Dual of Ordered Semigroup is Ordered Semigroup

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Theorem

Let $\struct {S, \circ, \preccurlyeq}$ be an ordered semigroup.

Then its dual $\struct {S, \circ, \succcurlyeq}$ is also an ordered semigroup.


Proof

From Dual Ordering is Ordering, we have that $\struct {S, \succcurlyeq}$ is an ordered set.

We also note from the definition that $\struct {S, \circ}$ is a semigroup.


It remains to be demonstrated that $\succcurlyeq$ is compatible with $\circ$.


Recall that $\struct {S, \circ, \preccurlyeq}$ is an ordered semigroup.

Hence a fortiori $\preccurlyeq$ is compatible with $\circ$.


Let $x, y \in S$ be arbitrary such that $x \succcurlyeq y$.

We have:

\(\ds x\) \(\succcurlyeq\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds y\) \(\preccurlyeq\) \(\ds x\) Definition of Dual Ordering
\(\ds \leadsto \ \ \) \(\ds \paren {y \circ z}\) \(\preccurlyeq\) \(\ds \paren {x \circ z}\) Definition of Relation Compatible with Operation
\(\ds \leadsto \ \ \) \(\ds \paren {x \circ z}\) \(\succcurlyeq\) \(\ds \paren {y \circ z}\)

and similarly:

\(\ds x\) \(\succcurlyeq\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds y\) \(\preccurlyeq\) \(\ds x\) Definition of Dual Ordering
\(\ds \leadsto \ \ \) \(\ds \paren {z \circ y}\) \(\preccurlyeq\) \(\ds \paren {z \circ x}\) Definition of Relation Compatible with Operation
\(\ds \leadsto \ \ \) \(\ds \paren {z \circ x}\) \(\succcurlyeq\) \(\ds \paren {z \circ y}\)

Hence the result.

$\blacksquare$


Sources

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