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