Ordering Induced by Join Semilattice

Theorem
Let $\left({S, \vee, \preceq}\right)$ be a join semilattice.

By Join Semilattice is Semilattice, $\left({S, \vee}\right)$ is a semilattice.

By Semilattice Induces Ordering, $\left({S, \vee}\right)$ induces an ordering $\preceq'$ on $S$, by:


 * $a \preceq' b$ iff $a \vee b = b$

for all $a, b \in S$.

The ordering $\preceq'$ coincides with the original ordering $\preceq$.

Proof
It is to be shown that, for all $a, b \in S$:


 * $a \preceq b$ iff $b = \sup \left\{{a, b}\right\}$

by definition of join.

Here $\sup$ denotes supremum.

Since any upper bound $c$ of $\left\{{a, b}\right\}$ must satisfy:


 * $b \preceq c$

it suffices to verify that:


 * $a \preceq b$ iff $b$ is an upper bound for $\left\{{a, b}\right\}$

Since $\preceq$ is reflexive, we know that:


 * $b \preceq b$

and therefore said equivalence is established.

We conclude that $\preceq'$ and $\preceq$ coincide.