Cancellability in Naturally Ordered Semigroup

Theorem

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

Then:

Ordering of Naturally Ordered Semigroup is Strongly Compatible

$\forall m, n, p \in S: m \preceq n \iff m \circ p \preceq n \circ p$

Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible

$\forall m, n, p \in S: m \prec n \iff m \circ p \prec n \circ p$