry `i` corresponds to `GeneratorsOfSemigroups(S)[i]`. More specifically,

    EvaluateWord(GeneratorsOfSemigroup(S), MinimalFactorization(S, x)) = x;

`MinimalFactorization` involves exhaustively enumerating S until the element x is found, and so `MinimalFactorization` may be less efficient than `Factorization` (11.5-2) for some semigroups.

Unlike `Factorization` (11.5-2) this operation does not distinguish between semigroups and inverse semigroups. See also `EvaluateWord` (11.5-1) and `GeneratorsOfSemigroup` (Reference: GeneratorsOfSemigroup).

    gap> S := Semigroup(Transformation([2, 2, 9, 7, 4, 9, 5, 5, 4, 8]),
    >                   Transformation([4, 10, 5, 6, 4, 1, 2, 7, 1, 2]));
    <transformation semigroup of degree 10 with 2 generators>
    gap> x := Transformation([8, 8, 2, 2, 9, 2, 8, 8, 9, 9]);
    Transformation( [ 8, 8, 2, 2, 9, 2, 8, 8, 9, 9 ] )
    gap> Factorization(S, x);
    [ 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1 ]
    gap> MinimalFactorization(S, x);
    [ 1, 2, 1, 1, 1, 1, 2, 2, 1 ]

##### 11.5-4 NonTrivialFactorization

`‣ NonTrivialFactorization`( S, x ) | ( operation )

Returns: A non-trivial word in the generators, or `fail`.

When S is a semigroup and x belongs to S, this operation returns a non-trivial word in the generators of the semigroup S that equals x, if one exists. The definition of a word in the generators is the same as given in `Factorization` (11.5-2) for semigroups and inverse semigroups. A word is non-trivial if it has length two or more.

If no non-trivial word for x exists, then x is an indecomposable element of S and this operation returns `fail`; see `IndecomposableElements` (11.6-6).

When x does not belong to `GeneratorsOfSemigroup(S)`, any factorization of x is non-trivial. In this case, `NonTrivialFactorization` returns the same word as `Factorization` (11.5-2).

See also `EvaluateWord` (11.5-1) and `GeneratorsOfSemigroup` (Reference: GeneratorsOfSemigroup).

    gap> x := Transformation([5, 4, 2, 1, 3]);;
    gap> y := Transformation([4, 4, 2, 4, 1]);;
    gap> S := Semigroup([x, y]);
    <transformation semigroup of degree 5 with 2 generators>
    gap> NonTrivialFactorization(S, x * y);
    [ 1, 2 ]
    gap> Factorization(S, x);
    [ 1 ]
    gap> NonTrivialFactorization(S, x);
    [ 1, 1, 1, 1, 1, 1 ]
    gap> Factorization(S, y);
    [ 2 ]
    gap> NonTrivialFactorization(S, y);
    [ 2, 1, 1, 1, 1, 1 ]
    gap> z := PartialPerm([2]);;
    gap> S := Semigroup(z);
    <commutative partial perm semigroup of rank 1 with 1 generator>
    gap> NonTrivialFactorization(S, z);
    fail

#### 11.6 Generating sets

##### 11.6-1 Generators

`‣ Generators`( S ) | ( attribute )

Returns: A list of generators.

`Generators` returns a generating set that can be used to define the semigroup S. The generators of a monoid or inverse semigroup S, say, can be defined in several ways, for example, including or excluding the identity element, including or not the inverses of the generators. `Generators` uses the definition that returns the least number of generators. If no generating set for S is known, then `GeneratorsOfSemigroup` is used by default.

for a group

  `Generators(S)` is a synonym for `GeneratorsOfGroup` (Reference: GeneratorsOfGroup).

for an ideal of semigroup

  `Generators(S)` is a synonym for `GeneratorsOfSemigroupIdeal` (9.2-1).

for a semigroup

  `Generators(S)` is a synonym for `GeneratorsOfSemigroup` (Reference: GeneratorsOfSemigroup).

for a monoid

  `Generators(S)` is a synonym for `GeneratorsOfMonoid` (Reference: GeneratorsOfMonoid).

for an inverse semigroup

  `Generators(S)` is a synonym for `GeneratorsOfInverseSemigroup` (Reference: GeneratorsOfInverseSemigroup).

for an inverse monoid

  `Generators(S)` is a synonym for `GeneratorsOfInverseMonoid` (Reference: GeneratorsOfInverseMonoid).

    gap> M := Monoid([
    > Transformation([1, 4, 6, 2, 5, 3, 7, 8, 9, 9]),
    > Transformation([6, 3, 2, 7, 5, 1, 8, 8, 9, 9])]);;
    gap> GeneratorsOfSemigroup(M);
    [ IdentityTransformation,
      Transformation( [ 1, 4, 6, 2, 5, 3, 7, 8, 9, 9 ] ),
      Transformation( [ 6, 3, 2, 7, 5, 1, 8, 8, 9, 9 ] ) ]
    gap> GeneratorsOfMonoid(M);
    [ Transformation( [ 1, 4, 6, 2, 5, 3, 7, 8, 9, 9 ] ),
      Transformation( [ 6, 3, 2, 7, 5, 1, 8, 8, 9, 9 ] ) ]
    gap> Generators(M);
    [ Transformation( [ 1, 4, 6, 2, 5, 3, 7, 8, 9, 9 ] ),
      Transformation( [ 6, 3, 2, 7, 5, 1, 8, 8, 9, 9 ] ) ]
    gap> S := Semigroup(Generators(M));;
    gap> Generators(S);
    [ Transformation( [ 1, 4, 6, 2, 5, 3, 7, 8, 9, 9 ] ),
      Transformation( [ 6, 3, 2, 7, 5, 1, 8, 8, 9, 9 ] ) ]
    gap> GeneratorsOfSemigroup(S);
    [ Transformation( [ 1, 4, 6, 2, 5, 3, 7, 8, 9, 9 ] ),
      Transformation( [ 6, 3, 2, 7, 5, 1, 8, 8, 9, 9 ] ) ]

##### 11.6-2 SmallGeneratingSet

`‣ SmallGeneratingSet`( coll ) | ( attribute )

`‣ SmallSemigroupGeneratingSet`( coll ) | ( attribute )

`‣ SmallMonoidGeneratingSet`( coll ) | ( attribute )

`‣ SmallInverseSemigroupGeneratingSet`( coll ) | ( attribute )

`‣ SmallInverseMonoidGeneratingSet`( coll ) | ( attribute )

Returns: A small generating set for a semigroup.

The attributes `SmallXGeneratingSet` return a relatively small generating subset of the collection of elements coll, which can also be a semigroup. The returned value of `SmallXGeneratingSet`, where applicable, has the property that

          X(SmallXGeneratingSet(coll)) = X(coll);

where `X` is any of `Semigroup` (Reference: Semigroup), `Monoid` (Reference: Monoid), `InverseSemigroup` (Reference: InverseSemigroup), or `InverseMonoid` (Reference: InverseMonoid).

If the number of generators for S is already relatively small, then these functions will often return the original generating set. These functions may return different results in different GAP sessions.

`SmallGeneratingSet` returns the smallest of the returned values of `SmallXGeneratingSet` which is applicable to coll; see `Generators` (11.6-1).

As neither irredundancy, nor minimal length are proven, these functions usually return an answer much more quickly than `IrredundantGeneratingSubset` (11.6-3). These functions can be used whenever a small generating set is desired which does not necessarily needs to be minimal.

    gap> S := Semigroup([
    > Transformation([1, 2, 3, 2, 4]),
    > Transformation([1, 5, 4, 3, 2]),
    > Transformation([2, 1, 4, 2, 2]),
    > Transformation([2, 4, 4, 2, 1]),
    > Transformation([3, 1, 4, 3, 2]),
    > Transformation([3, 2, 3, 4, 1]),
    > Transformation([4, 4, 3, 3, 5]),
    > Transformation([5, 1, 5, 5, 3]),
    > Transformation([5, 4, 3, 5, 2]),
    > Transformation([5, 5, 4, 5, 5])]);;
    gap> SmallGeneratingSet(S);
    [ Transformation( [ 1, 5, 4, 3, 2 ] ), Transformation( [ 3, 2, 3, 4, 1 ] ),
      Transformation( [ 5, 4, 3, 5, 2 ] ), Transformation( [ 1, 2, 3, 2, 4 ] ),
      Transformation( [ 4, 4, 3, 3, 5 ] ) ]
    gap> S := RandomInverseMonoid(IsPartialPermMonoid, 10000, 10);;
    gap> SmallGeneratingSet(S);
    [ [ 1 .. 10 ] -> [ 3, 2, 4, 5, 6, 1, 7, 10, 9, 8 ],
      [ 1 .. 10 ] -> [ 5, 10, 8, 9, 3, 2, 4, 7, 6, 1 ],
      [ 1, 3, 4, 5, 6, 7, 8, 9, 10 ] -> [ 1, 6, 4, 8, 2, 10, 7, 3, 9 ] ]
    gap> M := MathieuGroup(24);;
    gap> mat := List([1 .. 1000], x -> Random(M));;
    gap> Append(mat, [1 .. 1000] * 0);
    gap> mat := List([1 .. 138], x -> List([1 .. 57], x -> Random(mat)));;
    gap> R := ReesZeroMatrixSemigroup(M, mat);;
    gap> U := Semigroup(List([1 .. 200], x -> Random(R)));
    <subsemigroup of 57x138 Rees 0-matrix semigroup with 100 generators>
    gap> Length(SmallGeneratingSet(U));
    84
    gap> S := RandomSemigroup(IsBipartitionSemigroup, 100, 4);
    <bipartition semigroup of degree 4 with 96 generators>
    gap> Length(SmallGeneratingSet(S));
    13

##### 11.6-3 IrredundantGeneratingSubset

`‣ IrredundantGeneratingSubset`( coll ) | ( operation )

Returns: A list of irredundant generators.

