The aim of this note is to demonstrate that Kaplansky–Hilbert lattices and injective Banach lattices may be produced from each other by means of the well known convexification procedure. This is done via the Boolean valued analysis approach. The subject gives a good opportunity to discuss also the relationship between the Kantorovich’s heuristic principle and the Boolean value transfer principl...

A Π1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩C. Cholak, Coles, Downey and Herrmann [7] proved that a Π1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is the free Boolean algebra, then all such think classes are automorphic in the lattice of Π1 classes under i...

In [1] and [2] we have discussed the problem of existence of an independent set of generators for a countably generated filter in an atomless Boolean algebra and for any filter in a free Boolean algebra. In particular, we have proved that if F is a filter in a free Boolean algebra then F is freely generated provided the minimal cardinality of the set of generators of F is not a singular cardina...

0. Notation. The basic notation for Boolean algebras is the standard one (for instance, see [l]). Throughout this paper, the letter O will denote the two-element Boolean algebra; whenever a topology is assumed to be defined on 0, then this topology is the discrete one. If 73 is a Boolean algebra, then a valuation of B is a homomorphism from 73 onto 0. We make precise the terminology we shall be...

We present a groupoid which can be converted into a Boolean algebra with respect to term operations. Also conversely, every Boolean algebra can be reached in this way.

where (~xn) ∈ B and B is a Boolean algebra. Only finite Boolean algebras are considered. The Boolean logic presented is based on the abstract Boolean algebra given by Huntington’s [14] postulates and on a special canonical form for a Boolean function given by Archie Blake [2]. The theory of reasoning is divided into functional and general parts, both of which seek antecedents or consequents of ...

Every group is isomorphic to the automorphism group of a Kripke structure with Boolean part equal to a power set Boolean algebra. More generally, we prove that the category of Kripke structures with Boolean part equal to a power set Boolean algebra and morphisms with complete Boolean part is alg-universal, which means that it contains any category of universal algebras as a full subcategory.