Axiom:Infinite Join Distributive Law

Definition

Let $\struct {L, \preceq}$ be a complete lattice.

$\struct {L, \preceq}$ is a frame if and only if $\struct {L, \preceq}$ satisfies the axiom:

\(\ds \forall a \in L, S \subseteq L:\)

\(\ds a \wedge \bigvee S = \bigvee \set {a \wedge s : S \in S} \)

where $\bigvee S$ denotes the supremum $\sup S$.

This criterion is called the infinite join distributive law.

Also see

• Definition:Frame (Lattice Theory)

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to Locales, $\S 1.1$