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Z-Continuous Posets and Their Topological Manifestation

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Abstract

A subset selection Z assigns to each partially ordered set P a certain collection Z P of subsets. The theory of topological and of algebraic (i.e. finitary) closure spaces extends to the general Z-level, by replacing finite or directed sets, respectively, with arbitrary ‘Z-sets’. This leads to a theory of Z-union completeness, Z-arity, Z-soberness etc. Order-theoretical notions such as complete distributivity and continuity of lattices or posets extend to the general Z-setting as well. For example, we characterize Z-distributive posets and Z-continuous posets by certain homomorphism properties and adjunctions. It turns out that for arbitrary subset selections Z, a poset P is strongly Z-continuous iff its Z-join ideal completion Z∨ P is Z-ary and completely distributive. Using that characterization, we show that the category of strongly Z-continuous posets (with interpolation) is concretely isomorphic to the category of Z-ary Z-complete core spaces. For suitable subset selections Y and Z, these are precisely the Y-sober core spaces.

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  1. Institut für Mathematik, Universität Hannover, D-30167, Hannover, Germany

    Marcel Erné

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Erné, M. Z-Continuous Posets and Their Topological Manifestation. Applied Categorical Structures 7, 31–70 (1999). https://doi.org/10.1023/A:1008657800278

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