The embedding of an ordered semigroup into an le-semigroup

In this paper we prove the following: If $S$ is an ordered semigroup, then the set $\mathcal P(S)$ of all subsets of $S$ with the multiplication "$\circ$" on $\mathcal P(S)$ defined by "$A\circ B\colon=(AB]$ if $A,B\in\mathcal P(S)$, $A\neq\emptyset$, $B\neq\emptyset$ and $A\circ B\colon=\emptyset$ if $A=\emptyset$ or $B=\emptyset$ is an le-semigroup having a zero element and $S$ is embedded in $\mathcal P(S)$.