Definition:Naturally Ordered Semigroup/Axioms

Definition

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 $

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers

\((\text {NO} 1)\)	$:$	$S$ 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	\(\ds \exists m, n \in S:\)	\(\ds m \ne n \)