Any MV-algebra M can be embedded as a lattice in the Boolean algebra B(M) that is R-generated by M . We relate the study of states on an MV-algebra M to the study of finitely additive probabilities on B(M). In particular, we show that each state on M can be uniquely extended to a finitely additive probability on B(M). In case that M is a PMV-algebra, the conditional state s(a|b) defined for a, ...

MV-algebras of various kinds have been heavily investigated in the recent times. The isomorphism theorems between the MV-algebras and the interval MV-algebras in the corresponding lattice-ordered algebraic structures support research that utilizes the established properties of these structures in order to obtain specific information about the initial MV-algebras. In this talk we will discuss a ...

Some well-known algebras of logic or other algebraic models of non-classical reasoning, such as MV-algebras or orthomodular lattices, can be regarded as lattices with antitone involutions, i.e., bounded lattices with the property that all principal filters (or ideals) are equipped with antitone involutions. This simple observation is behind the so-called “basic algebras” that were introduced in...

In this paper we will introduce the variety of state MV-algebras by enlarging the language of MV-algebras with an unary operator σ, and by adding equations ensuring σ to satisfy some basic properties of a state on an MV-algebra in the sense of [4]. The main result of this paper says us that there is a standard way to define a state MV-algebra starting from a state on an MV-algebra and vice-vers...

We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a "‡at" quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra o...

We further study the filter theory of pseudo-BL algebras.We give some equivalent conditions of filter, normal filter and Boolean filter. We introduce the notion of pseudo MV-filter, pseudo-G filter and characterize Boolean algebras, pseudo-MV algebras and pseudo Gödel algebras (i.e. Gödel algebras) in pseudo-BL algebras. We establish the connections between BCC-algebras, pseudo-BCK algebras, ps...

An MV-pair is a pair (B, G) where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain conditions. Let ∼G be the equivalence relation on B naturally associated with G. We prove that for every MV-pair (B, G), the effect algebra B/ ∼G is an MV-effect algebra. Moreover, for every MV-effect algebra M there is an MV-pair (B, G) such that M is isomorphic to B/ ∼G.

An MV-pair is a pair (B, G) where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain conditions. Let ∼G be the equivalence relation on B naturally associated with G. We prove that for every MV-pair (B, G), the effect algebra B/ ∼G is an MV-effect algebra. Moreover, for every MV-effect algebra M there is an MV-pair (B, G) such that M is isomorphic to B/ ∼G.