If coll is a collection of elements of a semigroup, then this function returns a subset `U` of coll such that no element of `U` is generated by the other elements of `U`.

    gap> S := Semigroup([
    > Transformation([5, 1, 4, 6, 2, 3]),
    > Transformation([1, 2, 3, 4, 5, 6]),
    > Transformation([4, 6, 3, 4, 2, 5]),
    > Transformation([5, 4, 6, 3, 1, 3]),
    > Transformation([2, 2, 6, 5, 4, 3]),
    > Transformation([3, 5, 5, 1, 2, 4]),
    > Transformation([6, 5, 1, 3, 3, 4]),
    > Transformation([1, 3, 4, 3, 2, 1])]);;
    gap> IrredundantGeneratingSubset(S);
    [ Transformation( [ 1, 3, 4, 3, 2, 1 ] ),
      Transformation( [ 2, 2, 6, 5, 4, 3 ] ),
      Transformation( [ 3, 5, 5, 1, 2, 4 ] ),
      Transformation( [ 5, 1, 4, 6, 2, 3 ] ),
      Transformation( [ 5, 4, 6, 3, 1, 3 ] ),
      Transformation( [ 6, 5, 1, 3, 3, 4 ] ) ]
    gap> S := RandomInverseMonoid(IsPartialPermMonoid, 1000, 10);
    <inverse partial perm monoid of degree 10 with 1000 generators>
    gap> SmallGeneratingSet(S);
    [ [ 1 .. 10 ] -> [ 6, 5, 1, 9, 8, 3, 10, 4, 7, 2 ],
      [ 1 .. 10 ] -> [ 1, 4, 6, 2, 8, 5, 7, 10, 3, 9 ],
      [ 1, 2, 3, 4, 6, 7, 8, 9 ] -> [ 7, 5, 10, 1, 8, 4, 9, 6 ]
      [ 1 .. 9 ] -> [ 4, 3, 5, 7, 10, 9, 1, 6, 8 ] ]
    gap> IrredundantGeneratingSubset(last);
    [ [ 1 .. 9 ] -> [ 4, 3, 5, 7, 10, 9, 1, 6, 8 ],
      [ 1 .. 10 ] -> [ 1, 4, 6, 2, 8, 5, 7, 10, 3, 9 ],
      [ 1 .. 10 ] -> [ 6, 5, 1, 9, 8, 3, 10, 4, 7, 2 ] ]
    gap> S := RandomSemigroup(IsBipartitionSemigroup, 1000, 4);
    <bipartition semigroup of degree 4 with 749 generators>
    gap> SmallGeneratingSet(S);
    [ <bipartition: [ 1, -3 ], [ 2, -2 ], [ 3, -1 ], [ 4, -4 ]>,
      <bipartition: [ 1, 3, -2 ], [ 2, -1, -3 ], [ 4, -4 ]>,
      <bipartition: [ 1, -4 ], [ 2, 4, -1, -3 ], [ 3, -2 ]>,
      <bipartition: [ 1, -1, -3 ], [ 2, -4 ], [ 3, 4, -2 ]>,
      <bipartition: [ 1, -2, -4 ], [ 2 ], [ 3, -3 ], [ 4, -1 ]>,
      <bipartition: [ 1, -2 ], [ 2, -1, -3 ], [ 3, 4, -4 ]>,
      <bipartition: [ 1, 3, -1 ], [ 2, -3 ], [ 4, -2, -4 ]>,
      <bipartition: [ 1, -1 ], [ 2, 4, -4 ], [ 3, -2, -3 ]>,
      <bipartition: [ 1, 3, -1 ], [ 2, -2 ], [ 4, -3, -4 ]>,
      <bipartition: [ 1, 2, -2 ], [ 3, -1, -4 ], [ 4, -3 ]>,
      <bipartition: [ 1, -2, -3 ], [ 2, -4 ], [ 3 ], [ 4, -1 ]>,
      <bipartition: [ 1, -1 ], [ 2, 4, -3 ], [ 3, -2 ], [ -4 ]>,
      <bipartition: [ 1, -3 ], [ 2, -1 ], [ 3, 4, -4 ], [ -2 ]>,
      <bipartition: [ 1, 2, -4 ], [ 3, -1 ], [ 4, -2 ], [ -3 ]>,
      <bipartition: [ 1, -3 ], [ 2, -4 ], [ 3, -1, -2 ], [ 4 ]> ]
    gap> IrredundantGeneratingSubset(last);
    [ <bipartition: [ 1, 2, -4 ], [ 3, -1 ], [ 4, -2 ], [ -3 ]>,
      <bipartition: [ 1, 3, -1 ], [ 2, -2 ], [ 4, -3, -4 ]>,
      <bipartition: [ 1, 3, -2 ], [ 2, -1, -3 ], [ 4, -4 ]>,
      <bipartition: [ 1, -1 ], [ 2, 4, -3 ], [ 3, -2 ], [ -4 ]>,
      <bipartition: [ 1, -3 ], [ 2, -1 ], [ 3, 4, -4 ], [ -2 ]>,
      <bipartition: [ 1, -3 ], [ 2, -2 ], [ 3, -1 ], [ 4, -4 ]>,
      <bipartition: [ 1, -3 ], [ 2, -4 ], [ 3, -1, -2 ], [ 4 ]>,
      <bipartition: [ 1, -2, -3 ], [ 2, -4 ], [ 3 ], [ 4, -1 ]>,
      <bipartition: [ 1, -2, -4 ], [ 2 ], [ 3, -3 ], [ 4, -1 ]> ]

##### 11.6-4 MinimalSemigroupGeneratingSet

`‣ MinimalSemigroupGeneratingSet`( S ) | ( attribute )

`‣ MinimalMonoidGeneratingSet`( S ) | ( attribute )

`‣ MinimalInverseSemigroupGeneratingSet`( S ) | ( attribute )

`‣ MinimalInverseMonoidGeneratingSet`( S ) | ( attribute )

Returns: A minimal generating set for a semigroup.

The attribute `MinimalXGeneratingSet` returns a minimal generating set for the semigroup S, with respect to length. The returned value of `MinimalXGeneratingSet`, where applicable, is a minimal-length list of elements of S with the property that

          X(MinimalXGeneratingSet(S)) = S;

where `X` is one of `Semigroup` (Reference: Semigroup), `Monoid` (Reference: Monoid), `InverseSemigroup` (Reference: InverseSemigroup), or `InverseMonoid` (Reference: InverseMonoid).

For many types of semigroup, it is not currently possible to find a `MinimalXGeneratingSet` with the Semigroups package.

See also `SmallGeneratingSet` (11.6-2) and `IrredundantGeneratingSubset` (11.6-3).

    gap> S := MonogenicSemigroup(3, 6);;
    gap> MinimalSemigroupGeneratingSet(S);
    [ Transformation( [ 2, 3, 4, 5, 6, 1, 6, 7, 8 ] ) ]
    gap> S := FullTransformationMonoid(4);;
    gap> MinimalSemigroupGeneratingSet(S);
    [ Transformation( [ 1, 4, 2, 3 ] ), Transformation( [ 4, 3, 1, 2 ] ),
      Transformation( [ 1, 2, 3, 1 ] ) ]
    gap> S := Monoid([
    >  PartialPerm([2, 3, 4, 5, 1, 0, 6, 7]),
    >  PartialPerm([3, 4, 5, 1, 2, 0, 0, 6])]);
    <partial perm monoid of rank 8 with 2 generators>
    gap> IsMonogenicMonoid(S);
    true
    gap> MinimalMonoidGeneratingSet(S);
    [ [8,7,6](1,2,3,4,5) ]

##### 11.6-5 GeneratorsSmallest

`‣ GeneratorsSmallest`( S ) | ( attribute )

Returns: A set of elements.

For a semigroup S, `GeneratorsSmallest` returns the lexicographically least set of elements `X` such that `X` generates S as a semigroup, and such that `X` is lexicographically ordered and has the property that each `X[i]` is not generated by `X[1], X[2], ..., X[i-1]`.

It can be difficult to find the set of generators `X`, and it might contain a substantial proportion of the elements of S.

Two semigroups have the same set of elements if and only if their smallest generating sets are equal. However, due to the complexity of determining the `GeneratorsSmallest`, this is not the method used by the Semigroups package when comparing semigroups.

    gap> S := Monoid([
    >  Transformation([1, 3, 4, 1]),
    >  Transformation([2, 4, 1, 2]),
    >  Transformation([3, 1, 1, 3]),
    >  Transformation([3, 3, 4, 1])]);
    <transformation monoid of degree 4 with 4 generators>
    gap> GeneratorsSmallest(S);
    [ Transformation( [ 1, 1, 1, 1 ] ), Transformation( [ 1, 1, 1, 2 ] ),
      Transformation( [ 1, 1, 1, 3 ] ), Transformation( [ 1, 1, 1 ] ),
      Transformation( [ 1, 1, 2, 1 ] ), Transformation( [ 1, 1, 2, 2 ] ),
      Transformation( [ 1, 1, 3, 1 ] ), Transformation( [ 1, 1, 3, 3 ] ),
      Transformation( [ 1, 1 ] ), Transformation( [ 1, 1, 4, 1 ] ),
      Transformation( [ 1, 2, 1, 1 ] ), Transformation( [ 1, 2, 2, 1 ] ),
      IdentityTransformation, Transformation( [ 1, 3, 1, 1 ] ),
      Transformation( [ 1, 3, 4, 1 ] ), Transformation( [ 2, 1, 1, 2 ] ),
      Transformation( [ 2, 2, 2 ] ), Transformation( [ 2, 4, 1, 2 ] ),
      Transformation( [ 3, 3, 3 ] ), Transformation( [ 3, 3, 4, 1 ] ) ]
    gap> T := Semigroup(Bipartition([[1, 2, 3], [4, -1], [-2], [-3], [-4]]),
    >                   Bipartition([[1, -3, -4], [2, 3, 4, -2], [-1]]),
    >                   Bipartition([[1, 2, 3, 4, -2], [-1, -4], [-3]]),
    >                   Bipartition([[1, 2, 3, 4], [-1], [-2], [-3, -4]]),
    >                   Bipartition([[1, 2, -1, -2], [3, 4, -3], [-4]]));
    <bipartition semigroup of degree 4 with 5 generators>
    gap> GeneratorsSmallest(T);
    [ <bipartition: [ 1, 2, 3, 4, -1, -2, -3 ], [ -4 ]>,
      <bipartition: [ 1, 2, 3, 4, -1, -2 ], [ -3 ], [ -4 ]>,
      <bipartition: [ 1, 2, 3, 4, -1 ], [ -2 ], [ -3 ], [ -4 ]>,
      <bipartition: [ 1, 2, 3, 4, -2, -3, -4 ], [ -1 ]>,
      <bipartition: [ 1, 2, 3, 4, -2 ], [ -1, -4 ], [ -3 ]>,
      <bipartition: [ 1, 2, 3, 4, -2 ], [ -1 ], [ -3, -4 ]>,
      <bipartition: [ 1, 2, 3, 4, -3 ], [ -1, -2 ], [ -4 ]>,
      <bipartition: [ 1, 2, 3, 4 ], [ -1, -2, -3 ], [ -4 ]>,
      <bipartition: [ 1, 2, 3, 4, -3, -4 ], [ -1 ], [ -2 ]>,
      <bipartition: [ 1, 2, 3 ], [ 4, -1, -2, -3 ], [ -4 ]>,
      <bipartition: [ 1, 2, -1, -2 ], [ 3, 4, -3 ], [ -4 ]>,
      <bipartition: [ 1, -3 ], [ 2, 3, 4, -1, -2 ], [ -4 ]>,
      <bipartition: [ 1, -3, -4 ], [ 2, 3, 4, -2 ], [ -1 ]> ]

##### 11.6-6 IndecomposableElements

`‣ IndecomposableElements`( S ) | ( attribute )

Returns: A list of elements.

If S is a semigroup, then this attribute returns the set of elements of S that are not decomposable. A element of S is decomposable if it can be written as the product of two elements in S. An element of S is indecomposable if it is not decomposable.

See also `IsSurjectiveSemigroup` (12.1-6).

