Group-symmetric semigroups

In the previous post the nature of Clifford semigroups was briefly discussed. We noticed that Clifford semigroups are divisibility commutative (because L=R). But due to complete regularity the Green's relations of Clifford semigroups have the further property that all non-trivial H classes form subgroups. We can therefore form a special subclass of the class of divisibility commutative semigroups that captures the divisibility properties of Clifford semigroups.

Definition. a semigroup is called group-symmetric if it is divisibility commutative and there are no non-trivial H classes that do not form groups.

The reason that I call these semigroups "group-symmetric" is that all the non-trivial components of the factorization preordering are a result of subgroups. Since subgroups are responsible for all factorization symmetry, antisymmetry is equivalent to aperiodicity. Here are three basic theorems about these semigroups:

• Clifford semigroups are group-symmetric
• All commutative semigroups of order four or less are group-symmetric
• Finite monogenic semigroups are group-symmetric

To see (1) first recall that Clifford semigroups are divisibility commutative by a previous theorem. Then by complete relugarity all H classes form subgroups, so there are no non-trivial H classes that do not form groups. To see (2) consider that Green's theorem demonstrates that a non-trivial H class that does not contain an idempotent must not contain the iterations of any of its own elements, so there must be a third element that is an element of these two. Additionally, the semigroup cannot be aperiodic because then it would be H-trivial since it is finite. So there needs to be a group which means at least two other elements need to exist to contain the group leading to a minimum of five elements. To see (3) compute the H classes of the finite monogenic semigroup. Clearly there is an H class containing the idempotent, but then every other element is H-trivial because a lesser iterate cannot be obtained from a larger one outside the group.

Starting with order five, there are commutative semigroups which are not group-symmetric. But these semigroups are not group-free because it is proven that all finite commutative aperiodic semigroups are antisymmetric and hence trivially group-symmetric. So even in those cases the symmetry in the semigroup is because of some subgroup. Those non-trivial H classes that are not subgroups are externally symmetric because they emerge from some group outside of themselves which operates on them to create some symmetry in the semigroup, which absent everything else would be antisymmetric like a finite commutative aperiodic semigroup.

Definition. the group elements of a semigroup are all those that are contained in some subgroup and the non-group elements are all those elements are not contained in a subgroup

The order of any finite semigroup is equal to the sum of the group elements count and the non-group elements count. So for example, example monogenic semigroups are classified by their period and index. The only difference is that their sum doesn't equal to the order of the semigroup because the index is never zero even for cyclic groups. So the non-group elements count is equal to the index minus one. The non-group elements count determines how far the semigroup is from being completely regular, and therefore in the group-symmetric case from being Clifford.

We saw how, particularly in the finite case, J-trivial semigroups generalize semilattices. Group-symmetric semigroups allow us to do the same thing for Clifford semigroups, which are often called semilattices of groups. So for example, given a group-symmetric semigroup we can form a Clifford semigroup from it which contains its ordered group structure. In the opposite direction, given a semilattice of groups we can form different group-symmetric semigroups which maintain the Cliffordic structure of that semigroup.