Ordering Induced by Join Semilattice

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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.

$\blacksquare$