Relation Compatibility in Totally Ordered Semigroup
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Theorem
Let $\left({S, \circ, \preceq}\right)$ be an ordered semigroup such that:
- $(1): \quad$ All the elements of $\left({S, \circ, \preceq}\right)$ are cancellable for $\circ$
- $(2): \quad \preceq$ is a total ordering.
Then:
- $\forall x, y, z \in S: x \circ z \preceq y \circ z \iff x \preceq y$
Proof
From Strict Ordering Preserved under Cancellability in Totally Ordered Semigroup:
- $x \circ z \prec y \circ z \implies x \prec y$
From the definition of cancellable element:
- $x \circ z = y \circ z \implies x = y$
$\blacksquare$