Definition:Naturally Ordered Semigroup
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Definition
The concept of a naturally ordered semigroup is intended to capture the behaviour of the natural numbers $\N$, addition $+$ and the ordering $\le$ as they pertain to $\N$.
Naturally Ordered Semigroup Axioms
A naturally ordered semigroup is a (totally) ordered commutative semigroup $\struct {S, \circ, \preceq}$ satisfying:
| \((\text {NO} 1)\) | $:$ | $S$ is well-ordered by $\preceq$ | \(\ds \forall T \subseteq S:\) | \(\ds T = \O \lor \exists m \in T: \forall n \in T: m \preceq n \) | ||||
| \((\text {NO} 2)\) | $:$ | $\circ$ is cancellable in $S$ | \(\ds \forall m, n, p \in S:\) | \(\ds m \circ p = n \circ p \implies m = n \) | ||||
| \(\ds p \circ m = p \circ n \implies m = n \) | ||||||||
| \((\text {NO} 3)\) | $:$ | Existence of product | \(\ds \forall m, n \in S:\) | \(\ds m \preceq n \implies \exists p \in S: m \circ p = n \) | ||||
| \((\text {NO} 4)\) | $:$ | $S$ has at least two distinct elements | \(\ds \exists m, n \in S:\) | \(\ds m \ne n \) |
Addition
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
The operation $\circ$ in $\struct {S, \circ, \preceq}$ is called addition.
Ordering
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
The relation $\preceq$ in $\struct {S, \circ, \preceq}$ is called the ordering.
Also see
- Results about naturally ordered semigroups can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers