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.


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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