nLab compact element in a lattice

Idea

This notion is a variant of a notion of a compact space (every cover has a finite subcover) and more specifically a special case of compact element for orders. However this entry is written in parallel and with a view toward another generalization – compact elements in quantales.

Definition

Let $L$ be a complete lattice. An element $c\in L$ is compact iff any of the following equivalent conditions hold:

(i) for every $S\subset L$, such that $c\leq sup(S)$, there is a finite subset $F\subset S$ such that $c\leq sup(F)$

(ii) for every directed subset $S\subset L$ such that $c\leq S$ there is $s\in S$ with $c\leq s$.

Frames with properties concerning compact elements

A frame $L$ is algebraic if every element of $L$ is the sup of some set of compact elements. (Cf. also entry algebraic lattice). In that case, one can take the set of all elements below or equal the given element. The latter definition is accepted also for quantales.

$L$ is coherent if it is algebraic and the finite inf of a set of compact elements is compact.

Theorem. A frame $L$ is coherent iff $L$ is isomorphic to the frame of ideals in some distributive lattice.

Literature

A quantale is precoherent if it is algebraic and binary products of compact elements are compact. Precoherent quantale is coherent if the truth is a compact element.

• K. I. Rosenthal, Quantales and their applications, Longman 1990

• Peter Johnstone, Stone spaces