# Category:Naturally Ordered Semigroup

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This category contains results about **the naturally ordered semigroup**.

Definitions specific to this category can be found in **Definitions/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 | \(\ds \exists m, n \in S:\) | \(\ds m \ne n \) |

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Naturally Ordered Semigroup"

The following 25 pages are in this category, out of 25 total.

### C

### N

- Natural Numbers with Extension fulfil Naturally Ordered Semigroup Axioms 1, 3 and 4
- Natural Numbers without 1 fulfil Naturally Ordered Semigroup Axioms 1, 2 and 4
- Naturally Ordered Semigroup Axioms are Independent
- Naturally Ordered Semigroup Axioms imply Commutativity
- Naturally Ordered Semigroup Exists
- Naturally Ordered Semigroup forms Peano Structure
- Naturally Ordered Semigroup is Unique

### P

- Positive Rational Numbers under Addition fulfil Naturally Ordered Semigroup Axioms 2 to 4
- Principle of Induction applied to Interval of Naturally Ordered Semigroup
- Principle of Mathematical Induction for Naturally Ordered Semigroup
- Principle of Mathematical Induction/Naturally Ordered Semigroup
- Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result