Orbit-finite 0-simple semigroups

We ask the question of how the Rees-Suskevitch theorem should be generalised to provide representations of orbit-finite 0-simple semigroups. We believe that orbit-finite groupoids will be useful for this.

Semigroups

A semigroup is a set S associated with an associative operation S\times S\to S.  By S^1 we denote the monoid obtained by adjoining S with a neutral element. We say that:

  • x is a prefix of y if y=xs for some s\in S^1,
  • x is a suffix of y if y=sx for some t\in S^1,
  • x is an infix of y if y=sxt for some t\in S^1.

These three relations define preorders on S. We call the corresponding equivalence relations prefix equivalence, suffix equivalence and similarity. We also define bi-equivalence to be the intersection of prefix and suffix equivalence, i.e., x and y are bi-equivalent if they are both prefix and suffix equivalent.

We call the equivalence classes of these relations, respectively, prefix classes, suffix classes, similarity classes and bi-classes.

Remark. In semigroup theory, those equivalence relations are denoted \mathcal R, \mathcal L,\mathcal J,\mathcal H, respectively, and are called Green’s relations

Semigroups from groupoids

Every groupoid (see the previous post) M defines a semigroup, whose elements are the arrows of M together with an additional element 0, and f\cdot g=f;g if the composition f;g is defined and 0 otherwise.

Example 1. Consider the semigroup S associated to a groupoid M. Let s,t be two nonzero elements in S, i.e., two arrows in M. They are similar iff s factorizes through t and vice-versa. They are prefix-equivalent iff they have the same source, suffix-equivalent iff they have the same target, and are bi-equivalent iff they have both the same source and targets. Bi-classes are of the form M(x,y), where x,y are objects of M. All bi-classes of similar elements are in bijection, by Lemma 1.

0-simple semigroups

The semigroup S associated to a connected groupoid M has several special properties:

  • It is has a zero element, i.e., an element 0 such that 0\cdot s=s\cdot 0=0 for all elements s\in S,
  • It is a 0-simple semigroup, i.e., all non-zero elements are similar.

Below is another example of a 0-simple semigroup.

Example 2. Fix two sets I,J and a group G. Let m:J\times I\to G\cup \set{\bot} be a fixed mapping. We define a semigroup structure on I\times G\times J\cup\set{\bot} as follows:

    \[\!\!\!\!\!\!(i,g,j)\cdot (i',g',j')=\\\begin{cases}(i,g\cdot m(j,i')\cdot g',j')&\text{if }m(j,i')\neq \bot}\\\bot&\text{otherwise}.\]

We denote this semigroup by R(I,J,m). This is a 0-simple semigroup. Note that two elements are prefix equivalent iff their first coordinate agrees and are suffix equivalent iff their last coordinate agrees.

Theorem (Rees-Suskevitch). Let S be a finite 0-simple semigroup. Then S is isomorphic to a semigroup of the form R(I,J,m), where I is the set of prefix-classes of S, J is the set of suffix classes of S, G is some group and m:J\times I\to G\cup\set{\bot} is some function.

The finiteness assumption in the theorem can be replaced by the assumption that the semigroup is completely 0-simple (has a minimal left ideal and a minimal right ideal), which is the case in all orbit-finite semigroups.

Example 3. Consider the set S of all partial bijections of \atoms whose domain and range have exactly two elements, extended by the element \bot. This forms a semigroup:

    \[f\cdot g=\begin{cases} f;g&\text{if \textrm{rg}(f)=\textrm{dom}(f)}\\\bot&\text{otherwise}.\]

The semigroup S is 0-simple, and in fact is isomorphic to the semigroup associated to the groupoid \textrm{Iso}_{A} (see previous post) where A is the alphabet consisting of unordered pairs of atoms.

To represent the semigroup S as a semigroup of the form R(I,J,m), set I=J=P_2(\atoms), G=\mathbb Z_2,  and m(j,i)=0 for all j\in J,i\in I. The isomorphism between S and R(I,J,m) is not finitely supported.

The problem we would like to resolve is:

Problem: Find a generalization of the Rees-Suskevitch theorem which provides an equivariant representation of an equivariant 0-simple semigroup.

We believe that orbit-finite groupoids will be useful for obtaining such a result. Below is an example where it is unclear how a representation might look like.

Example 4. Let X be the set consisting of all sets of the form \bar \alpha=\set{\alpha_1,\alpha_2,\alpha_3}, (the number 3 is irrelevant here) where \alpha_1,\alpha_2,\alpha_3 are bijections between two-element sets of atoms, all with the same range but pairwise-distinct domains. Define \set{\alpha_1,\alpha_2,\alpha_3}\cdot \set{\beta_1,\beta_2,\beta_3} to be equal to \set{\alpha_1;\beta_i, \alpha_2;\beta_i, \alpha_3;\beta_i,} if the range of \alpha_1,\alpha_2,\alpha_3 is equal to the domain of \beta_i, and \bot otherwise. This forms a 0-simple semigroup. The suffix class of \set{\alpha_1,\alpha_2,\alpha_3} is identified by the common range of the three mappings. The prefix class, on the other hand, is identified by the set of mappings \set{\alpha_i;\alpha_j^{-1}: 1\le i<j\le 3}.

Schützenberger groupoids

One way to obtain a groupoid from a 0-simple semigroup S is as follows. For s\in S, denote by \rho_s:S\to S the mapping defined by \rho_s(t)=t\cdot s, and by \lambda_s:S\to S the mapping defined by \lambda_s(t)=s\cdot t.

Denote by G_R the following category:

  • The objects are bi-classes of S
  • The arrows are of the form H_1\stackrel {f} \rightarrow H_2, where f:H_1\to H_2 is the restriction of a mapping \rho_s to H_1 for some s such that \rho_s(H_1)\subset H_2, i.e., H_1\cdot s\subseteq H_2.

Lemma. The category G_R is a groupoid. Its connected components are the prefix-classes of S.

We call G_R the (right) Schützenberger groupoid associated to S. The left one is defined symmetrically, using \lambda_s instead of \rho_s, and is denoted G_L, and its connected components are the suffix classes of S.

Lemma. Let x,y be bi-classes of S. The groups G_R(x,x) and G_L(y,y) are anti-isomorphic.

Note that the groupoids G_R and G_L are usually not anti-isomorphic as they have different sets of connected components.The group G_L(x,x) is (isomorphic to) the group G which appears in the Rees-Suskevitch theorem.

This will be studied in detail in the folowing post.

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