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Completely distributive completions of posets. (English) Zbl 1438.06011

In a poset \(P\), let \(\mathcal{F}\) stand for a collection of upsets containing all principal upsets and \(\mathcal{I}\), for a collection of downsets containing all principal downsets. The author deals with \((\mathcal{F}, \mathcal{I})\)-completions in the sense of M. Gehrke et al. [Order 30, No. 1, 39–64 (2013; Zbl 1317.06002)]. The results presented in the paper under review are organized in several groups.
(1) If the pair \((\mathcal{F}, \mathcal{I})\) has a certain separation property (P) (for every \(F \in \mathcal{F}\) and every \(I \in \mathcal{I}\), if \( \mathcal{F} \cap \mathcal{I} = \emptyset\), then there exists \( \mathcal{H} \in \mathcal{F}\) with \(\ \mathcal{F}^c \in \mathcal{I}\) such that \( \mathcal{F} \subseteq \mathcal{H}\) and \( \mathcal{H} \cap \mathcal{I} = \emptyset)\), then the \((\mathcal{F}, \mathcal{I})\)-completion of \(P\) is a completely distributive algebraic lattice (c.d.a.l., for short).
(2) It is shown how, for \(\mathcal{F}\) a distributive lattice, choose \(\mathcal{I}\) to force the \((\mathcal{F}, \mathcal{I})\)-completion of \(P\) to be a c.d.a.l. In this way, the canonical extension of a distributive meet semilattice is demonstrated to be such a lattice.
(3) Studied are also extensions of additional \(n\)-ary operations on \(P\) to their corresponding \((\mathcal{F}, \mathcal{I})\)-completions.
(4) The previous results are used to obtain appropriate \((\mathcal{F}, \mathcal{I})\)-completions for Tarski algebras, for Hilbert algebras, and for algebras that are associated with filter distributive finitary congruential logics. For the latter case, such extensions, known as the \(\mathcal{S}\)-canonical extensions, have already been defined and investigated by M. Gehrke et al. [Ann. Pure Appl. Logic 161, No. 12, 1502–1519 (2010; Zbl 1238.03051)]. The approach developed in the paper under review, though also involving abstract algebraic logic, seems to be more straightforward.

MSC:

06B23 Complete lattices, completions
06A06 Partial orders, general
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D10 Complete distributivity
03G25 Other algebras related to logic
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