# 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 $\left({S, \circ, \preceq}\right)$ satisfying:

\((NO 1)\) | $:$ | $S$ is well-ordered by $\preceq$ | \(\displaystyle \forall T \subseteq S:\) | \(\displaystyle T = \varnothing \lor \exists m \in T: \forall n \in T: m \preceq n \) | ||||

\((NO 2)\) | $:$ | $\circ$ is cancellable in $S$ | \(\displaystyle \forall m, n, p \in S:\) | \(\displaystyle m \circ p = n \circ p \implies m = n \) | ||||

\(\displaystyle p \circ m = p \circ n \implies m = n \) | ||||||||

\((NO 3)\) | $:$ | Existence of product | \(\displaystyle \forall m, n \in S:\) | \(\displaystyle m \preceq n \implies \exists p \in S: m \circ p = n \) | ||||

\((NO 4)\) | $:$ | $S$ 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

## Also see

- Results about
**naturally ordered semigroups**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 16$