# 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 $\struct {S, \circ, \preceq}$ 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

• Results about naturally ordered semigroups can be found here.