Representation Of MV-algebras Research Articles

Representation of MV-algebras by regular ultrapowers of [0, 1

We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real interval [0,1]. This result also implies the existence, for any cardinal α, of a single MV-algebra in which all infinite MV-algebras of cardinality at most α embed. Recasting the above construction with iterated ultrapowers, we show how to construct such an algebra of values in a definable way, thus providing a sort of “canonical” set of values for the functional representation.

Local algebras in the representation of MV-algebras

In this paper we present several results about local MV-algebras, extending existing results given for MV-chains. The role of local MV-algebras in sheaf representation and weak boolean product is stressed and the relationship of local MV-algebras with varieties of MV-algebras is analyzed.

Prime spectra of non-commutative generalizations of MV-algebras

Non-commutative generalizations of MV-algebras were introduced by G. Georgescu and A. Iorgulesco as well as by the author; the generalizations are equivalent and are called GMV-algebras. We show that GMV-algebras can be considered as special cases of Grishin algebras. As MV-algebras are algebraic models of the Łukasiewicz logic and Grishin algebras have the analogous role for the classical bilinear logic, GMV-algebras correspond to a non-commutative logic between the above logics. Further, by A. Dvurecenskij, any GMV-algebra is isomorphic to an interval of an l-group, which in general is not commutative. This generalizes D. Mundici's representation of MV-algebras by means of intervals of abelian l-groups. In the paper (using this representation) we describe the properties of prime ideal spectra of GMV-algebras and of their factor algebras and ideals and prove that the spectrum of closed ideals of any GMV-algebra is homeomorphic to that of a completely distributive GMV-algebra.

A Common Abstraction of MV-Algebras and Abelian l-groups

We investigate the class of strongly distributive pregroups, a common abstraction of MV-algebras and Abelian l-groups which was introduced by E.Casari. The main result of the paper is a representation theorem which yields both Chang's representation of MV-algebras and Clifford's representation of Abelian l-groups as immediate corollaries.