Title: Idempotent ordered semigroup,
Abstract: An element e of an ordered semigroup $(S,\cdot,\leq)$ is called an ordered
idempotent if $e\leq e^2$. We call an ordered semigroup $S$ idempotent ordered
semigroup if every element of $S$ is an ordered idempotent. Every idempotent
semigroup is a complete semilattice of rectangular idempotent semigroups and in
this way we arrive to many other important classes of idempotent ordered
semigroups.