Upper Bound is Lower Bound for Inverse Ordering

Definition

Let $\left({S, \preceq}\right)$ be a poset.

Let $T \subseteq S$.

Let $M$ be an upper bound for $\left({T, \preceq}\right)$.

Let $\succeq$ be the inverse of $\preceq$.

Then $M$ is a lower bound for $\left({T, \succeq}\right)$.

Corollary

Let $m$ be a lower bound for $\left({T, \preceq}\right)$.

Then $m$ is an upper bound for $\left({T, \succeq}\right)$.

Proof

Let $M$ be an upper bound for $\left({T, \preceq}\right)$.

We have from Inverse of Ordering is Ordering that $\succeq$ is also an ordering.

That is:

$\forall a \in T: a \preceq M$

By definition of inverse relation, it follows that:

$\forall a \in T: M \succeq a$

That is, $M$ is a lower bound for $\left({T, \succeq}\right)$.

$\blacksquare$

Proof of Corollary

We have that $\succeq$ is an ordering whose inverse is $\preceq$.

The result follows by reversing the argument of the main proof.

$\blacksquare$