\({\mathcal Z}\)-continuous posets and their topological manifestation. (English) Zbl 0939.06005

Erné, Marcel

Appl. Categ. Struct. 7, No. 1-2, 31-70 (1999).

A subset collection \({\mathcal Z}\) assigns to each poset \(P\) a certain collection \({\mathcal Z}P\) of subsets. The theory of topological or algebraic closure spaces extends to the general \({\mathcal Z}\)-level, by replacing directed or finite sets with arbitrary \({\mathcal Z}\)-sets and leads to a theory of \({\mathcal Z}\)-union completeness, \({\mathcal Z}\)-arity, \({\mathcal Z}\)-soberness etc.
The author shows that the category of strongly \({\mathcal Z}\)-continuous posets (with interpolation) is concretely isomorphic to the category of \({\mathcal Z}\)-ary \({\mathcal Z}\)-complete core spaces. His paper contains the theory of \({\mathcal Z}\)-continuous posets, the fundamental setting of subset selections, various connections between union completeness, \({\mathcal Z}\)-arity and generalized soberness, the algebraic theory of \({\mathcal Z}\)-distributivity and \({\mathcal Z}\)-continuous posets and some topological aspects.

Reviewer: B.F.Šmarda (Brno)

MSC:

06B35	Continuous lattices and posets, applications
54F05	Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54A05	Topological spaces and generalizations (closure spaces, etc.)
54H10	Topological representations of algebraic systems

Keywords:

closure space; completion; sober space; category of strongly \({\mathcal Z}\)-continuous posets; interpolation; \({\mathcal Z}\)-complete core spaces; subset selections; \({\mathcal Z}\)-arity; generalized soberness