compact element in a lattice

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.

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

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

(ii) for every directed subset SLS\subset L such that cSc\leq S there is sSs\in S with csc\leq s

A frame LL is algebraic if every element of LL 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.

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

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

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.

Last revised on April 10, 2014 at 02:00:49. See the history of this page for a list of all contributions to it.