Note that any generating set for S contains each indecomposable element of S. Thus `IndecomposableElements(S)` is a subset of `GeneratorsOfSemigroup(S)`.

    gap> S := Semigroup([
    > Transformation([1, 1, 2, 3]),
    > Transformation([1, 1, 1, 2])]);
    <transformation semigroup of degree 4 with 2 generators>
    gap> x := IndecomposableElements(S);
    [ Transformation( [ 1, 1, 2, 3 ] ) ]
    gap> IsSubset(GeneratorsOfSemigroup(S), x);
    true
    gap> T := FullTransformationMonoid(10);
    <full transformation monoid of degree 10>
    gap> IndecomposableElements(T);
    [  ]
    gap> B := ZeroSemigroup(IsBipartitionSemigroup, 3);
    <commutative non-regular bipartition semigroup of size 3, degree 4
     with 2 generators>
    gap> IndecomposableElements(B);
    [ <bipartition: [ 1, 2, 3, -1 ], [ 4, -2 ], [ -3 ], [ -4 ]>,
      <bipartition: [ 1, 2, 4, -1 ], [ 3, -2 ], [ -3 ], [ -4 ]> ]

#### 11.7 Minimal ideals and multiplicative zeros

In this section we describe the attributes of a semigroup that can be found using the Semigroups package.

##### 11.7-1 MinimalIdeal

`‣ MinimalIdeal`( S ) | ( attribute )

Returns: The minimal ideal of a semigroup.

The minimal ideal of a semigroup is the least ideal with respect to containment.

It is significantly easier to find the minimal \(\mathscr{D}\)-class of a semigroup, than to find its \(\mathscr{D}\)-classes.

See also `RepresentativeOfMinimalIdeal` (11.7-2), `PartialOrderOfDClasses` (10.1-10), `IsGreensLessThanOrEqual` (Reference: IsGreensLessThanOrEqual), and `MinimalDClass` (10.1-6).

    gap> S := Semigroup(
    > Transformation([3, 4, 1, 3, 6, 3, 4, 6, 10, 1]),
    > Transformation([8, 2, 3, 8, 4, 1, 3, 4, 9, 7]));;
    gap> MinimalIdeal(S);
    <simple transformation semigroup ideal of degree 10 with 1 generator>
    gap> Elements(MinimalIdeal(S));
    [ Transformation( [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
      Transformation( [ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 ] ),
      Transformation( [ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 ] ),
      Transformation( [ 6, 6, 6, 6, 6, 6, 6, 6, 6, 6 ] ),
      Transformation( [ 8, 8, 8, 8, 8, 8, 8, 8, 8, 8 ] ) ]
    gap> x := Transformation([8, 8, 8, 8, 8, 8, 8, 8, 8, 8]);;
    gap> D := DClass(S, x);;
    gap> ForAll(GreensDClasses(S), x -> IsGreensLessThanOrEqual(D, x));
    true
    gap> MinimalIdeal(POI(10));
    <partial perm group of rank 0>
    gap> MinimalIdeal(BrauerMonoid(6));
    <simple bipartition *-semigroup ideal of degree 6 with 1 generator>

##### 11.7-2 RepresentativeOfMinimalIdeal

`‣ RepresentativeOfMinimalIdeal`( S ) | ( attribute )

`‣ RepresentativeOfMinimalDClass`( S ) | ( attribute )

Returns: An element of the minimal ideal of a semigroup.

The minimal ideal of a semigroup is the least ideal with respect to containment.

This method returns a representative element of the minimal ideal of S without having to create the minimal ideal itself. In general, beyond being a member of the minimal ideal, the returned element is not guaranteed to have any special properties. However, the element will coincide with the zero element of S if one exists.

This method works particularly well if S is a semigroup of transformations or partial permutations.

See also `MinimalIdeal` (11.7-1) and `MinimalDClass` (10.1-6).

    gap> S := SymmetricInverseSemigroup(10);;
    gap> RepresentativeOfMinimalIdeal(S);
    <empty partial perm>
    gap> B := Semigroup([
    > Bipartition([[1, 2], [3, 6, -2], [4, 5, -3, -4], [-1, -6], [-5]]),
    > Bipartition([[1, -1], [2], [3], [4, -3], [5, 6, -5, -6],
    >   [-2, -4]])]);;
    gap> RepresentativeOfMinimalIdeal(B);
    <bipartition: [ 1, 2 ], [ 3, 6 ], [ 4, 5 ], [ -1, -5, -6 ],
     [ -2, -4 ], [ -3 ]>
    gap> S := Semigroup(Transformation([5, 1, 6, 2, 2, 4]),
    >                   Transformation([3, 5, 5, 1, 6, 2]));;
    gap> RepresentativeOfMinimalDClass(S);
    Transformation( [ 5, 6, 6, 3, 3, 5 ] )
    gap> MinimalDClass(S);
    <Green's D-class: Transformation( [ 5, 6, 6, 3, 3, 5 ] )>

##### 11.7-3 MultiplicativeZero

`‣ MultiplicativeZero`( S ) | ( attribute )

Returns: The zero element of a semigroup.

`MultiplicativeZero` returns the zero element of the semigroup S if it exists and `fail` if it does not. See also `MultiplicativeZero` (Reference: MultiplicativeZero).

    gap> S := Semigroup(Transformation([1, 4, 2, 6, 6, 5, 2]),
    >                   Transformation([1, 6, 3, 6, 2, 1, 6]));;
    gap> MultiplicativeZero(S);
    Transformation( [ 1, 1, 1, 1, 1, 1, 1 ] )
    gap> S := Semigroup(Transformation([2, 8, 3, 7, 1, 5, 2, 6]),
    >                   Transformation([3, 5, 7, 2, 5, 6, 3, 8]),
    >                   Transformation([6, 7, 4, 1, 4, 1, 6, 2]),
    >                   Transformation([8, 8, 5, 1, 7, 5, 2, 8]));;
    gap> MultiplicativeZero(S);
    fail
    gap> S := InverseSemigroup(
    > PartialPerm([1, 3, 4], [5, 3, 1]),
    > PartialPerm([1, 2, 3, 4], [4, 3, 1, 2]),
    > PartialPerm([1, 3, 4, 5], [2, 4, 5, 3]));;
    gap> MultiplicativeZero(S);
    <empty partial perm>
    gap> S := PartitionMonoid(6);
    <regular bipartition *-monoid of size 4213597, degree 6 with 4
     generators>
    gap> MultiplicativeZero(S);
    fail
    gap> S := DualSymmetricInverseMonoid(6);
    <inverse block bijection monoid of degree 6 with 3 generators>
    gap> MultiplicativeZero(S);
    <block bijection: [ 1, 2, 3, 4, 5, 6, -1, -2, -3, -4, -5, -6 ]>

##### 11.7-4 UnderlyingSemigroupOfSemigroupWithAdjoinedZero

`‣ UnderlyingSemigroupOfSemigroupWithAdjoinedZero`( S ) | ( attribute )

Returns: A semigroup, or `fail`.

If S is a semigroup for which the property `IsSemigroupWithAdjoinedZero` (12.1-20) is true, (i.e. S has a `MultiplicativeZero` (11.7-3) and the set \(\textit{S} \setminus \{ 0 \}\) is a subsemigroup of S), then this method returns the semigroup \(\textit{S} \setminus \{ 0 \}\).

Otherwise, if S is a semigroup for which the property `IsSemigroupWithAdjoinedZero` (12.1-20) is `false`, then this method returns `fail`.

    gap> S := Semigroup([
    > Transformation([2, 3, 4, 5, 1, 6]),
    > Transformation([2, 1, 3, 4, 5, 6]),
    > Transformation([6, 6, 6, 6, 6, 6])]);
    <transformation semigroup of degree 6 with 3 generators>
    gap> MultiplicativeZero(S);
    Transformation( [ 6, 6, 6, 6, 6, 6 ] )
    gap> G := UnderlyingSemigroupOfSemigroupWithAdjoinedZero(S);
    <transformation semigroup of degree 5 with 2 generators>
    gap> IsGroupAsSemigroup(G);
    true
    gap> IsZeroGroup(S);
    true
    gap> S := SymmetricInverseMonoid(6);;
    gap> MultiplicativeZero(S);
    <empty partial perm>
    gap> G := UnderlyingSemigroupOfSemigroupWithAdjoinedZero(S);
    fail

#### 11.8 Group of units and identity elements

##### 11.8-1 GroupOfUnits

`‣ GroupOfUnits`( S ) | ( attribute )

Returns: The group of units of a semigroup or `fail`.

`GroupOfUnits` returns the group of units of the semigroup S as a subsemigroup of S if it exists and returns `fail` if it does not. Use `IsomorphismPermGroup` (6.5-5) if you require a permutation representation of the group of units.

If a semigroup S has an identity `e`, then the group of units of S is the set of those `s` in S such that there exists `t` in S where `s*t=t*s=e`. Equivalently, the group of units is the \(\mathscr{H}\)-class of the identity of S.

See also `GreensHClassOfElement` (Reference: GreensHClassOfElement), `IsMonoidAsSemigroup` (12.1-13), and `MultiplicativeNeutralElement` (Reference: MultiplicativeNeutralElement).

    gap> S := Semigroup(
    > Transformation([1, 2, 5, 4, 3, 8, 7, 6]),
    > Transformation([1, 6, 3, 4, 7, 2, 5, 8]),
    > Transformation([2, 1, 6, 7, 8, 3, 4, 5]),
    > Transformation([3, 2, 3, 6, 1, 6, 1, 2]),
    > Transformation([5, 2, 3, 6, 3, 4, 7, 4]));;
    gap> Size(S);
    5304
    gap> StructureDescription(GroupOfUnits(S));
    "C2 x S4"
    gap> S := InverseSemigroup(
    > PartialPerm([1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
    >             [2, 4, 5, 3, 6, 7, 10, 9, 8, 1]),
    > PartialPerm([1, 2, 3, 4, 5, 6, 7, 8, 10],
    >             [8, 2, 3, 1, 4, 5, 10, 6, 9]));;
    gap> StructureDescription(GroupOfUnits(S));
    "C8"
    gap> S := InverseSemigroup(
    > PartialPerm([1, 3, 4], [4, 3, 5]),
    > PartialPerm([1, 2, 3, 5], [3, 1, 5, 2]));;
    gap> GroupOfUnits(S);
    fail
    gap> S := Semigroup(
    > Bipartition([[1, 2, 3, -1, -3], [-2]]),
    > Bipartition([[1, -1], [2, 3, -2, -3]]),
    > Bipartition([[1, -2], [2, -3], [3, -1]]),
    > Bipartition([[1], [2, 3, -2], [-1, -3]]));;
    gap> StructureDescription(GroupOfUnits(S));
    "C3"

#### 11.9 Idempotents

##### 11.9-1 Idempotents

`‣ Idempotents`( obj[, n] ) | ( attribute )

Returns: A list of idempotents.

The argument obj should be a semigroup, \(\mathscr{D}\)-class, \(\mathscr{H}\)-class, \(\mathscr{L}\)-class, or \(\mathscr{R}\)-class.

If the optional second argument n is present and obj is a semigroup, then a list of the idempotents in obj of rank n is returned. If you are only interested in the idempotents of a given rank, then the second version of the function will probably be faster. However, if the optional second argument is present, then nothing is stored in obj and so every time the function is called the computation must be repeated.

This functions produce essentially the same output as the GAP library function with the same name; see `Idempotents` (Reference: Idempotents). The main difference is that this function can be applied to a wider class of objects as described above.

