Ordering Induced by Join Semilattice

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Let $\struct {S, \vee, \preceq}$ be a join semilattice.

By Join Semilattice is Semilattice, $\struct {S, \vee}$ is a semilattice.

By Semilattice Induces Ordering, $\struct {S, \vee}$ induces an ordering $\preceq'$ on $S$, by:

$a \preceq' b$ if and only if $a \vee b = b$

for all $a, b \in S$.

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


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

$a \preceq b$ if and only if $b = \sup \set {a, b}$

by definition of join.

Here $\sup$ denotes supremum.

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

$b \preceq c$

it suffices to verify that:

$a \preceq b$ if and only if $b$ is an upper bound for $\set {a, b}$

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

$b \preceq b$

and therefore said equivalence is established.

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