Definition:Ordered Subsemigroup

Definition
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 $\struct {S, \circ, \preceq}$.

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

Also see

 * Definition:Ordered Structure