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.

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