# Category:Definitions/Naturally Ordered Semigroup

This category contains definitions related to Naturally Ordered Semigroup.
Related results can be found in Category:Naturally Ordered Semigroup.

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 $\ds \exists m, n \in S:$ $\ds m \ne n$

## Pages in category "Definitions/Naturally Ordered Semigroup"

The following 15 pages are in this category, out of 15 total.