Ordering of Inverses in Ordered Monoid

Theorem

Let $\struct {S, \circ, \preceq}$ be an ordered monoid whose identity is $e$.

Let $x, y \in S$ be invertible.

Then:

$x \prec y \iff y^{-1} \prec x^{-1}$

Proof

Necessary Condition

\(\ds x\)

\(\prec\)

\(\ds y\)

\(\ds \leadsto \ \ \)

\(\ds e\)

\(=\)

\(\ds x^{-1} \circ x \prec x^{-1} \circ y\)

Strict Ordering Preserved under Product with Cancellable Element

\(\ds \leadsto \ \ \)

\(\ds y^{-1}\)

\(=\)

\(\ds e \circ y^{-1} \prec x^{-1} \circ y \circ y^{-1} = x^{-1}\)

\(\ds \leadsto \ \ \)

\(\ds y^{-1}\)

\(\prec\)

\(\ds x^{-1}\)

$\Box$

Sufficient Condition

\(\ds y^{-1}\)

\(\prec\)

\(\ds x^{-1}\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds \paren {x^{-1} }^{-1} \prec \paren {y^{-1} }^{-1} = y\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\prec\)

\(\ds y\)

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups: Theorem $15.2$