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 $\preceq$	\(\ds \forall T \subseteq S:\)	\(\ds T = \O \lor \exists m \in T: \forall n \in T: m \preceq n \)
\((\text {NO} 2)\)	$:$		$\circ$ 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	\(\ds \exists m, n \in S:\)	\(\ds m \ne n \)

Addition

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

The operation $\circ$ in $\struct {S, \circ, \preceq}$ 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

• Naturally Ordered Semigroup Axioms are Independent

• Naturally Ordered Semigroup Axioms imply Commutativity

• Results about naturally ordered semigroups can be found here.

Sources

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