Miroslav Ploščica

Dual spaces of some congruence lattices

Abstract For any set X and any n ⩾3 we define a topological space L n (X) and characterize its closed subspaces if |X|⩽ℵ 1 . As an application we obtain a characterization of congruence lattices of algebras in some varieties of lattices. The spaces L n (X) are close to Boolean spaces, but they are not Hausdorff.

Affine complete distributive lattices

We prove a characterization theorem for affine complete distributive lattices. To do so we introduce the notions of almost principal ideal and almost principal.

Affine complete Stone algebras

In [1] R. Beazer characterized affine complete Stone algebras having a smallest dense element. We remove this latter assumption and describe affine complete algebras in the class of all Stone algebras.

Iterative separation in distributive congruence lattices

In [PLOŠČICA, M.: Separation in distributive congruence lattices, Algebra Universalis 49 (2003), 1–12] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in . Now we generalize these results using the concept of separable mappings (defined on some trees) and apply them to some lattice varieties.

Affine completions of distributive lattices

For any distributive lattice L we construct its extension ℑ(ı(L)) with the property that every isotone compatible function on L can be interpolated by a polynomial of ℑ(ı(L)). Further, we characterize all extensions with this property and show that our construction is in some sense the simplest possible.

Professor Ján Jakubík nonagenarian

This issue of Mathematica Slovaca is in honour of Prof. Ján Jakubík’s 90th birthday. We present here a brief account of his research (to date) and a personal sketch of the gentleman.

On a characterization of distributive lattices by the betweenness relation

We construct an example of a ternary structure satisfying certain conditions due to M. Kolibiar, which is not a betweenness relation of any lattice. This answers a question posed by J. Hedlíková and T. Katriňák.

Compact Intersection Property and description of congruence lattices

We say that a variety V of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every A ∈ V is closed under intersection. We investigate the congruence lattices of algebras in locally finite congruence-distributive CIP varieties. We prove some general results and obtain a complete characterization for some types of such varieties. We provide two kinds of description of congruence lattices: via direct limits and via Priestley duality.