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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups