Definition:Naturally Ordered Semigroup

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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      \(\ds \exists m, n \in S:\) \(\ds m \ne n \)             


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

The operation $\circ$ in $\struct {S, \circ, \preceq}$ is called addition.


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.