\begin{definition}[Definition:Ordered Subsemigroup]
Let $\struct {S, \circ, \preceq}$ be an ordered structure.
Let $T \subseteq S$ be a subset of $S$ such that:
:$\struct {T, \circ_T, \preceq_T}$ is an ordered semigroup
where:
:$\circ_T$ denotes the operation induced on $T$ by $\circ$
:$\preceq_T$ denotes the restriction of $\preceq$ to $T \times T$.
Then $\struct {T, \circ_T, \preceq_T}$ is an '''ordered subsemigroup''' of $\struct {S, \circ, \preceq}$.
\end{definition}