The embedding of an ordered semigroup in a simple one with identity (Q1815354)

Let \((S,\cdot,\leq)\) be a partially ordered semigroup. A non-empty subset \(I\) of \(S\) is called an ideal, if it is downward closed in \((S,\leq)\) and closed under multiplication from the left and right by elements from \(S\). Then \(S\) is called simple, if it has no proper ideal. The authors show that any partially ordered semigroup can be embedded into one having an identity element.