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 be a complete lattice. An element is compact iff any of the following equivalent conditions hold:
(i) for every , such that , there is a finite subset such that
(ii) for every directed subset such that there is with .
A frame is algebraic if every element of 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.
is coherent if it is algebraic and the finite inf of a set of compact elements is compact.
Theorem. A frame is coherent iff 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.
K. I. Rosenthal, Quantales and their applications, Longman 1990
Last revised on October 10, 2023 at 11:18:53. See the history of this page for a list of all contributions to it.