Quasi-decompositions and quasidirect products of Hilbert algebras

A quasi-decomposition of a Hilbert algebra A is a pair (C, D) of its subalgebras such that (i) every element a ∈ A is a meet c ∧ d with c ∈ C, d ∈ D, where c and d are compatible (i.e., c → d = c → (c ∧ d)), and (ii) d → c = c (then c is uniquely defined). Quasi-decompositions are intimately related to the so-called triple construction of Hilbert algebras, which we reinterpret as a construction of quasidirect products. We show that it can be viewed as a generalization of the semidirect product construction, that quasidirect products has a certain universal property and that they can be characterised in terms of short exact sequences. We also discuss four classes of Hilbert algebras and give for each of them conditions on a quasi-decomposition of an arbitrary Hilbert algebra A under which A belongs to this class.

Dualities for Bounded Prelinear Hilbert Algebras

This paper deals about dualities for bounded prelinear Hilbert algebras. In particular, we give an Esakia-style duality between the algebraic category of bounded prelinear Hilbert algebras and a category of H-spaces whose morphisms are certain continuous p-morphisms.

Fuzzy filters of Sheffer stroke Hilbert algebras

The aim of this study is to introduce fuzzy filters of Sheffer stroke Hilbert algebra. After defining fuzzy filters of Sheffer stroke Hilbert algebra, it is shown that a quotient structure of this algebra is described by its fuzzy filter. In addition to this, the level filter of a Sheffer stroke Hilbert algebra is determined by its fuzzy filter. Some fuzzy filters of a Sheffer stroke Hilbert algebra are defined by a homomorphism. Normal and maximal fuzzy filters of a Sheffer stroke Hilbert algebra and the relation between them are presented. By giving the Cartesian product of fuzzy filters of a Sheffer stroke Hilbert algebra, various properties are examined.

On Fuzzy Deductive Systems of Hilbert Algebras

In this paper, we study fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law. Several characterizations are obtained for fuzzy deductive systems generated by a fuzzy set. It is also proved that the class of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system. Furthermore, we obtain a lattice isomorphism between the class of fuzzy deductive systems and the class of fuzzy congruence relations in the variety of Hilbert algebras.

On GE-Algebras

Hilbert algebras are important tools for certain investigations in intuitionistic logic and other non-classical logic and as a generalization of Hilbert algebra a new algebraic structure, called a GE-algebra (generalized exchange algebra), is introduced and studied its properties. We consider filters, upper sets and congruence kernels in a GE-algebra. We also characterize congruence kernels of transitive GE-algebras.