# Definition: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 $\preceq$ $\displaystyle \forall T \subseteq S:$ $\displaystyle T = \varnothing \lor \exists m \in T: \forall n \in T: m \preceq n$ $(\text {NO} 2)$ $:$ $\circ$ 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 $\displaystyle \exists m, n \in S:$ $\displaystyle m \ne n$