Condensible semigroups

I have identified condensation as the most important part of the structure theory of commutative semigroups. The most natural generalization of commutative semigroups is therefore all semigroups for which condensation is possible, in other words semigroups for which J is a congruence. As these includes all divisibility commutative semigroups for which J = H, this subsumes the theory of divisibility commutativity.

Condensation theory means that every commutative semigroup is a commutative J trivial semigroup of symmetric components, which when closed are always groups. In the more general case when condensation is possible, a semigroup is a J-trivial semigroup of J-total semigroups and empty J classes ($J^2 \cap J = \emptyset$). Condensible semigroups are basically semigroups with a clean J class structure.

We can classify condensible semigroups by their condensation. The property of having commutative condensation makes a semigroup one step closer to being commutative. A special case is semigroups having semilattice condensation. It is known [1] that completely regular semigroups are an example. Bands for example, are semilattices of rectangular bands. If a semigroup has commutative condensation, the origin of its non-commutative components lies entirely in its non-commutative simple components.

In the special case of Clifford semigroups, which are semilattices of groups the non-commutativity of Clifford semigroups generally comes from its non-commutative subgroups. As groups tend to be highly commutative (as a consequence of Cauchy's theorem for example) it is not hard to see why Clifford semigroups tend to be highly commutative in some sense. If a semigroup has all commutative symmetric components, then its non-commutative might come from its condensation. In this way, we have identified the two sources of non-commutativity in condensible semigroups. Not all semigroups are condensible. The only condensible 0-simple semigroups are null semigroups or simple semigroups with zero. In particular, the Brandt semigroups of inverse semigroups are not condensible, and so all condensible inverse semigroups are Clifford. Condensible semigroups form a very broad class that includes all the most important generalizations of commutativity, but it doesn't include everything.

In addition to the J class structure of a semigroup, its condensation, and any closed symmetric components we need to consider the different ways that a given condensation combines its components together. For example, Clifford semigroups with the same semilattice of groups can be non-isomorphic. This provides a final non-trivial component in the structure theory of any condensible semigroup.

Every semigroup is inherently preordered. Every preorder can be condensed to form a partial order, and by analogy any condensible semigroup can be condensed to form a J-trivial semigroup, which is partially ordered. This suggests that J-trivial semigroups can be studied using order theory, and that they can be understood as different bound-producing functions on a partial order. Condensible semigroups operate to some extent by these order-theoretic J-trivial semigroups.

In order theory and monoid theory we can condense structures to get their underlying partially ordered components. Perhaps the most analogous thing in category theory is to get the skeleton of a category, which is a category without any extraneous isomorphisms. The skeleton of a category has such importance to category theory, that the equivalence of categories is determined by it. The condensation of a preorder, the condensation of a semigroup, and the skeleton of a category are all the same idea in different forms in three related subjects.

References:
[1] Fundamentals of semigroup theory