# 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 $\left({S, \circ, \preceq}\right)$ 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 $\displaystyle \exists m, n \in S:$ $\displaystyle m \ne n$

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.

## Also see

• Results about naturally ordered semigroups can be found here.