See also `IsRegularDClass` (Reference: IsRegularDClass), `IsRegularGreensClass` (10.3-2) `IsGroupHClass` (Reference: IsGroupHClass), `NrIdempotents` (11.9-2), and `GroupHClass` (10.4-1).

    gap> S := Semigroup(Transformation([2, 3, 4, 1]),
    >                   Transformation([3, 3, 1, 1]));;
    gap> Idempotents(S, 1);
    [  ]
    gap> AsSet(Idempotents(S, 2));
    [ Transformation( [ 1, 1, 3, 3 ] ), Transformation( [ 1, 3, 3, 1 ] ),
      Transformation( [ 2, 2, 4, 4 ] ), Transformation( [ 4, 2, 2, 4 ] ) ]
    gap> AsSet(Idempotents(S));
    [ Transformation( [ 1, 1, 3, 3 ] ), IdentityTransformation,
      Transformation( [ 1, 3, 3, 1 ] ), Transformation( [ 2, 2, 4, 4 ] ),
      Transformation( [ 4, 2, 2, 4 ] ) ]
    gap> x := Transformation([2, 2, 4, 4]);;
    gap> R := GreensRClassOfElement(S, x);;
    gap> Idempotents(R);
    [ Transformation( [ 1, 1, 3, 3 ] ), Transformation( [ 2, 2, 4, 4 ] ) ]
    gap> x := Transformation([4, 2, 2, 4]);;
    gap> L := GreensLClassOfElement(S, x);;
    gap> AsSet(Idempotents(L));
    [ Transformation( [ 2, 2, 4, 4 ] ), Transformation( [ 4, 2, 2, 4 ] ) ]
    gap> D := DClassOfLClass(L);;
    gap> AsSet(Idempotents(D));
    [ Transformation( [ 1, 1, 3, 3 ] ), Transformation( [ 1, 3, 3, 1 ] ),
      Transformation( [ 2, 2, 4, 4 ] ), Transformation( [ 4, 2, 2, 4 ] ) ]
    gap> L := GreensLClassOfElement(S, Transformation([3, 1, 1, 3]));;
    gap> AsSet(Idempotents(L));
    [ Transformation( [ 1, 1, 3, 3 ] ), Transformation( [ 1, 3, 3, 1 ] ) ]
    gap> H := GroupHClass(D);
    <Green's H-class: Transformation( [ 1, 1, 3, 3 ] )>
    gap> Idempotents(H);
    [ Transformation( [ 1, 1, 3, 3 ] ) ]
    gap> S := InverseSemigroup(
    > PartialPerm([10, 6, 3, 4, 9, 0, 1]),
    > PartialPerm([6, 10, 7, 4, 8, 2, 9, 1]));;
    gap> Idempotents(S, 1);
    [ <identity partial perm on [ 4 ]> ]
    gap> Idempotents(S, 0);
    [  ]

##### 11.9-2 NrIdempotents

`‣ NrIdempotents`( obj ) | ( attribute )

Returns: A positive integer.

This function returns the number of idempotents in obj where obj can be a semigroup, \(\mathscr{D}\)-, \(\mathscr{L}\)-, \(\mathscr{H}\)-, or \(\mathscr{R}\)-class. If the actual idempotents are not required, then it is more efficient to use `NrIdempotents(obj)` than `Length(Idempotents(obj))` since the idempotents themselves are not created when `NrIdempotents` is called.

See also `Idempotents` (Reference: Idempotents) and `Idempotents` (11.9-1), `IsRegularDClass` (Reference: IsRegularDClass), `IsRegularGreensClass` (10.3-2) `IsGroupHClass` (Reference: IsGroupHClass), and `GroupHClass` (10.4-1).

    gap> S := Semigroup(Transformation([2, 3, 4, 1]),
    >                   Transformation([3, 3, 1, 1]));;
    gap> NrIdempotents(S);
    5
    gap> f := Transformation([2, 2, 4, 4]);;
    gap> R := GreensRClassOfElement(S, f);;
    gap> NrIdempotents(R);
    2
    gap> f := Transformation([4, 2, 2, 4]);;
    gap> L := GreensLClassOfElement(S, f);;
    gap> NrIdempotents(L);
    2
    gap> D := DClassOfLClass(L);;
    gap> NrIdempotents(D);
    4
    gap> L := GreensLClassOfElement(S, Transformation([3, 1, 1, 3]));;
    gap> NrIdempotents(L);
    2
    gap> H := GroupHClass(D);;
    gap> NrIdempotents(H);
    1
    gap> S := InverseSemigroup(
    > PartialPerm([1, 2, 3, 5, 7, 9, 10],
    >              [6, 7, 2, 9, 1, 5, 3]),
    > PartialPerm([1, 2, 3, 5, 6, 7, 9, 10],
    >             [8, 1, 9, 4, 10, 5, 6, 7]));;
    gap> NrIdempotents(S);
    236
    gap> f := PartialPerm([2, 3, 7, 9, 10],
    >                     [7, 2, 1, 5, 3]);;
    gap> D := DClassNC(S, f);;
    gap> NrIdempotents(D);
    13

##### 11.9-3 IdempotentGeneratedSubsemigroup

`‣ IdempotentGeneratedSubsemigroup`( S ) | ( attribute )

Returns: A semigroup.

`IdempotentGeneratedSubsemigroup` returns the subsemigroup of the semigroup S generated by the idempotents of S.

See also `Idempotents` (11.9-1) and `SmallGeneratingSet` (11.6-2).

    gap> S := Semigroup(Transformation([1, 1]),
    >                   Transformation([2, 1]),
    >                   Transformation([1, 2, 2]),
    >                   Transformation([1, 2, 3, 4, 5, 1]),
    >                   Transformation([1, 2, 3, 4, 5, 5]),
    >                   Transformation([1, 2, 3, 4, 6, 5]),
    >                   Transformation([1, 2, 3, 5, 4]),
    >                   Transformation([1, 2, 3, 7, 4, 5, 7]),
    >                   Transformation([1, 2, 4, 8, 8, 3, 8, 7]),
    >                   Transformation([1, 2, 8, 4, 5, 6, 7, 8]),
    >                   Transformation([7, 7, 7, 4, 5, 6, 1]));;
    gap> IdempotentGeneratedSubsemigroup(S) =
    > Monoid(Transformation([1, 1]),
    >        Transformation([1, 2, 1]),
    >        Transformation([1, 2, 2]),
    >        Transformation([1, 2, 3, 1]),
    >        Transformation([1, 2, 3, 2]),
    >        Transformation([1, 2, 3, 4, 1]),
    >        Transformation([1, 2, 3, 4, 2]),
    >        Transformation([1, 2, 3, 4, 4]),
    >        Transformation([1, 2, 3, 4, 5, 1]),
    >        Transformation([1, 2, 3, 4, 5, 2]),
    >        Transformation([1, 2, 3, 4, 5, 5]),
    >        Transformation([1, 2, 3, 4, 5, 7, 7]),
    >        Transformation([1, 2, 3, 4, 7, 6, 7]),
    >        Transformation([1, 2, 3, 6, 5, 6]),
    >        Transformation([1, 2, 3, 7, 5, 6, 7]),
    >        Transformation([1, 2, 8, 4, 5, 6, 7, 8]),
    >        Transformation([2, 2]));
    true
    gap> S := SymmetricInverseSemigroup(5);
    <symmetric inverse monoid of degree 5>
    gap> IdempotentGeneratedSubsemigroup(S);
    <inverse partial perm monoid of rank 5 with 5 generators>
    gap> S := DualSymmetricInverseSemigroup(5);
    <inverse block bijection monoid of degree 5 with 3 generators>
    gap> IdempotentGeneratedSubsemigroup(S);
    <inverse block bijection monoid of degree 5 with 10 generators>
    gap> IsSemilattice(last);
    true

#### 11.10 Maximal subsemigroups

The Semigroups package provides methods to calculate the maximal subsemigroups of a finite semigroup, subject to various conditions. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup of the semigroup.

When computing the maximal subsemigroups of a regular Rees (0-)matrix semigroup over a group, additional functionality is available. As described in [GGR68], a maximal subsemigroup of a finite regular Rees (0-)matrix semigroup over a group is one of 6 possible types. Using the Semigroups package, it is possible to search for only those maximal subsemigroups of certain types.

A maximal subsemigroup of such a Rees (0-)matrix semigroup `R` over a group `G` is either:

1. `{0};`

2. formed by removing `0`;

3. formed by removing a column (a non-zero \(\mathscr{L}\)-class);

4. formed by removing a row (a non-zero \(\mathscr{R}\)-class);

5. formed by removing a set of both rows and columns;

6. isomorphic to a Rees (0-)matrix semigroup of the same dimensions over a maximal subgroup of `G` (in particular, the maximal subsemigroup intersects every \(\mathscr{H}\)-class of `R`).

Note that if `R` is a Rees matrix semigroup then it has no maximal subsemigroups of types 1, 2, or 5. Only types 3, 4, and 6 are relevant to a Rees matrix semigroup.

##### 11.10-1 MaximalSubsemigroups

`‣ MaximalSubsemigroups`( S ) | ( attribute )

`‣ MaximalSubsemigroups`( S, opts ) | ( operation )

Returns: The maximal subsemigroups of S.

If S is a finite semigroup, then the attribute `MaximalSubsemigroups` returns a list of the non-empty maximal subsemigroups of S. The methods used by `MaximalSubsemigroups` are based on [GGR68], and are described in [DMW18].

It is computationally expensive to search for the maximal subsemigroups of a semigroup, and so computations involving `MaximalSubsemigroups` may be very lengthy. A substantial amount of information on the progress of `MaximalSubsemigroups` is provided through the info class `InfoSemigroups` (2.5-1), with increasingly detailed information given at levels 1, 2, and 3.

The behaviour of `MaximalSubsemigroups` can be altered via the second argument opts, which should be a record. The optional components of opts are:

gens (a boolean)

  If `opts.gens` is `false` or unspecified, then the maximal subsemigroups themselves are returned and not just generating sets for these subsemigroups.

  It can be more computationally expensive to return the generating sets for the maximal subsemigroups, than to return the maximal subsemigroups themselves.

contain (a list)

  If `opts.contain` is duplicate-free list of elements of S, then `MaximalSubsemigroups` will search for the maximal subsemigroups of S which contain those elements.

D (a \(\mathscr{D}\)-class)

  For a maximal subsemigroup `M` of a finite semigroup S, there exists a unique \(\mathscr{D}\)-class which contains the complement of `M` in S. In other words, the elements of S which `M` lacks are contained in a unique \(\mathscr{D}\)-class.

  If `opts.D` is a \(\mathscr{D}\)-class of S, then `MaximalSubsemigroups` will search exclusively for those maximal subsemigroups of S whose complement is contained in `opts.D`.

types (a list)

  This option is relevant only if S is a regular Rees (0-)matrix semigroup over a group.

  As described at the start of this subsection, 11.10, a maximal subsemigroup of a regular Rees (0-)matrix semigroup over a group is one of 6 possible types.

  If S is a regular Rees (0-)matrix semigroup over a group and `opts.types` is a subset of `[1 .. 6]`, then `MaximalSubsemigroups` will search for those maximal subsemigroups of S of the types enumerated by `opts.types`.

