The function lattice, or generalized Boolean algebra, is the set of `-tuples with the ith coordinate an integer between 0 and a bound ni. Two `-tuples t-intersect if they have at least t common nonzero coordinates. We prove a Hilton–Milner type theorem for systems of t-intersecting `-tuples.

If κ is an infinite cardinal, a complete Boolean algebra B is called κ-supported if and only if for each sequence 〈bβ : β < κ〉 of elements of B there holds V α<κ W β>α bβ = W A∈[κ]κ V β∈A bβ. We consider the problem of the existence of a non-atomic complete Boolean algebra B which is h2(B)-supported, where h2(B) is the minimal cardinal κ such that B is not (κ, 2)-distributive.

1. Introduction. Let U be a Boolean algebra of a t least two elements .

For every regular cardinal κ there exists a simple complete Boolean algebra with κ generators.

For every regular cardinal κ there exists a simple complete Boolean algebra with κ generators.

Double Boolean algebras are algebras (D,u,t, , ,⊥,>) of type (2, 2, 1, 1, 0, 0). They have been introduced to capture the equational theory of the algebra of protoconcepts. A filter (resp. an ideal) of a double Boolean algebra D is an upper set F (resp. down set I) closed under u (resp. t). A filter F is called primary if F 6= ∅ and for all x ∈ D we have x ∈ F or x ∈ F . In this note we prove t...

We study the set of depths of relative algebras of countable Boolean algebras, in particular the extent to which this set may not be downward closed within the countable ordinals for a fixed countable Boolean algebra. Doing so, we exhibit a structural difference between the class of arbitrary rank countable Boolean algebras and the class of rank one countable Boolean algebras.

In this article we consider homogeneity properties of Boolean algebras that have nonprincipal ultrafilters which are countably generated. It is shown that a Boolean algebra B is homogeneous if it is the union of countably generated nonprincipal ultrafilters and has a dense subset D such that for every a ∈ D the relative algebra B 1 a := {b ∈ B : b ≤ a} is isomorphic to B. In particular, the fre...