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\({\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}\)-Complete Partially Ordered Sets and Their Representations by \(\boldsymbol{\mathcal{Q}}\)-Spaces

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Abstract

The present paper proposes a general theory for \(\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) \)-complete partially ordered sets (alias \(\mathcal{Z} _{1}\)-join complete and \(\mathcal{Z}_{2}\)-meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections \(\mathcal{Z}_{i}\) (i = 1,...,4) and \(\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) \), the category \(\mathcal{Q}\) P of \(\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) \)-complete partially ordered sets and \(\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) \)-continuous (alias \(\mathcal{ Z}_{3}\)-join preserving and \(\mathcal{Z}_{4}\)-meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category \(\mathcal{Q}\) S of \(\mathcal{Q}\)-spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory \( \mathcal{Q}\) P s of \(\mathcal{Q}\) P of all \(\mathcal{Q}\)-spatial objects and the full subcategory \(\mathcal{Q}\) S s of \(\mathcal{Q}\) S of all \(\mathcal{Q}\)-sober objects. Here \(\mathcal{Q}\)-spatiality and \(\mathcal{Q}\)-sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to \(\mathcal{Z}\)-compact generation and \(\mathcal{Z}\)-sobriety have also been pointed out in this paper.

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  1. Faculty of Sciences, Department of Mathematics, Akdeniz University, 07058, Antalya, Turkey

    Mustafa Demirci

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Demirci, M. \({\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}\)-Complete Partially Ordered Sets and Their Representations by \(\boldsymbol{\mathcal{Q}}\)-Spaces. Appl Categor Struct 21, 703–723 (2013). https://doi.org/10.1007/s10485-012-9277-4

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