  The default value for this option is `[1 .. 6]` (i.e. no restriction).

    gap> S := FullTransformationSemigroup(3);
    <full transformation monoid of degree 3>
    gap> MaximalSubsemigroups(S);
    [ <transformation semigroup of degree 3 with 7 generators>,
      <transformation semigroup of degree 3 with 7 generators>,
      <transformation semigroup of degree 3 with 7 generators>,
      <transformation semigroup of degree 3 with 7 generators>,
      <transformation monoid of degree 3 with 5 generators> ]
    gap> MaximalSubsemigroups(S,
    > rec(gens := true, D := DClass(S, Transformation([2, 2, 3]))));
    [ [ Transformation( [ 1, 1, 1 ] ), Transformation( [ 3, 3, 3 ] ),
          Transformation( [ 2, 2, 2 ] ), IdentityTransformation,
          Transformation( [ 2, 3, 1 ] ), Transformation( [ 2, 1 ] ) ] ]
    gap> MaximalSubsemigroups(S,
    > rec(contain := [Transformation([2, 3, 1])]));
    [ <transformation semigroup of degree 3 with 7 generators>,
      <transformation monoid of degree 3 with 5 generators> ]
    gap> R := PrincipalFactor(
    > DClass(FullTransformationMonoid(4), Transformation([2, 2])));
    <Rees 0-matrix semigroup 6x4 over Group([ (2,3,4), (2,4) ])>
    gap> MaximalSubsemigroups(R, rec(types := [5],
    >   contain := [RMSElement(R, 1, (), 1),
    >               RMSElement(R, 1, (2, 3), 2)]));
    [ <subsemigroup of 6x4 Rees 0-matrix semigroup with 10 generators>,
      <subsemigroup of 6x4 Rees 0-matrix semigroup with 10 generators>,
      <subsemigroup of 6x4 Rees 0-matrix semigroup with 10 generators>,
      <subsemigroup of 6x4 Rees 0-matrix semigroup with 10 generators> ]

##### 11.10-2 NrMaximalSubsemigroups

`‣ NrMaximalSubsemigroups`( S ) | ( attribute )

Returns: The number of maximal subsemigroups of S.

If S is a finite semigroup, then `NrMaximalSubsemigroups` returns the number of non-empty maximal subsemigroups of S. The methods used by `MaximalSubsemigroups` are based on [GGR68], and are described in [DMW18].

It can be significantly faster to find the number of maximal subsemigroups of a semigroup than to find the maximal subsemigroups themselves.

Unless the maximal subsemigroups of S are already known, the command `NrMaximalSubsemigroups(S)` simply calls the command `MaximalSubsemigroups(S, rec(number := true))`.

For more information about searching for maximal subsemigroups of a finite semigroup in the Semigroups package, and for information about the options available to alter the search, see `MaximalSubsemigroups` (11.10-1). By supplying the additional option `opts.number :=` `true`, the number of maximal subsemigroups will be returned rather than the subsemigroups themselves.

    gap> S := FullTransformationSemigroup(3);
    <full transformation monoid of degree 3>
    gap> NrMaximalSubsemigroups(S);
    5
    gap> S := RectangularBand(3, 4);;
    gap> NrMaximalSubsemigroups(S);
    7
    gap> R := PrincipalFactor(
    > DClass(FullTransformationMonoid(4), Transformation([2, 2])));
    <Rees 0-matrix semigroup 6x4 over Group([ (2,3,4), (2,4) ])>
    gap> MaximalSubsemigroups(R, rec(number := true, types := [3, 4]));
    10

##### 11.10-3 IsMaximalSubsemigroup

`‣ IsMaximalSubsemigroup`( S, T ) | ( operation )

Returns: `true` or `false`.

If S and T are semigroups, then `IsMaximalSubsemigroup` returns `true` if and only if T is a maximal subsemigroup of S.

A maximal subsemigroup of S is a proper subsemigroup of S which is contained in no other proper subsemigroup of S.

    gap> S := ZeroSemigroup(2);;
    gap> IsMaximalSubsemigroup(S, Semigroup(MultiplicativeZero(S)));
    true
    gap> S := FullTransformationSemigroup(4);
    <full transformation monoid of degree 4>
    gap> T := Semigroup(Transformation([3, 4, 1, 2]),
    >                   Transformation([1, 4, 2, 3]),
    >                   Transformation([2, 1, 1, 3]));
    <transformation semigroup of degree 4 with 3 generators>
    gap> IsMaximalSubsemigroup(S, T);
    true
    gap> R := Semigroup(Transformation([3, 4, 1, 2]),
    >                   Transformation([1, 4, 2, 2]),
    >                   Transformation([2, 1, 1, 3]));
    <transformation semigroup of degree 4 with 3 generators>
    gap> IsMaximalSubsemigroup(S, R);
    false

#### 11.11 Attributes of transformations and transformation semigroups

##### 11.11-1 ComponentRepsOfTransformationSemigroup

`‣ ComponentRepsOfTransformationSemigroup`( S ) | ( attribute )

Returns: The representatives of components of a transformation semigroup.

This function returns the representatives of the components of the action of the transformation semigroup S on the set of positive integers not greater than the degree of S.

The representatives are the least set of points such that every point can be reached from some representative under the action of S.

    gap> S := Semigroup(
    > Transformation([11, 11, 9, 6, 4, 1, 4, 1, 6, 7, 12, 5]),
    > Transformation([12, 10, 7, 10, 4, 1, 12, 9, 11, 9, 1, 12]));;
    gap> ComponentRepsOfTransformationSemigroup(S);
    [ 2, 3, 8 ]

##### 11.11-2 ComponentsOfTransformationSemigroup

`‣ ComponentsOfTransformationSemigroup`( S ) | ( attribute )

Returns: The components of a transformation semigroup.

This function returns the components of the action of the transformation semigroup S on the set of positive integers not greater than the degree of S; the components of S partition this set.

    gap> S := Semigroup(
    > Transformation([11, 11, 9, 6, 4, 1, 4, 1, 6, 7, 12, 5]),
    > Transformation([12, 10, 7, 10, 4, 1, 12, 9, 11, 9, 1, 12]));;
    gap> ComponentsOfTransformationSemigroup(S);
    [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] ]

##### 11.11-3 CyclesOfTransformationSemigroup

`‣ CyclesOfTransformationSemigroup`( S ) | ( attribute )

Returns: The cycles of a transformation semigroup.

This function returns the cycles, or strongly connected components, of the action of the transformation semigroup S on the set of positive integers not greater than the degree of S.

    gap> S := Semigroup(
    > Transformation([11, 11, 9, 6, 4, 1, 4, 1, 6, 7, 12, 5]),
    > Transformation([12, 10, 7, 10, 4, 1, 12, 9, 11, 9, 1, 12]));;
    gap> CyclesOfTransformationSemigroup(S);
    [ [ 1, 11, 12, 5, 4, 6, 10, 7, 9 ], [ 2 ], [ 3 ], [ 8 ] ]

##### 11.11-4 DigraphOfAction

`‣ DigraphOfAction`( S, list, act ) | ( operation )

Returns: A digraph, or `fail`.

If S is a transformation semigroup and list is list such that S acts on the items in list via the function act, then `DigraphOfAction` returns a digraph representing the action of S on the items in list and any further items output by `act(list[i], S.j)`.

If `act(list[i], S.j)` is the `k`-th item in list, then in the output digraph there is an edge from the vertex `i` to the vertex `k` labelled by `j`.

The values in list and the additional values generated are stored in the vertex labels of the output digraph; see `DigraphVertexLabels` (Digraphs: DigraphVertexLabels), and the edge labels are stored in the `DigraphEdgeLabels` (Digraphs: DigraphEdgeLabels)

The digraph returned by `DigraphOfAction` has no multiple edges; see `IsMultiDigraph` (Digraphs: IsMultiDigraph).

    gap> S := Semigroup(Transformation([2, 4, 3, 4, 7, 1, 6]),
    >                   Transformation([3, 3, 2, 3, 5, 1, 5]));
    <transformation semigroup of degree 7 with 2 generators>
    gap> list := Concatenation(List([1 .. 7], x -> [x]),
    >                          Combinations([1 .. 7], 2));
    [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ], [ 1, 2 ],
      [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 2, 3 ],
      [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 4 ], [ 3, 5 ],
      [ 3, 6 ], [ 3, 7 ], [ 4, 5 ], [ 4, 6 ], [ 4, 7 ], [ 5, 6 ],
      [ 5, 7 ], [ 6, 7 ] ]
    gap> D := DigraphOfAction(S, list, OnSets);
    <immutable digraph with 28 vertices, 54 edges>
    gap> OnSets([2, 5], S.1);
    [ 4, 7 ]
    gap> Position(DigraphVertexLabels(D), [2, 5]);
    16
    gap> DigraphVertexLabel(D, 25);
    [ 4, 7 ]
    gap> DigraphEdgeLabel(D, 16, 25);
    1

##### 11.11-5 DigraphOfActionOnPoints

`‣ DigraphOfActionOnPoints`( S ) | ( attribute )

`‣ DigraphOfActionOnPoints`( S, n ) | ( attribute )

Returns: A digraph.

If S is a transformation semigroup and n is a non-negative integer, then `DigraphOfActionOnPoints(S, n)` returns a digraph representing the `OnPoints` (Reference: OnPoints) action of S on the set `[1 .. n]`.

If the optional argument n is not specified, then by default the degree of S will be chosen for n; see `DegreeOfTransformationSemigroup` (Reference: DegreeOfTransformationSemigroup).

The digraph returned by `DigraphOfActionOnPoints` has n vertices, where the vertex `i` corresponds to the point `i`. For each point `i` in `[1 .. n]`, and for each generator `f` in `GeneratorsOfSemigroup(S)`, there is an edge from the vertex `i` to the vertex `i ^ f`. See `GeneratorsOfSemigroup` (Reference: GeneratorsOfSemigroup) for further information.

    gap> S := Semigroup(Transformation([2, 4, 2, 4, 7, 1, 6]),
    >                   Transformation([3, 3, 2, 3, 5, 1, 5]));
    <transformation semigroup of degree 7 with 2 generators>
    gap> D1 := DigraphOfActionOnPoints(S);
    <immutable digraph with 7 vertices, 12 edges>
    gap> OnPoints(2, S.1);
    4
    gap> D2 := DigraphOfActionOnPoints(S, 4);
    <immutable digraph with 4 vertices, 7 edges>
    gap> D2 = InducedSubdigraph(D1, [1 .. 4]);
    true
    gap> DigraphOfActionOnPoints(S, 5);
    <immutable digraph with 5 vertices, 8 edges>

##### 11.11-6 FixedPointsOfTransformationSemigroup

`‣ FixedPointsOfTransformationSemigroup`( S ) | ( attribute )

Returns: A set of positive integers.

