Semigroup compatibility conditions

Let $S$ be a semigroup, then $S^1$ is the monoid created from $S$ by adjoining an identity. We can then form an $S^1$-set $S$ by left actions and an $(S^1)^{op}$-set $S$ by right actions, where these are respectively the left and right self induced monoid actions of a semigroup. The topos of monoid actions has products defined for any corresponding family of objects, so we can use this to form monoid actions on $S^2$.

Monoid actions on ordered pairs:
The self-induced actions of a monoid on an ordered pair are defined by their respective products in their topoi of monoid actions:

• The left action of $S^1$ on $(a,b)$ takes an $m \in S^1$ to $(ma,mb)$
• The right action of $(S^1)^{op}$ on $(a,b)$ takes $m \in (S^1)^{op}$ to $(am,bm)$.

The familiar left and right compatability conditions are now equivalent to action closed sets in $S^2$ with respect to these left/right monoid actions. Congruences are clearly left/right compatible.

Congruences and divisibility comutative semigroups:
While my basic thinking here is to highlight another example of a monoid action in semigroup theory, these left and right compatibility conditions may be useful in the theory of congruences. The key thing is an equivalence is a congruence iff it is left/right compatible.

Theorem. an equivalence relation $P$ is a semigroup congruence if it is left/right compatible

Proof. let $(a,b) \in P$ and $(c,d) \in P$ then by compatibility we now have $(ac,bc) \in P$ and $(bc,bd) \in P$. Then by transitivity this implies $(ac,bd) \in P$, so $P$ is a congruence. $\square$

This by itself isn't that interesting, but one key observation is that with respect to the Green's relations of a semigroup $L$ is right compatible and $R$ is left compatible. The compatibility conditions are in the opposite orders of the Green's relations, because then they don't effect the validity of the relations.

Theorem. in a semigroup $L$ is right compatible and $R$ is left compatible

Proof. suppose $S^1a = S^1b$ then by a right action by $x$ we have $S^1ax = S^1bx$. In the other direction, $aS^1 = bS^1$ implies that $xaS^1 = xbS^1$. $\square$

This leads to an immediate corollary that is of relevance in the theory of divisibility-commutative semigroups. As $L = R = H$ in a divisibility commutative semigroup, then $H$ is both left and right compatible, which means it is a congruence.

Corollary. in a divisibility commutative semigroup ($L$ = $R$) then $H$ is a congruence

It follows that we can form the condensation $\frac{S}{H}$ of a divisibility commutative semigroup, even when it is not commutative. While this condensation is of note in the theory of commutative semigroups, as it generalizes the condensation of preorders we now see that there is a broader context that it can be applied in. So divisibility commutative semigroups share much of the structure of commutative semigroups, and so they are just the right generalizations of commutativity just as Clifford semigroups are just the right generalizations of groups.

Corollary. a semigroup is a Clifford semigroup iff it is divisibility commutative and $\frac{S}{H}$ is a semilattice.

It is well known that the condensation of a Clifford semigroup is a semilattice, and that $L = R$ in Clifford semigroups. Suppose on the other hand we have that the condensation is a semilattice, then this simply means that each H class of the semigroup has an idempotent quotient, but that means that the H class is a group by Green's theorem. So the semigroup is a semilattice of groups, which is a Clifford semigroup. Therefore, this is an alternative characterization of Clifford semigroups.

The defining aspect of non-Clifford divisibility commutative semigroups is that they have other J-trivial semigroups for their condensations. The effect of this condensation theory on the theory of divisibility commutative semigroups is that it relates them to the theory of J-trivial semigroups which are partially ordered by their algebraic preordering. Therefore, J-trivial semigroups are upper bound functions on partial orders: in this case the partial order is precisely the condensation of the preorder of the semigroup and hence the two notions of condensation coincide.

J-trivial semigroups form a much broader class of semigroups then either semilattices or commutative J-trivial semigroups. The unique smallest non-commutative divisibility commutative semigroup is a J-trivial semigroup on a three element total order: the smallest non-commutative totally ordered semigroup. There are J-trivial semigroups on tree partial orders that produce either the immediate ancestor of two elements or a further ancestor up the tree dependent upon the argument order, and so on. So J-trivial semigroups produce a wider variety of activity on partially orderd sets.