Cancellability in Naturally Ordered Semigroup
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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
- $\forall m, n, p \in S: m \prec n \iff m \circ p \prec n \circ p$