Definition:Naturally Ordered Semigroup
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 $\preceq$ | \(\displaystyle \forall T \subseteq S:\) | \(\displaystyle T = \varnothing \lor \exists m \in T: \forall n \in T: m \preceq n \) | ||||
| \((\text {NO} 2)\) | $:$ | $\circ$ is cancellable in $S$ | \(\displaystyle \forall m, n, p \in S:\) | \(\displaystyle m \circ p = n \circ p \implies m = n \) | ||||
| \(\displaystyle p \circ m = p \circ n \implies m = n \) | ||||||||
| \((\text {NO} 3)\) | $:$ | Existence of product | \(\displaystyle \forall m, n \in S:\) | \(\displaystyle m \preceq n \implies \exists p \in S: m \circ p = n \) | ||||
| \((\text {NO} 4)\) | $:$ | $S$ has at least two distinct elements | \(\displaystyle \exists m, n \in S:\) | \(\displaystyle m \ne n \) |
Addition
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.
The operation $\circ$ in $\left({S, \circ, \preceq}\right)$ is called addition.
Ordering
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.
The relation $\preceq$ in $\left({S, \circ, \preceq}\right)$ is called the ordering.
Comment
It can be proved that each of the properties that defines the natural ordered semigroup is independent of the others. You can not drop one and still uniquely define a naturally ordered semigroup, neither can you deduce one from the other three.
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It can also be proved that if $\left({S, \circ, \preceq}\right)$ satisfies conditions NO 1, NO 2 and NO 3, then $\circ$ is commutative on $S$. Therefore specifying commutativity is not strictly speaking necessary.
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Also see
- Results about naturally ordered semigroups can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 16$
