Definition:Complete Join Semilattice

Definition

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

Then $\struct {S, \preceq}$ is a complete join semilattice if and only if:

$\forall S' \subseteq S: \sup S' \in S$

That is, if and only if all subsets of $S$ have a supremum.

Also see

• Definition:Complete Meet Semilattice

• Definition:Complete Lattice

• Equivalence of Complete Semilattice and Complete Lattice

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text I$: Preliminaries, $\S4.3$