The embedding of an ordered semigroup into an le-semigroup (Q1422278)

Let \(S\) be an ordered semigroup, \(\mathcal {P}(S)\) be the set of all subsets of \(S\). The authors prove that the set \(\mathcal{P}(S)\) with the multiplication \(\circ\) on \(\mathcal{P}(S)\) defined by \(A\circ B: = (AB]\) if \(A,B\in \mathcal{P}(S)\), \(A\not=\emptyset, B\not= \emptyset,\) and \(A\circ B: = \emptyset\) if \(A =\emptyset,\) or \(B = \emptyset\), is an le-semigroup having a zero element. Furthermore, \(S\) is embedded in \(\mathcal {P} (S)\).