Ideal and MacNeille completions of subordination algebras

$\mathsf{S5}$-subordination algebras were recently introduced as a generalization of de Vries algebras, and it was proved that the category $\mathsf{SubS5^S}$ of $\mathsf{S5}$-subordination algebras and compatible subordination relations between them is equivalent to the category of compact Hausdorff spaces and closed relations. We generalize MacNeille completions of boolean algebras to the setting of $\mathsf{S5}$-subordination algebras, and utilize the relational nature of the morphisms in $\mathsf{SubS5^S}$ to prove that the MacNeille completion functor establishes an equivalence between $\mathsf{SubS5^S}$ and its full subcategory consisting of de Vries algebras. We also generalize ideal completions of boolean algebras to the setting of $\mathsf{S5}$-subordination algebras and prove that the ideal completion functor establishes a dual equivalence between $\mathsf{SubS5^S}$ and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stone-like dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of $\mathsf{SubS5^S}$ corresponding to continuous relations and continuous functions between compact Hausdorff spaces

Partial orderings with the weak Freese-Nation property

A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping f:P ---> [P]^{<= aleph_0} such that, for any a, b in P, if a <= b then there exists c in f(a) cap f(b) such that a <= c <= b. In this note, we study the WFN and some of its generalizations. Some features of the class of BAs with the WFN seem to be quite sensitive to additional axioms of set theory: e.g., under CH, every ccc cBA has this property while, under b >= aleph_2, there exists no cBA with the WFN