Completely distributive completions of posets

Abstract

A $$\Delta_1$$ -completion of a poset is a completion for which, simultaneously, every element is reachable as a join of meets and a meet of joins from the original poset. We focus our attention on $$\Delta_1$$ -completions that can be obtained from polarities $$\langle\mathcal{F},\mathcal{I},\mathcal{R}\rangle$$ where $$\mathcal{F}$$ is a collection of upsets containing the principal upsets, $$\mathcal{I}$$ is a collection of downsets containing the principal downsets of the original poset, and $$R\subseteq\mathcal{F}\times\mathcal{I}$$ is the relation of nonempty intersection. These $$\Delta_1$$ -completions are called $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions, and they satisfy a compactness property. In this paper, we show that if a pair $$\langle\mathcal{F},\mathcal{I}\rangle$$ satisfies a separating condition (similar to the Prime Filter Theorem for distributive lattices), then the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of the original poset is a completely distributive algebraic lattice. Given a poset P and an algebraic closure system $$\mathcal{F}$$ of upsets of P satisfying a distributivity condition, we show how to choose a collection of downsets $$\mathcal{I}$$ of P such that the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of P is a completely distributive algebraic lattice. Then, we study the extensions of additional operations on posets to their corresponding $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions. Finally, we use the previous results to obtain adequate $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions for the classes of Tarski algebras and Hilbert algebras, and for the classes of algebras that are canonically associated (in the sense of abstract algebraic logic) with some propositional logics.