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

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
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