If S is a transformation semigroup, then `FixedPointsOfTransformationSemigroup(S)` returns the set of points `i` in `[1 .. DegreeOfTransformationSemigroup(S)]` such that `i ^ f = i` for all `f` in S.

    gap> f := Transformation([1, 4, 2, 4, 3, 7, 7]);
    Transformation( [ 1, 4, 2, 4, 3, 7, 7 ] )
    gap> S := Semigroup(f);
    <commutative transformation semigroup of degree 7 with 1 generator>
    gap> FixedPointsOfTransformationSemigroup(S);
    [ 1, 4, 7 ]

##### 11.11-7 IsTransitive

`‣ IsTransitive`( S[, X] ) | ( property )

`‣ IsTransitive`( S[, n] ) | ( property )

Returns: `true` or `false`.

A transformation semigroup S is transitive or strongly connected on the set X if for every `i, j` in X there is an element `s` in S such that `i ^ s = j`.

If the optional second argument is a positive integer n, then `IsTransitive` returns `true` if S is transitive on `[1 .. n]`, and `false` if it is not.

If the optional second argument is not provided, then the degree of S is used by default; see `DegreeOfTransformationSemigroup` (Reference: DegreeOfTransformationSemigroup).

    gap> S := Semigroup([
    >  Bipartition([
    >    [1, 2], [3, 6, -2], [4, 5, -3, -4], [-1, -6], [-5]]),
    >  Bipartition([
    >    [1, -4], [2, 3, 4, 5], [6], [-1, -6], [-2, -3], [-5]])]);
    <bipartition semigroup of degree 6 with 2 generators>
    gap> AsSemigroup(IsTransformationSemigroup, S);
    <transformation semigroup of size 11, degree 12 with 2 generators>
    gap> IsTransitive(last);
    false
    gap> IsTransitive(AsSemigroup(Group((1, 2, 3))));
    true

##### 11.11-8 SmallestElementSemigroup

`‣ SmallestElementSemigroup`( S ) | ( attribute )

`‣ LargestElementSemigroup`( S ) | ( attribute )

Returns: A transformation.

These attributes return the smallest and largest element of the transformation semigroup S, respectively. Smallest means the first element in the sorted set of elements of S and largest means the last element in the set of elements.

It is not necessary to find the elements of the semigroup to determine the smallest or largest element, and this function has considerable better performance than the equivalent `Elements(S)[1]` and `Elements(S)[Size(S)]`.

    gap> S := Monoid(
    > Transformation([1, 4, 11, 11, 7, 2, 6, 2, 5, 5, 10]),
    > Transformation([2, 4, 4, 2, 10, 5, 11, 11, 11, 6, 7]));
    <transformation monoid of degree 11 with 2 generators>
    gap> SmallestElementSemigroup(S);
    IdentityTransformation
    gap> LargestElementSemigroup(S);
    Transformation( [ 11, 11, 10, 10, 7, 6, 5, 6, 2, 2, 4 ] )

##### 11.11-9 CanonicalTransformation

`‣ CanonicalTransformation`( trans[, n] ) | ( function )

Returns: A transformation.

If trans is a transformation, and n is a non-negative integer such that the restriction of trans to `[1 .. n]` defines a transformation of `[1 .. n]`, then `CanonicalTransformation` returns a canonical representative of the transformation trans restricted to `[1 .. n]`.

More specifically, let `C(n)` be a class of transformations of degree n such that `AsDigraph` returns isomorphic digraphs for every pair of element elements in `C(n)`. Recall that for a transformation trans and integer n the function `AsDigraph` returns a digraph with n vertices and an edge with source `x` and range `x^trans` for every `x` in `[1 .. n]`. See `AsDigraph` (Digraphs: AsDigraph for a binary relation). Then `CanonicalTransformation` returns a canonical representative of the class `C(n)` that contains trans.

    gap> x := Transformation([5, 1, 4, 1, 1]);
    Transformation( [ 5, 1, 4, 1, 1 ] )
    gap> y := Transformation([3, 3, 2, 3, 1]);
    Transformation( [ 3, 3, 2, 3, 1 ] )
    gap> CanonicalTransformation(x);
    Transformation( [ 3, 5, 2, 2, 2 ] )
    gap> CanonicalTransformation(y);
    Transformation( [ 3, 5, 2, 2, 2 ] )

##### 11.11-10 IsConnectedTransformationSemigroup

`‣ IsConnectedTransformationSemigroup`( S ) | ( property )

Returns: `true` or `false`.

A transformation semigroup S is connected if the digraph returned by the function `DigraphOfActionOnPoints` is connected. See `IsConnectedDigraph` (Digraphs: IsConnectedDigraph) and `DigraphOfActionOnPoints` (11.11-5). The function `IsConnectedTransformationSemigroup` returns `true` if the semigroup S is connected and `false` otherwise.

    gap> S := Semigroup([
    >  Transformation([2, 4, 3, 4]),
    >  Transformation([3, 3, 2, 3, 3])]);
    <transformation semigroup of degree 5 with 2 generators>
    gap> IsConnectedTransformationSemigroup(S);
    true

#### 11.12 Attributes of partial perm semigroups

##### 11.12-1 ComponentRepsOfPartialPermSemigroup

`‣ ComponentRepsOfPartialPermSemigroup`( S ) | ( attribute )

Returns: The representatives of components of a partial perm semigroup.

This function returns the representatives of the components of the action of the partial perm semigroup S on the set of positive integers where it is defined.

The representatives are the least set of points such that every point can be reached from some representative under the action of S.

    gap> S := Semigroup([
    > PartialPerm([1, 2, 3, 5, 6, 7, 8, 11, 12, 16, 19],
    >             [9, 18, 20, 11, 5, 16, 8, 19, 14, 13, 1]),
    > PartialPerm([1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 18, 19, 20],
    >             [13, 1, 8, 5, 4, 14, 11, 12, 9, 20, 2, 18, 7, 3, 19])]);;
    gap> ComponentRepsOfPartialPermSemigroup(S);
    [ 6, 10, 15, 17 ]

##### 11.12-2 ComponentsOfPartialPermSemigroup

`‣ ComponentsOfPartialPermSemigroup`( S ) | ( attribute )

Returns: The components of a partial perm semigroup.

This function returns the components of the action of the partial perm semigroup S on the set of positive integers where it is defined; the components of S partition this set.

    gap> S := Semigroup([
    > PartialPerm([1, 2, 3, 5, 6, 7, 8, 11, 12, 16, 19],
    >             [9, 18, 20, 11, 5, 16, 8, 19, 14, 13, 1]),
    > PartialPerm([1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 18, 19, 20],
    >             [13, 1, 8, 5, 4, 14, 11, 12, 9, 20, 2, 18, 7, 3, 19])]);;
    gap> ComponentsOfPartialPermSemigroup(S);
    [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20 ],
      [ 15 ], [ 17 ] ]

##### 11.12-3 CyclesOfPartialPerm

`‣ CyclesOfPartialPerm`( x ) | ( attribute )

Returns: The cycles of a partial perm.

This function returns the cycles, or strongly connected components, of the action of the partial perm x on the set of positive integers where it is defined.

    gap> x := PartialPerm([3, 1, 4, 2, 5, 0, 0, 6, 0, 7]);
    [8,6][10,7](1,3,4,2)(5)
    gap> CyclesOfPartialPerm(x);
    [ [ 5 ], [ 1, 3, 4, 2 ] ]

##### 11.12-4 CyclesOfPartialPermSemigroup

`‣ CyclesOfPartialPermSemigroup`( S ) | ( attribute )

Returns: The cycles of a partial perm semigroup.

This function returns the cycles, or strongly connected components, of the action of the partial perm semigroup S on the set of positive integers where it is defined.

    gap> S := Semigroup([
    > PartialPerm([1, 2, 3, 5, 6, 7, 8, 11, 12, 16, 19],
    >             [9, 18, 20, 11, 5, 16, 8, 19, 14, 13, 1]),
    > PartialPerm([1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 18, 19, 20],
    >             [13, 1, 8, 5, 4, 14, 11, 12, 9, 20, 2, 18, 7, 3, 19])]);;
    gap> CyclesOfPartialPermSemigroup(S);
    [ [ 18, 7, 16 ], [ 1, 9, 12, 14, 2, 20, 19, 3, 8, 11 ], [ 4, 5 ] ]

#### 11.13 Attributes of Rees (0-)matrix semigroups

##### 11.13-1 RZMSDigraph

`‣ RZMSDigraph`( R ) | ( attribute )

Returns: A digraph.

If R is an \(n\) by \(m\) Rees 0-matrix semigroup \(M^{0}[I, T, \Lambda; P]\) (so that \(I = \{1,2,\ldots,n\}\) and \(\Lambda = \{1,2,\ldots,m\}\)) then `RZMSDigraph` returns a symmetric bipartite digraph with \(n+m\) vertices. An index \(i \in I\) corresponds to the vertex \(i\) and an index \(j \in \Lambda\) corresponds to the vertex \(j + n\).

Two vertices \(v\) and \(w\) in `RZMSDigraph(`R`)` are adjacent if and only if \(v\in I\), \(w - n\in \Lambda\), and `P[w - n][v]` \(\neq 0\).

This digraph is commonly called the Graham-Houghton graph of R.

    gap> R := PrincipalFactor(
    > DClass(FullTransformationMonoid(5),
    >        Transformation([2, 4, 1, 5, 5])));
    <Rees 0-matrix semigroup 10x5 over Group([ (1,2,3,4), (1,2) ])>
    gap> gr := RZMSDigraph(R);
    <immutable bipartite digraph with bicomponent sizes 10 and 5>
    gap> e := DigraphEdges(gr)[1];
    [ 1, 11 ]
    gap> Matrix(R)[e[2] - 10][e[1]] <> 0;
    true

##### 11.13-2 RZMSConnectedComponents

`‣ RZMSConnectedComponents`( R ) | ( attribute )

Returns: The connected components of a Rees 0-matrix semigroup.

If R is an \(n\) by \(m\) Rees 0-matrix semigroup \(M^{0}[I, T, \Lambda; P]\) (so that \(I = \{1,2,\ldots,n\}\) and \(\Lambda = \{1,2,\ldots,m\}\)) then `RZMSConnectedComponents` returns the connected components of R.

Connectedness is an equivalence relation on the indices of R: the equivalence classes of the relation are called the connected components of R, and two indices in \(I \cup \Lambda\) are connected if and only if their corresponding vertices in `RZMSDigraph(`R`)` are connected (see `RZMSDigraph` (11.13-1)). If R has \(n\) connected components, then `RZMSConnectedComponents` will return a list of pairs:

`[ [` \(I_{1}, \Lambda_{1}\)`],` \(\ldots\)`, [` \(I_{k}, \Lambda_{k}\)`] ]`

where \(I = I_1 \sqcup \cdots \sqcup I_k\), \(\Lambda = \Lambda_1 \sqcup \cdots \sqcup \Lambda_k\), and for each \(l\) the set \(I_{l}\cup\Lambda_{l}\) is a connected component of R. Note that at most one of \(I_l\) and \(\Lambda_l\) is possibly empty. The ordering of the connected components in the result in unspecified.

    gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(5),
    > [[(), 0, (1, 3), (4, 5), 0],
    >  [0, (), 0, 0, (1, 3, 4, 5)],
    >  [0, 0, (1, 5)(2, 3), 0, 0],
    >  [0, (2, 3)(1, 4), 0, 0, 0]]);
    <Rees 0-matrix semigroup 5x4 over Sym( [ 1 .. 5 ] )>
    gap> RZMSConnectedComponents(R);
    [ [ [ 1, 3, 4 ], [ 1, 3 ] ], [ [ 2, 5 ], [ 2, 4 ] ] ]

#### 11.14 Attributes of inverse semigroups

##### 11.14-1 NaturalLeqInverseSemigroup

`‣ NaturalLeqInverseSemigroup`( S ) | ( attribute )

Returns: An function.

`NaturalLeqInverseSemigroup` returns a function that, when given two elements `x, y` of the inverse semigroup S, returns `true` if `x` is less than or equal to `y` in the natural partial order on S.

    gap> S := Monoid(Transformation([1, 3, 4, 4]),
    >                Transformation([1, 4, 2, 4]));
    <transformation monoid of degree 4 with 2 generators>
    gap> IsInverseSemigroup(S);
    true
    gap> Size(S);
    6
    gap> NaturalPartialOrder(S);
    [ [ 2, 5, 6 ], [ 6 ], [ 6 ], [ 6 ], [ 6 ], [  ] ]

##### 11.14-2 JoinIrreducibleDClasses

`‣ JoinIrreducibleDClasses`( S ) | ( attribute )

Returns: A list of \(\mathscr{D}\)-classes.

`JoinIrreducibleDClasses` returns a list of the join irreducible \(\mathscr{D}\)-classes of the inverse semigroup of partial permutations, block bijections or partial permutation bipartitions S.

A join irreducible \(\mathscr{D}\)-class is a \(\mathscr{D}\)-class containing only join irreducible elements. See `IsJoinIrreducible` (12.2-7). If a \(\mathscr{D}\)-class contains one join irreducible element, then all of the elements in the \(\mathscr{D}\)-class are join irreducible.

    gap> S := SymmetricInverseSemigroup(3);
    <symmetric inverse monoid of degree 3>
    gap> JoinIrreducibleDClasses(S);
    [ <Green's D-class: <identity partial perm on [ 2 ]>> ]
    gap> T := InverseSemigroup([
    >   PartialPerm([1, 2, 4, 3]),
    >   PartialPerm([1]),
    >   PartialPerm([0, 2])]);
    <inverse partial perm semigroup of rank 4 with 3 generators>
    gap> JoinIrreducibleDClasses(T);
    [ <Green's D-class: <identity partial perm on [ 1, 2, 3, 4 ]>>,
      <Green's D-class: <identity partial perm on [ 1 ]>>,
      <Green's D-class: <identity partial perm on [ 2 ]>> ]
    gap> D := DualSymmetricInverseSemigroup(3);
    <inverse block bijection monoid of degree 3 with 3 generators>
    gap> JoinIrreducibleDClasses(D);
    [ <Green's D-class: <block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ]>> ]

##### 11.14-3 MajorantClosure

`‣ MajorantClosure`( S, T ) | ( operation )

Returns: A majorantly closed list of elements.

`MajorantClosure` returns a majorantly closed subset of an inverse semigroup of partial permutations, block bijections or partial permutation bipartitions, S, as a list. See `IsMajorantlyClosed` (12.2-8).

The result contains all elements of S which are greater than or equal to any element of T (with respect to the natural partial order `NaturalLeqPartialPerm` (Reference: NaturalLeqPartialPerm)). In particular, the result is a superset of T.

Note that T can be a subset of S or a subsemigroup of S.

    gap> S := SymmetricInverseSemigroup(4);
    <symmetric inverse monoid of degree 4>
    gap> T := [PartialPerm([1, 0, 3, 0])];
    [ <identity partial perm on [ 1, 3 ]> ]
    gap> U := MajorantClosure(S, T);
    [ <identity partial perm on [ 1, 3 ]>,
      <identity partial perm on [ 1, 2, 3 ]>, [2,4](1)(3), [4,2](1)(3),
      <identity partial perm on [ 1, 3, 4 ]>,
      <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2,4)(3) ]
    gap> B := InverseSemigroup([
    >  Bipartition([[1, -2], [2, -1], [3, -3], [4, 5, -4, -5]]),
    >  Bipartition([[1, -3], [2, -4], [3, -2], [4, -1], [5, -5]])]);;
    gap> T := [Bipartition([[1, -2], [2, 3, 5, -1, -3, -5], [4, -4]]),
    >  Bipartition([[1, -4], [2, 3, 5, -1, -3, -5], [4, -2]])];;
    gap> IsMajorantlyClosed(B, T);
    false
    gap> MajorantClosure(B, T);
    [ <block bijection: [ 1, -2 ], [ 2, 3, 5, -1, -3, -5 ], [ 4, -4 ]>,
      <block bijection: [ 1, -4 ], [ 2, 3, 5, -1, -3, -5 ], [ 4, -2 ]>,
      <block bijection: [ 1, -2 ], [ 2, 5, -1, -5 ], [ 3, -3 ], [ 4, -4 ]>
        , <block bijection: [ 1, -2 ], [ 2, -1 ], [ 3, 5, -3, -5 ],
         [ 4, -4 ]>,
      <block bijection: [ 1, -4 ], [ 2, 5, -3, -5 ], [ 3, -1 ], [ 4, -2 ]>
        , <block bijection: [ 1, -4 ], [ 2, -3 ], [ 3, 5, -1, -5 ],
         [ 4, -2 ]>, <block bijection: [ 1, -4 ], [ 2, -3 ], [ 3, -1 ],
         [ 4, -2 ], [ 5, -5 ]> ]
    gap> IsMajorantlyClosed(B, last);
    true

##### 11.14-4 Minorants

`‣ Minorants`( S, f ) | ( operation )

Returns: A list of elements.

`Minorants` takes an element f from an inverse semigroup of partial permutations, block bijections or partial permutation bipartitions S, and returns a list of the minorants of f in S.

A minorant of f is an element of S which is strictly less than f in the natural partial order of S. See `NaturalLeqPartialPerm` (Reference: NaturalLeqPartialPerm).

    gap> S := SymmetricInverseSemigroup(3);
    <symmetric inverse monoid of degree 3>
    gap> x := Elements(S)[13];
    [1,3](2)
    gap> Minorants(S, x);
    [ <empty partial perm>, [1,3], <identity partial perm on [ 2 ]> ]
    gap> x := PartialPerm([3, 2, 4, 0]);
    [1,3,4](2)
    gap> S := InverseSemigroup(x);
    <inverse partial perm semigroup of rank 4 with 1 generator>
    gap> Minorants(S, x);
    [ <identity partial perm on [ 2 ]>, [1,3](2), [3,4](2) ]

##### 11.14-5 PrimitiveIdempotents

`‣ PrimitiveIdempotents`( S ) | ( attribute )

Returns: A list of elements.

An idempotent in an inverse semigroup S is primitive if it is non-zero and minimal with respect to the `NaturalPartialOrder` (Reference: NaturalPartialOrder) on S. `PrimitiveIdempotents` returns the list of primitive idempotents in the inverse semigroup S.

    gap> S := InverseMonoid(
    > PartialPerm([1], [4]),
    > PartialPerm([1, 2, 3], [2, 1, 3]),
    > PartialPerm([1, 2, 3], [3, 1, 2]));;
    gap> MultiplicativeZero(S);
    <empty partial perm>
    gap> Set(PrimitiveIdempotents(S));
    [ <identity partial perm on [ 1 ]>, <identity partial perm on [ 2 ]>,
      <identity partial perm on [ 3 ]>, <identity partial perm on [ 4 ]> ]
    gap> S := DualSymmetricInverseMonoid(4);
    <inverse block bijection monoid of degree 4 with 3 generators>
    gap> Set(PrimitiveIdempotents(S));
    [ <block bijection: [ 1, 2, 3, -1, -2, -3 ], [ 4, -4 ]>,
      <block bijection: [ 1, 2, 4, -1, -2, -4 ], [ 3, -3 ]>,
      <block bijection: [ 1, 2, -1, -2 ], [ 3, 4, -3, -4 ]>,
      <block bijection: [ 1, 3, 4, -1, -3, -4 ], [ 2, -2 ]>,
      <block bijection: [ 1, 3, -1, -3 ], [ 2, 4, -2, -4 ]>,
      <block bijection: [ 1, 4, -1, -4 ], [ 2, 3, -2, -3 ]>,
      <block bijection: [ 1, -1 ], [ 2, 3, 4, -2, -3, -4 ]> ]

##### 11.14-6 RightCosetsOfInverseSemigroup

`‣ RightCosetsOfInverseSemigroup`( S, T ) | ( operation )

Returns: A list of lists of elements.

`RightCosetsOfInverseSemigroup` takes a majorantly closed inverse subsemigroup T of an inverse semigroup of partial permutations, block bijections or partial permutation bipartitions S. See `IsMajorantlyClosed` (12.2-8). The result is a list of the right cosets of T in S.

For \(s \in S\), the right coset \(\overline{Ts}\) is defined if and only if \(ss^{-1} \in T\), in which case it is defined to be the majorant closure of the set \(Ts\). See `MajorantClosure` (11.14-3). Distinct cosets are disjoint but do not necessarily partition S.

    gap> S := SymmetricInverseSemigroup(3);
    <symmetric inverse monoid of degree 3>
    gap> T := InverseSemigroup(MajorantClosure(S, [PartialPerm([1])]));
    <inverse partial perm monoid of rank 3 with 6 generators>
    gap> IsMajorantlyClosed(S, T);
    true
    gap> RC := RightCosetsOfInverseSemigroup(S, T);
    [ [ <identity partial perm on [ 1 ]>,
          <identity partial perm on [ 1, 2 ]>, [2,3](1), [3,2](1),
          <identity partial perm on [ 1, 3 ]>,
          <identity partial perm on [ 1, 2, 3 ]>, (1)(2,3) ],
      [ [1,3], [2,1,3], [1,3](2), (1,3), [1,3,2], (1,3,2), (1,3)(2) ],
      [ [1,2], (1,2), [1,2,3], [3,1,2], [1,2](3), (1,2)(3), (1,2,3) ] ]

##### 11.14-7 SameMinorantsSubgroup

`‣ SameMinorantsSubgroup`( H ) | ( attribute )

Returns: A list of elements of the group \(\mathscr{H}\)-class H.

Given a group \(\mathscr{H}\)-class H in an inverse semigroup of partial permutations, block bijections or partial permutation bipartitions `S`, `SameMinorantsSubgroup` returns a list of the elements of H which have the same strict minorants as the identity element of H. A strict minorant of `x` in H is an element of `S` which is less than `x` (with respect to the natural partial order), but is not equal to `x`.

