Definition:Naturally Ordered Semigroup/Axioms

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A naturally ordered semigroup is a (totally) ordered commutative semigroup $\struct {S, \circ, \preceq}$ satisfying:

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