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.
A semigroup is a set associated with an associative operation . By we denote the monoid obtained by adjoining with a neutral element. We say that:
- is a prefix of if for some ,
- is a suffix of if for some ,
- is an infix of if for some .
These three relations define preorders on . 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., and 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 , respectively, and are called Green’s relations.
Semigroups from groupoids
Every groupoid (see the previous post) defines a semigroup, whose elements are the arrows of together with an additional element , and if the composition is defined and otherwise.
Example 1. Consider the semigroup associated to a groupoid . Let be two nonzero elements in , i.e., two arrows in . They are similar iff factorizes through 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 , where are objects of . All bi-classes of similar elements are in bijection, by Lemma 1.
The semigroup associated to a connected groupoid has several special properties:
- It is has a zero element, i.e., an element such that for all elements ,
- It is a -simple semigroup, i.e., all non-zero elements are similar.
Below is another example of a -simple semigroup.
Example 2. Fix two sets and a group . Let be a fixed mapping. We define a semigroup structure on as follows:
We denote this semigroup by . This is a -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 be a finite -simple semigroup. Then is isomorphic to a semigroup of the form , where is the set of prefix-classes of , is the set of suffix classes of , is some group and is some function.
The finiteness assumption in the theorem can be replaced by the assumption that the semigroup is completely -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 of all partial bijections of whose domain and range have exactly two elements, extended by the element . This forms a semigroup:
The semigroup is -simple, and in fact is isomorphic to the semigroup associated to the groupoid (see previous post) where is the alphabet consisting of unordered pairs of atoms.
To represent the semigroup as a semigroup of the form , set , , and for all . The isomorphism between and 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 -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 be the set consisting of all sets of the form , (the number is irrelevant here) where are bijections between two-element sets of atoms, all with the same range but pairwise-distinct domains. Define to be equal to if the range of is equal to the domain of , and otherwise. This forms a -simple semigroup. The suffix class of is identified by the common range of the three mappings. The prefix class, on the other hand, is identified by the set of mappings .
One way to obtain a groupoid from a 0-simple semigroup is as follows. For , denote by the mapping defined by , and by the mapping defined by .
Denote by the following category:
- The objects are bi-classes of
- The arrows are of the form , where is the restriction of a mapping to for some such that , i.e., .
Lemma. The category is a groupoid. Its connected components are the prefix-classes of .
We call the (right) Schützenberger groupoid associated to . The left one is defined symmetrically, using instead of , and is denoted , and its connected components are the suffix classes of .
Lemma. Let be bi-classes of . The groups and are anti-isomorphic.
Note that the groupoids and are usually not anti-isomorphic as they have different sets of connected components.The group is (isomorphic to) the group which appears in the Rees-Suskevitch theorem.
This will be studied in detail in the folowing post.