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13 Attributes of inverse semigroups
 13.1 Attributes of inverse semigroups

13 Attributes of inverse semigroups

In this chapter we describe the attributes which are specific to inverse semigroups that can be determined using Semigroups.

The functions

were written by Wilf Wilson and Robert Hancock.

The function CharacterTableOfInverseSemigroup (13.1-10) was written by Jhevon Smith and Ben Steinberg.

13.1 Attributes of inverse semigroups

13.1-1 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 ], [  ] ]

13.1-2 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 (14.2-5). 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 semigroup on 3 pts>
gap> JoinIrreducibleDClasses(S);
[ <Green's D-class: <identity partial perm on [ 1 ]>> ]
gap> T := InverseSemigroup( 
> PartialPerm( [ 1, 2, 3, 4 ], [ 1, 2, 4, 3 ] ), 
> PartialPerm( [ 1 ], [ 1 ] ), PartialPerm( [ 2 ], [ 2 ] ) );
<inverse partial perm semigroup on 4 pts 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 bipartition monoid on 3 pts with 3 generators>
gap> JoinIrreducibleDClasses(D);
[ <Green's D-class: <block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ]>> ]

13.1-3 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 (14.2-6).

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 semigroup on 4 pts>
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

13.1-4 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 semigroup on 3 pts>
gap> f := Elements(s)[13];
[1,3](2)
gap> Minorants(s,f);
[ <empty partial perm>, [1,3], <identity partial perm on [ 2 ]> ]
gap> f := PartialPerm([3,2,4,0]);
[1,3,4](2)
gap> S := InverseSemigroup(f);
<inverse partial perm semigroup on 4 pts with 1 generator>
gap> Minorants(S,f);
[ <identity partial perm on [ 2 ]>, [1,3](2), [3,4](2) ]

13.1-5 PrimitiveIdempotents
‣ PrimitiveIdempotents( S )( attribute )

Returns: A list of idempotent partial permutations.

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 of partial permutations 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> PrimitiveIdempotents(S);
[ <identity partial perm on [ 4 ]>, <identity partial perm on [ 1 ]>, 
  <identity partial perm on [ 2 ]>, <identity partial perm on [ 3 ]> ]
gap> S := DualSymmetricInverseMonoid(4);
<inverse bipartition monoid on 4 pts with 3 generators>
gap> 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, -1 ], [ 2, 3, 4, -2, -3, -4 ]>, 
  <block bijection: [ 1, 2, -1, -2 ], [ 3, 4, -3, -4 ]>, 
  <block bijection: [ 1, 3, 4, -1, -3, -4 ], [ 2, -2 ]>, 
  <block bijection: [ 1, 4, -1, -4 ], [ 2, 3, -2, -3 ]>, 
  <block bijection: [ 1, 3, -1, -3 ], [ 2, 4, -2, -4 ]> ]

13.1-6 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 (14.2-6). The result is a list of the right cosets of T in S.

For s ∈ S, the right coset overlineTs is defined if and only if ss^-1 ∈ T, in which case it is defined to be the majorant closure of the set Ts. See MajorantClosure (13.1-3). Distinct cosets are disjoint but do not necessarily partition S.

gap> S := SymmetricInverseSemigroup(3);
<symmetric inverse semigroup on 3 pts>
gap> T := InverseSemigroup(MajorantClosure(S,[PartialPerm([1])]));
<inverse partial perm monoid on 3 pts 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) ] ]

13.1-7 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 semigroup on 3 pts>
gap> H := GroupHClass(GreensDClasses(S)[1]);
<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 on 4 pts 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(GreensDClasses(T)[1]);
<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> SameMinorantsSubgroup(x);
[ <identity partial perm on [ 1, 2, 3, 4 ]>, (1)(2)(3,4) ]

13.1-8 SmallerDegreePartialPermRepresentation
‣ SmallerDegreePartialPermRepresentation( S )( attribute )

Returns: An isomorphism.

SmallerDegreePartialPermRepresentation attempts to find an isomorphism from the inverse semigroup S of partial permutations to another inverse semigroup of partial permutations with smaller degree. If the function cannot reduce the degree, the identity mapping is returned.

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]));
<commutative inverse partial perm semigroup on 8 pts 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> T := SmallerDegreePartialPermRepresentation(S);
MappingByFunction( <partial perm group of size 2, 
 on 8 pts with 1 generator>
 , <commutative inverse partial perm semigroup on 2 pts
 with 1 generator>, function( x ) ... end, function( x ) ... end )
gap> R := Range(T);
<commutative inverse partial perm semigroup on 2 pts 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 on 6721 pts with 3 generators>
gap> SmallerDegreePartialPermRepresentation(T);
MappingByFunction( <inverse partial perm monoid on 6721 pts
 with 3 generators>, <inverse partial perm semigroup on 30 pts
 with 3 generators>, function( x ) ... end, function( x ) ... end )

13.1-9 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-> 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^-1x defined by f(x): sx^-1↦ sx^-1x.

In many cases, it is possible to find a smaller degree representation than that provided by VagnerPrestonRepresentation using IsomorphismPartialPermSemigroup (Reference: IsomorphismPartialPermSemigroup) or SmallerDegreePartialPermRepresentation (13.1-8).

gap> S := SymmetricInverseSemigroup(2);
<symmetric inverse semigroup on 2 pts>
gap> Size(S);
7
gap> iso := VagnerPrestonRepresentation(S);
MappingByFunction( <symmetric inverse semigroup on 2 pts>, <inverse pa\
rtial perm monoid on 7 pts
 with 2 generators>, function( x ) ... end, function( x ) ... end )
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 on 5 pts with 3 generators>
gap> IsInverseSemigroup(V);
true
gap> VagnerPrestonRepresentation(V);
MappingByFunction( <inverse bipartition semigroup of size 394, 
 on 5 pts with 3 generators>, <inverse partial perm semigroup 
on 394 pts
 with 5 generators>, function( x ) ... end, function( x ) ... end )

13.1-10 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 [ 1, 2 ]>, (1,2), 
      <identity partial perm on [ 1 ]>, <empty partial perm> ] ]
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