## 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

### Keywords:

poset; completion; extension of maps; complete distributivity

### Citations:

Zbl 1317.06002; Zbl 1238.03051
Full Text: