Complete Join Semilattice is Dual to Complete Meet Semilattice

Theorem

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

The following are dual statements:

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

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

Proof

By definition of complete join semilattice:

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

if and only if:

$\forall S' \subseteq S : \sup S' \in S$, where $\sup S'$ is the supremum of $S'$

The dual of this statement is:

$\forall S' \subseteq S : \inf S' \in S$, where $\inf S'$ is the infimum of $S'$ by Dual Pairs (Order Theory).

By definition of complete meet semilattice:

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

$\blacksquare$

Also see

• Duality Principle (Order Theory)