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 $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

The relation $\preceq$ in $\struct {S, \circ, \preceq}$ is called the ordering.

Also see

This article is complete as far as it goes, but it could do with expansion, in particular:
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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This article is complete as far as it goes, but it could do with expansion, in particular:
It can also be proved that if $\struct {S, \circ, \preceq}$ 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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