The Priestley duality for Wajsberg algebras is developed. The Wajsberg space is a De Morgan space endowed with a family of functions that are obtained in rather natural way.As a first application of this duality, a theorem about unicity of the structure is given.

In this paper, the Boolean representation of R0-algebras are investigated. In particular, we show that directly indecomposable R0-algebras are equivalent to local R0-algebras and any nontrivial R0-algebra is representable as a weak Boolean product of local R0-algebras.

We give a presentation of Post algebras of ordern+1 (n1) asn+1 bounded Wajsberg algebras with an additional constant, and we show that a Wajsberg algebra admits a P-algebra reduct if and only if it isn+1 bounded.

In this paper we show that the prime ideal space of an MV-algebra is the disjoint union of prime ideal spaces of suitable local MV-algebras. Some special classes of algebras are defined and their spaces are investigated. The space of minimal prime ideals is studied as well.

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate (...) for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens. (shrink)

The infinite-valued logic of ukasiewicz was originally defined by means of an infinite-valued matrix. ukasiewicz took special forms of negation and implication as basic connectives and proposed an axiom system that he conjectured would be sufficient to derive the valid formulas of the logic; this was eventually verified by M. Wajsberg. The algebraic counterparts of this logic have become know as Wajsberg algebras. In this paper we show that a Wajsberg algebra is complete and atomic (as a lattice) if and (...) only if it is a direct product of finite Wajsberg chains. The classical characterization of complete and atomic Boolean algebras as fields of sets is a particular case of this result. (shrink)