Definition:Ordered Semigroup
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Definition
An ordered semigroup is an ordered structure $\struct {S, \circ, \preceq}$ such that $\struct {S, \circ}$ is a semigroup.
Ordered Semigroup Axioms
The properties that define an ordered semigroup can be gathered together as follows:
An ordered semigroup is an algebraic system $\struct {S, \circ, \preceq}$ which satisfies the following properties:
\((\text {OS} 0)\) | $:$ | Closure | \(\ds \forall a, b \in S:\) | \(\ds a \circ b \in S \) | |||||
\((\text {OS} 1)\) | $:$ | Associativity | \(\ds \forall a, b, c \in S:\) | \(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \) | |||||
\((\text {OS} 2)\) | $:$ | Compatibility of $\preceq$ with $\circ$ | \(\ds \forall a, b, c \in S:\) | \(\ds a \preceq b \implies \paren {a \circ c} \preceq \paren {b \circ c} \) | |||||
where $\preceq$ is an ordering | \(\ds a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b} \) |
Also see
- Results about ordered semigroups can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups