Definition:Ordered Subsemigroup

Definition
An ordered subsemigroup $\left({T, \circ, \preceq}\right)$ of an ordered structure $\left({S, \circ, \preceq}\right)$ is an ordered semigroup such that the semigroup $\left({T, \circ}\right)$ is a subsemigroup of $\left({S, \circ}\right)$.

Also see

 * Ordered Structure