Definition:Locale (Lattice Theory)
Jump to navigation
Jump to search
Definition
Let $\mathbf{Loc}$ denote the category of locales.
An object of $\mathbf{Loc}$ is called a locale.
That is, a locale is a complete lattice $\struct {L, \preceq}$ satisfying the infinite join distributive law:
\(\ds \forall a \in L, S \subseteq L:\) | \(\ds a \wedge \bigvee S = \bigvee \set {a \wedge s : S \in S} \) |
where $\bigvee S$ denotes the supremum $\sup S$.
Also see
Sources
- 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to Locales, $\S 1.1$ Definition (b)