Exercise 15, § 1

A law $\tau$ on a set E is called idempotent if all the elements of E are idempotent (no. 4) under this law, that is if $x \tau x = x$ for all $x \in E$. Show that, if a law $\tau$ on E is associative, commutative and idempotent, the relation $x \tau y = y$ is an order relation on E; if we write $x \leq y$, any two elements $x, y$ of E admit a least upper bound (under this order relation) equal to $x \tau y$. Obtain the converse.