We prove a “supersaturation-type” extension of both Sperner’s Theorem (1928) and its generalization by Erdős (1945) to k-chains. Our result implies that a largest family whose size is x more than the size of a largest k-chain free family and that contains the minimum number of k-chains is the family formed by taking themiddle (k−1) rows of the Boolean lattice and x elements from the kthmiddle r...

Using the concept of the λ-lattice introduced recently by V. Snášel we define λ-lattices with antitone involutions. For them we establish a correspondence to ring-like structures similarly as it was done for ortholattices and pseudorings, for Boolean algebras and Boolean rings or for lattices with an antitone involution and the so-called Boolean quasirings.

In Part I of this paper, we introduced a method of making two isomorphic intervals of a bounded lattice congruence equivalent. In this paper, we make one interval dominate another one. Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let φ be a homomorphism of [a, b] onto [c, d]. We construct a bounded (convex) extension K of L such that a congruence Θ of L has an extens...

We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with intersections is dual to the category of surjective étale maps between Boolean spaces. We then extend the duality to skew Boolean algebras with in...

In this paper we present a comprehensive approach to automatic organization and hybrid navigation of text databases. An organizing stage first builds a particular lattice representation of the data, through text indexing followed by lattice clustering of the indexed texts. The lattice representation, then, supports the navigation stage of the system, a visual retrieval interface that combines t...

The problem of imbedding an abstract distributive lattice in a Boolean algebra by an algebraic extension was suggested to the writer by M. H. Stone in 1933. Hausdorfff had already given a solution of this problem for the case where the given distributive lattice was a ring of point sets. A solution for the abstract case was presented by the writer to the Harvard Mathematical Colloquium (1934), ...

Most work to date on Boolean approximation assumes that Boolean functions are represented by formulas in conjunctive normal form. That assumption is appropriate for the classical applications of Boolean approximation but potentially limits wider use. We revisit, in a lattice-theoretic setting, so-called envelopes and cores in propositional logic, identifying them with upper and lower closure op...

We prove that a Boolean algebra is countable iff its subalgebra lattice admits a continuous complementation.

The study of the lattice E of computably enumerable sets under inclusion has been one of the central tasks of computability theory since the 1960's. We investigate initial segments of the lattice L of 0 1 classes (of sets) under inclusion and we compare this lattice with E. It was recently proved by Nies that the theory of each interval of the lattice E which is not a boolean algebra interprets...

Let L be a bounded distributive lattice. For k 1, let Sk (L) be the lattice of k-ary functions on L with the congruence substitution property (Boolean functions); let S(L) be the lattice of all Boolean functions. The lattices that can arise as Sk (L) or S(L) for some bounded distributive lattice L are characterized in terms of their Priestley spaces of prime ideals. For bounded distributive lat...