Idempotent action posets

Let $S$ be an idempotent commutative semigroup, then its set of idempotents $E$ forms a subsemilattice. To see this, notice that the composition of any commuting pair of idempotents is again an idempotent. In general, for any subsemigroup $E$ of a semigroup $S$, we can form a preorder on $S$ by the action preorder of the $E^1$ monoid action acting on $S$. A special case is when $E$ is a subsemilattice.

Definition. let $S$ be a semigroup with a subsemilattice $E$ then the left $E$ action preorder on $S$ is defined by $a \subseteq b \Leftrightarrow \exists e \in E : ea = b$

This is merely a special case of the general construction of an action preorder from a subsemigroup. The interesting thing is that these subsemilattice action preorders are not only preorders, but partial orders as well.

Theorem. let $S$ be a semigroup with a subsemilattice $E$ then its left $E$ action preorder is antisymmetric.

Proof. let $a \subseteq b$ and $b \subseteq a$ then there exists idempotents $e,f \in E$ such that $ea = b$ and $fb = a$. We can substitute these into each equation to get: \[ b = ea = efb \] \[ a = fb = fea = fe^2a = fe(ea) = feb \] By simple substitutions applicable to any band action, we get that $a = feb$ and $b = efb$, so that the difference between $a$ and $b$ is only one of commutativity. Then by the fact that $E$ is a idempotent commutative we have that $a = efb = feb = b$. $\square$

The interesting thing is that this is applicable to any subsemilattice, so this generalizes the natural partial ordering of inverse semigroups to any semigroup with a subsemilattice. We can even form an idempotent action partial order for any commutative semigroup, which is clearly a subpreorder of the algebraic preorder.

I will conclude this post with a comparison to category theory. Semigroup theory and category theory are alike in almost every way, including in their use of action preorders. The natural partial order on an inverse semigroup is a lot like the poset of subobjects of a category. Both of them emerge from decreasing action preorders formed by subalgebras.

The inverse semigroup construction describes how partial idempotents restrict charts and the categorical one describes how monomorphisms tend to restrict images of functions or more generally the object of a morphism. The subalgebra of an inverse semigroup is its subsemilattice of idempotents, and in the case of a category it is its mono subcategory. Both of them are defined by decreasing action preorders, instead of increasing ones like those used for divisibility.

See also
Action preorders
Iteration as a monoid action