Rees Matrix Semigroups

In this section, we describe the functions in &GAP; for Rees matrix and 0-matrix semigroups and their subsemigroups. The importance of these semigroups lies in the fact that Rees matrix semigroups over groups are exactly the completely simple semigroups, and Rees 0-matrix semigroups over groups are the completely 0-simple semigroups.

Let I and J be sets, let S be a semigroup, and let P=(p_{ji})_{j\in J, i\in I} be a |J|\times |I| matrix with entries in S. Then the Rees matrix semigroup with underlying semigroup S and matrix P is just the direct product I\times S \times 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\times S\times J)\cup \{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 and every semigroup of Rees 0-matrix semigroup elements satisfies .

Rees matrix and 0-matrix semigroups can be created using the operations and , 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\times U\times J satisfies if and only if it is equal to I'\times U'\times J' where I'\subseteq I, J'\subseteq 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 . 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 .

Similarly, 0-simple subsemigroups of Rees 0-matrix semigroups do not have to satisfy . 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 .

Elements of a Rees matrix or 0-matrix semigroup belong to the categories and , respectively. Such elements can be created directly using the functions and .

A semigroup in &GAP; can either satisfies or but not both.

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) ])>
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>>
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 the argument is a 0-simple semigroup, then an isomorphism to a Rees 0-matrix semigroup over a group is returned.

See and . 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(IsomorphismReesMatrixSemigroup(U)); <Rees 0-matrix semigroup 2x2 over Group(())> gap> Rows(V); Columns(V); [ 1, 2 ] [ 1, 2 ]
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
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
true
or false.
Every semigroup consisting of elements of a Rees matrix semigroup satisfies the property and every semigroup of Rees 0-matrix semigroup elements satisfies .

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
true
or false.
A subsemigroup of a Rees matrix semigroup I\times U\times J satisfies if and only if it is equal to I'\times U'\times J' where I'\subseteq I, J'\subseteq 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 . 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 .

Similarly, 0-simple subsemigroups of Rees 0-matrix semigroups do not have to satisfy . 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 .

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 on 2 pts 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
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 or . 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 and columns
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 or . 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 ]
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 or . 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) ])
A Rees matrix or 0-matrix semigroup.
If D is a regular &D;-class of a finite semigroup S, then there is a standard way of associating a Rees matrix semigroup to D. If D is a subsemigroup of S, then D is simple and hence is isomorphic to a Rees matrix semigroup. In this case, the associated Rees matrix semigroup of D is just the Rees matrix semigroup isomorphic to D.

If D is not a subsemigroup of S, then we define a semigroup with elements D and a new element 0 with multiplication of x,y\in 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. xy equals the product of x and y if it belongs to D and 0 if it does not.

The semigroup thus defined is 0-simple and hence is isomorphic to a Rees 0-matrix semigroup. This semigroup can also be described as the Rees quotient of the ideal generated by D by it maximal subideal. The associated Rees matrix semigroup of D is just the Rees 0-matrix semigroup isomorphic to the semigroup defined above. 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) ])>