Definition:Ordered Subsemigroup

Definition
Let $\left({S, \circ, \preceq}\right)$ be an ordered structure.

Let $T \subseteq S$ be a subset of $S$ such that:
 * $\left({T, \circ_T, \preceq_T}\right)$ is an ordered semigroup

where:
 * $\circ_T$ is the operation induced on $H$ by $\circ$
 * $\preceq_t$ is the restriction of $\preceq$ to $T \times T$.

Then $\left({T, \circ_T, \preceq_T}\right)$ is an ordered subsemigroup of $\left({S, \circ, \preceq}\right)$.

Also denoted as
It is usual to drop the suffixes to denote the restrictions, and denote this as:
 * $\left({T, \circ, \preceq}\right)$

Also see

 * Definition:Ordered Structure