The algebraic theory of quasi-MV algebras, generalizations of MV algebras arising in quantum computation, is by now rather well-developed. Although it is possible to define several interesting logics from these structures, so far this aspect has not been investigated. The present article aims at filling this gap.

In this paper we prove algebraic equivalents of the conservative extension theorems of Lukasiewicz propositional logic to rational Pavelka propositional logic. The results concerns the conservative extension and the very conservative extension of MV-algebras to Pavelka algebras.

By studying two unsharp quantum structures, namely extended lattice ordered effect algebras and lattice ordered QMV algebras, we obtain some characteristic theorems of MV algebras. We go on to discuss automata theory based on these two unsharp quantum structures. In particular, we prove that an extended lattice ordered effect algebra (or a lattice ordered QMV algebra) is an MV algebra if and on...

We show that the theory of MV-algebras is Morita-equivalent to that of abelian `-groups with strong unit. This generalizes the well-known equivalence between the categories of set-based models of the two theories established by D. Mundici in 1986, and allows to transfer properties and results across them by using the methods of topos theory. We discuss several applications, including a sheaf-th...

This material is studying MV-algebras and BL-algebras in terms of applicability in Boole algebra. Boole algebras are expanded use in fuzzy logic.

In this paper we enlarge the language of MV-algebras by a unary operation r equationally described so as to preserve the basic properties of a state in its original meaning. The resulting class of algebras will be called MV-algebras with internal state (or SMV-algebras for short). After discussing some basic algebraic properties of SMV-algebras, we apply them to the study of the coherence probl...

Motivated by the concept of quantifier (in the sense of P. Halmos) on different algebraic structures (Boolean algebras, Heyting algebras, MV-algebras, orthomodular lattices, bounded distributive lattices) and the resulting notion of monadic algebra, the paper introduces the concept of a monadic quantale algebra, considers its properties and provides several representation theorems for the new s...

Effect algebras have been introduced by Foulis and Bennett [2] (see also [3, 4] for equivalent definitions) for modeling unsharp measurements in quantum mechanical systems [5]. They are a generalization of many structures which arise in the axiomatization of quantum mechanics (Hilbert space effects [7]), noncommutative measure theory and probability (orthomodular lattices and posets, [6]), fuzz...

We develop alternative semantics for à Lukasiewicz logic and for cancellative hoop logic according to the following idea. We formalize statements reflecting an inexact knowledge of certain (sharp) properties; we assume that all what can be known about a property is its expressive strength. To this end, we consider a Boolean algebra endowed with an automorphism group or, alternatively, with a me...

1 Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia E-mail: [email protected] In the frames of quantum structures, states are a very important notion that model probability on an algebraic structure. Already G. Boole observed that to calculate probability of event of the structure M , it is important to know only which two events A and B we c...