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 $\left({S, \circ, \preceq}\right)$ satisfying:

\((NO 1)\)   $:$   $S$ is well-ordered by $\preceq$      \(\displaystyle \forall T \subseteq S:\) \(\displaystyle T = \varnothing \lor \exists m \in T: \forall n \in T: m \preceq n \)             
\((NO 2)\)   $:$   $\circ$ is cancellable in $S$      \(\displaystyle \forall m, n, p \in S:\) \(\displaystyle m \circ p = n \circ p \implies m = n \)             
\(\displaystyle p \circ m = p \circ n \implies m = n \)             
\((NO 3)\)   $:$   Existence of product      \(\displaystyle \forall m, n \in S:\) \(\displaystyle m \preceq n \implies \exists p \in S: m \circ p = n \)             
\((NO 4)\)   $:$   $S$ has at least two distinct elements      \(\displaystyle \exists m, n \in S:\) \(\displaystyle m \ne n \)             


Addition

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.


The operation $\circ$ in $\left({S, \circ, \preceq}\right)$ is called addition.


Ordering

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.


The relation $\preceq$ in $\left({S, \circ, \preceq}\right)$ is called the ordering.


Comment



Also see

  • Results about naturally ordered semigroups can be found here.


Sources