Join Semilattice is Dual to Meet Semilattice

Theorem

Let $\struct {S, \preceq}$ be an ordered set.

The following are dual statements:

$\struct{S, \vee, \preceq}$ is a join semilattice, where $\vee$ is the join for all $a,b \in S$

$\struct{S, \wedge, \preceq}$ is a meet semilattice, where $\wedge$ is the meet for all $a,b \in S$

Proof

By definition of join semilattice:

$\struct{S, \vee, \preceq}$ is a join semilattice

if and only if:

$\forall a,b \in S : a \vee b \in S$, where $a \vee b$ is the join of $a$ and $b$

The dual of this statement is:

$\forall a,b \in S : a \wedge b \in S$, where $a \wedge b$ is the meet of $a$ and $b$

by Dual Pairs (Order Theory).

By definition of meet semilattice:

$\struct{S, \wedge, \preceq}$ is a meet semilattice

$\blacksquare$

Also see

• Duality Principle (Order Theory)