## A natural partial order on the set of idempotents in a semigroup

Let $S$ be a semigroup and let $E(S)$ denote the set of idempotents in $S$. Define an order $\leq$ on $E(S)$ by $e \leq f$ iff $e = ef = fe$. Show that $\leq$ is a partial order.

Recall that a partial order is a binary relation which is reflexive, antisymmetric, and transitive.

If $e \in E(S)$, then $e$ is idempotent, so that $e = ee$. So $e \leq e$.

Suppose $e,f \in E(S)$ such that $e \leq f$ and $f \leq e$. Then $e = ef = fe$ and $f = fe = ef$, so that $e = f$.

Suppose $e,f,h \in E(S)$ such that $e \leq f$ and $f \leq h$. Now $e = ef = fe$ and $f = fh = hf$. So we have $e = ef = efh = eh$ and $e = fe = hfe = he$, and thus $e \leq h$.