The returned list of elements of H describe a subgroup of H.

    gap> S := SymmetricInverseSemigroup(3);
    <symmetric inverse monoid of degree 3>
    gap> H := GroupHClass(DClass(S, PartialPerm([1, 2, 3])));
    <Green's H-class: <identity partial perm on [ 1, 2, 3 ]>>
    gap> Elements(H);
    [ <identity partial perm on [ 1, 2, 3 ]>, (1)(2,3), (1,2)(3),
      (1,2,3), (1,3,2), (1,3)(2) ]
    gap> SameMinorantsSubgroup(H);
    [ <identity partial perm on [ 1, 2, 3 ]> ]
    gap> T := InverseSemigroup(
    > PartialPerm([1, 2, 3, 4], [1, 2, 4, 3]),
    > PartialPerm([1], [1]), PartialPerm([2], [2]));
    <inverse partial perm semigroup of rank 4 with 3 generators>
    gap> Elements(T);
    [ <empty partial perm>, <identity partial perm on [ 1 ]>,
      <identity partial perm on [ 2 ]>,
      <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4) ]
    gap> x := GroupHClass(DClass(T, PartialPerm([1, 2, 3, 4])));
    <Green's H-class: <identity partial perm on [ 1, 2, 3, 4 ]>>
    gap> Elements(x);
    [ <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4) ]
    gap> AsSet(SameMinorantsSubgroup(x));
    [ <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4) ]

##### 11.14-8 SmallerDegreePartialPermRepresentation

`‣ SmallerDegreePartialPermRepresentation`( S ) | ( attribute )

Returns: An isomorphism.

`SmallerDegreePartialPermRepresentation` attempts to find an isomorphism from the inverse semigroup S to an inverse semigroup of partial permutations with small degree. If S is already a partial permutation semigroup, and the function cannot reduce the degree, the identity mapping is returned.

There is no guarantee that the smallest possible degree representation is returned. For more information see [Sch92].

    gap> S := InverseSemigroup(PartialPerm([2, 1, 4, 3, 6, 5, 8, 7]));
    <partial perm group of rank 8 with 1 generator>
    gap> Elements(S);
    [ <identity partial perm on [ 1, 2, 3, 4, 5, 6, 7, 8 ]>,
      (1,2)(3,4)(5,6)(7,8) ]
    gap> iso := SmallerDegreePartialPermRepresentation(S);;
    gap> Source(iso) = S;
    true
    gap> R := Range(iso);
    <partial perm group of rank 2 with 1 generator>
    gap> Elements(R);
    [ <identity partial perm on [ 1, 2 ]>, (1,2) ]
    gap> S := DualSymmetricInverseMonoid(5);;
    gap> T := Range(IsomorphismPartialPermSemigroup(S));
    <inverse partial perm monoid of size 6721, rank 6721 with 3
     generators>
    gap> SmallerDegreePartialPermRepresentation(T);
    <inverse partial perm monoid of size 6721, rank 6721 with 3
      generators> -> <inverse partial perm monoid of rank 30 with 3
      generators>

##### 11.14-9 VagnerPrestonRepresentation

`‣ VagnerPrestonRepresentation`( S ) | ( attribute )

Returns: An isomorphism to an inverse semigroup of partial permutations.

`VagnerPrestonRepresentation` returns an isomorphism from an inverse semigroup S where the elements of S have a unique semigroup inverse accessible via `Inverse` (Reference: inverse), to the inverse semigroup of partial permutations T of degree equal to the size of S, which is obtained using the Vagner-Preston Representation Theorem.

More precisely, if \(f:S\to T\) is the isomorphism returned by `VagnerPrestonRepresentation(S)` and \(x\) is in S, then \(f(x)\) is the partial permutation with domain \(Sx^{-1}\) and range \(Sx^{-1}x\) defined by \(f(x): sx^{-1}\mapsto sx^{-1}x\).

In many cases, it is possible to find a smaller degree representation than that provided by `VagnerPrestonRepresentation` using `IsomorphismPartialPermSemigroup` (Reference: IsomorphismPartialPermSemigroup) or `SmallerDegreePartialPermRepresentation` (11.14-8).

    gap> S := SymmetricInverseSemigroup(2);
    <symmetric inverse monoid of degree 2>
    gap> Size(S);
    7
    gap> iso := VagnerPrestonRepresentation(S);
    <symmetric inverse monoid of degree 2> ->
    <inverse partial perm monoid of rank 7 with 2 generators>
    gap> RespectsMultiplication(iso);
    true
    gap> inv := InverseGeneralMapping(iso);;
    gap> ForAll(S, x -> (x ^ iso) ^ inv = x);
    true
    gap> V := InverseSemigroup(
    > Bipartition([[1, -4], [2, -1], [3, -5], [4], [5], [-2], [-3]]),
    > Bipartition([[1, -5], [2, -1], [3, -3], [4], [5], [-2], [-4]]),
    > Bipartition([[1, -2], [2, -4], [3, -5], [4, -1], [5, -3]]));
    <inverse bipartition semigroup of degree 5 with 3 generators>
    gap> IsInverseSemigroup(V);
    true
    gap> VagnerPrestonRepresentation(V);
    <inverse bipartition semigroup of size 394, degree 5 with 3
      generators> -> <inverse partial perm semigroup of rank 394 with 5
      generators>

##### 11.14-10 CharacterTableOfInverseSemigroup

`‣ CharacterTableOfInverseSemigroup`( S ) | ( attribute )

Returns: The character table of the inverse semigroup S and a list of conjugacy class representatives of S.

Returns a list with two entries: the first entry being the character table of the inverse semigroup S as a matrix, while the second entry is a list of conjugacy class representatives of S.

The order of the columns in the character table matrix follows the order of the conjugacy class representatives list. The conjugacy representatives are grouped by \(\mathscr{D}\)-class and then sorted by rank. Also, as is typical of character tables, the rows of the matrix correspond to the irreducible characters and the columns correspond to the conjugacy classes.

This function was contributed by Jhevon Smith and Ben Steinberg.

    gap> S := InverseMonoid([
    >   PartialPerm([1, 2], [3, 1]),
    >  PartialPerm([1, 2, 3], [1, 3, 4]),
    >  PartialPerm([1, 2, 3], [2, 4, 1]),
    >  PartialPerm([1, 3, 4], [3, 4, 1])]);;
    gap> CharacterTableOfInverseSemigroup(S);
    [ [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 3, 1, 1, 1, 0, 0, 0, 0 ],
          [ 3, 1, E(3), E(3)^2, 0, 0, 0, 0 ],
          [ 3, 1, E(3)^2, E(3), 0, 0, 0, 0 ], [ 6, 3, 0, 0, 1, -1, 0, 0 ],
          [ 6, 3, 0, 0, 1, 1, 0, 0 ], [ 4, 3, 0, 0, 2, 0, 1, 0 ],
          [ 1, 1, 1, 1, 1, 1, 1, 1 ] ],
      [ <identity partial perm on [ 1, 2, 3, 4 ]>,
          <identity partial perm on [ 1, 3, 4 ]>, (1,3,4), (1,4,3),
          <identity partial perm on [ 1, 3 ]>, (1,3),
          <identity partial perm on [ 3 ]>, <empty partial perm> ] ]
    gap> S := SymmetricInverseMonoid(4);;
    gap> CharacterTableOfInverseSemigroup(S);
    [ [ [ 1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0 ],
          [ 3, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0 ],
          [ 2, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0 ],
          [ 3, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0 ],
          [ 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0 ],
          [ 4, -2, 1, 0, 0, 1, -1, 1, 0, 0, 0, 0 ],
          [ 8, 0, -1, 0, 0, 2, 0, -1, 0, 0, 0, 0 ],
          [ 4, 2, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0 ],
          [ 6, 0, 0, -2, 0, 3, -1, 0, 1, -1, 0, 0 ],
          [ 6, 2, 0, 2, 0, 3, 1, 0, 1, 1, 0, 0 ],
          [ 4, 2, 1, 0, 0, 3, 1, 0, 2, 0, 1, 0 ],
          [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ],
      [ <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4),
          (1)(2,3,4), (1,2)(3,4), (1,2,3,4),
          <identity partial perm on [ 1, 2, 3 ]>, (1)(2,3), (1,2,3),
          <identity partial perm on [ 2, 3 ]>, (2,3),
          <identity partial perm on [ 1 ]>, <empty partial perm> ] ]

##### 11.14-11 EUnitaryInverseCover

`‣ EUnitaryInverseCover`( S ) | ( attribute )

Returns: A homomorphism between semigroups.

If the argument S is an inverse semigroup then this function returns a finite E-unitary inverse cover of S. A finite E-unitary cover of S is a surjective idempotent separating homomorphism from a finite semigroup satisfying `IsEUnitaryInverseSemigroup` (12.2-3) to S. A semigroup homomorphism is said to be idempotent separating if no two idempotents are mapped to the same element of the image.

    gap> S := InverseSemigroup([PartialPermNC([1, 2], [2, 1]),
    > PartialPermNC([1], [1])]);
    <inverse partial perm semigroup of rank 2 with 2 generators>
    gap> cov := EUnitaryInverseCover(S);
    <inverse partial perm semigroup of rank 4 with 2 generators> ->
    <inverse partial perm semigroup of rank 2 with 2 generators>
    gap> IsEUnitaryInverseSemigroup(Source(cov));
    true
    gap> S = Range(cov);
    true

#### 11.15 Nambooripad partial order

##### 11.15-1 NambooripadLeqRegularSemigroup

`‣ NambooripadLeqRegularSemigroup`( S ) | ( attribute )

Returns: A function.

`NambooripadLeqRegularSemigroup` returns a function that, when given two elements `x, y` of the regular semigroup S, returns `true` if `x` is less than or equal to `y` in the Nambooripad partial order on S. See also `NambooripadPartialOrder` (11.15-2).

    gap> S := BrauerMonoid(3);
    <regular bipartition *-monoid of degree 3 with 3 generators>
    gap> IsRegularSemigroup(S);
    true
    gap> Size(S);
    15
    gap> NambooripadPartialOrder(S);
    [ [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ],
      [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ],
      [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ],
      [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ]
    gap> NambooripadLeqRegularSemigroup(S)(Elements(S)[3], Elements(S)[9]);
    false
    gap> NambooripadLeqRegularSemigroup(S)(Elements(S)[2], Elements(S)[15]);
    true

##### 11.15-2 NambooripadPartialOrder

`‣ NambooripadPartialOrder`( S ) | ( attribute )

Returns: The Nambooripad partial order on a regular semigroup.

The Nambooripad partial order \(\leq\) on a regular semigroup S is defined by `s`\(\leq\)`t` if the principal right ideal of `S` generated by `s` is contained in the principal right ideal of `S` generated by `t` and there is an idempotent `e` in the \(\mathscr{R}\)-class of `s` such that `s`\(=\)`et`. The Nambooripad partial order coincides with the natural partial order when considering inverse semigroups `NaturalPartialOrder` (Reference: NaturalPartialOrder).

`NambooripadPartialOrder` returns the Nambooripad partial order on the regular semigroup S as a list of sets of positive integers where entry `i` in `NaturalPartialOrder(S)` is the set of positions in `Elements(S)` of elements which are less than `Elements(S)[i]`. See also `NambooripadLeqRegularSemigroup` (11.15-1).

    gap> S := BrauerMonoid(3);
    <regular bipartition *-monoid of degree 3 with 3 generators>
    gap> IsRegularSemigroup(S);
    true
    gap> Size(S);
    15
    gap> NambooripadPartialOrder(S);
    [ [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ],
      [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ],
      [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ],
      [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ]
    gap> NambooripadLeqRegularSemigroup(S)(Elements(S)[3], Elements(S)[9]);
    false
    gap> NambooripadLeqRegularSemigroup(S)(Elements(S)[2], Elements(S)[15]);
    true

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Bib Ind

generated by GAPDoc2HTML