Partial Order proof with operation on a set Let X be a set and let $f$ be an operation on X (i.e. it is a function from X $\times$ X to X), which we will denote with $f(x, y) = xy$. In addition, $x \le y$ iff $f(x, y) = x$. Suppose further that $f$ is commutative, associative, and idempotent.
Prove that $\le$ is a partial order on X.
I can see how to prove that it is reflexive and anti-symmetric, but I'm having a hard time proving that it is transitive. This is what I have so far for transitivity (I omit the my answers to proving that it is reflexive and anti-symmetric):
$\forall a, b, c \in X$, if $a \le b \land b \le c$, then $a = f(a, b) \land b = f(b, c)$
I know that we want to get to that $a = f(a, c)$ to show that $a \le c$...
Any suggestions? Thanks!

HINT: You haven't used the fact that $f$ is associative.
