Filters of a lattice, maximal filters (ultrafilters), operation to create a filter generating by an element or by a nonempty set of elements of the lattice are discussed. Besides, there are introduced implicative lattices such that for every two elements there is an element being pseudo-complement of them. Some facts concerning these concepts are presented too, i.e. for any proper filter there ...

We discuss some notions of compactness relative to a specified family F of subsets of some topological space X . In particular, we relativize to F the notions ofD-compactness, CAPλ, and [μ, λ]-compactness. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in which F is the family of all singletons of X , in which case we get back the mor...

The functor taking global elements of Boolean algebras in the topos ShB of sheaves on a complete Boolean algebra B is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in B-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial...

In the paper we show equivalence of the convergence of filters on a topological space and the convergence of nets in the space. We also give, five characterizations of compactness. Namely, for any topological space T we proved that following condition are equivalent: • T is compact, • every ultrafilter on T is convergent, • every proper filter on T has cluster point, • every net in T has cluste...

Applying Weglorz’ models of set theory without the axiom of choice, we investigate Arrow-type social welfare functions for infinite societies with restricted coalition algebras. We show that there is a reasonable, nondictatorial social welfare function satisfying “finite discrimination”, if and only if in Weglorz’ model there is a free ultrafilter on a set representing the individuals. AMS-subj...

We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a p-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of p-points and selective ultrafilters. Results fall into three main categories: comparison to a basis element for selective ultrafilters, ...

Filters of a lattice, maximal filters (ultrafilters), operation to create a filter generating by an element or by a non-empty subset of the lattice are discussed. Besides, there are introduced implicative lattices such that for every two elements there is an element being pseudo-complement of them. Some facts concerning these concepts are presented too, i.e. for any proper filter there exists a...

We continue our study of the class C (D), where D is a uniform ultrafilter on a cardinal κ and C (D) is the class of all pairs (θ1, θ2), where (θ1, θ2) is the cofinality of a cut in Jκ/D and J is some (θ1 + θ2)-saturated dense linear order. We give a combinatorial characterization of the class C (D). We also show that if (θ1, θ2) ∈ C (D) and D is א1-complete or θ1 + θ2 > 2κ, then θ1 = θ2.

In the present report, we study collections of contact relations on a fixed Boolean algebra and show that they can be provided with a rich lattice structure. This follows from a representation theorem which associates with each contact relation a closed graph on the ultrafilter space of the underlying Boolean algebra and vice versa. We also consider collections of special contact relations whic...