Revaz Grigolia

On monadic MV-algebras

Abstract We define and study monadic MV -algebras as pairs of MV -algebras one of which is a special case of relatively complete subalgebra named m -relatively complete. An m -relatively complete subalgebra determines a unique monadic operator. A necessary and sufficient condition is given for a subalgebra to be m -relatively complete. A description of the free cyclic monadic MV -algebra is also given.

Finitely generated free MV-algebras and their automorphism groups

The MV-algebra Smw is obtained from the (m+1)-valued Łukasiewicz chain by adding infinitesimals, in the same way as Changs algebra is obtained from the two-valued chain. These algebras were introduced by Komori in his study of varieties of MV-algebras. In this paper we describe the finitely generated totally ordered algebras in the variety MVmw generated by Smw. This yields an easy description of the free MVmw-algebras over one generator. We characterize the automorphism groups of the free MV-algebras over finitely many generators.

Representations of monadic MV -algebras

Representations of monadic MV -algebra, the characterization of locally finite monadic MV -algebras, with axiomatization of them, definability of non-trivial monadic operators on finitely generated free MV -algebras are given. Moreover, it is shown that finitely generated m-relatively complete subalgebra of finitely generated free MV -algebra is projective.

MV-algebras in duality with labelled root systems

The category of labelled root systems is defined which is dually equivalent to a certain category of finitely generated MV(C)-algebras with finite spectrum, i.e. MV-algebras (with finite spectrum) from the variety generated by perfect MV-algebras.

Gödel spaces and perfect MV-algebras

The category of Godel spaces GS (with strongly isotone maps as morphisms), which are dually equivalent to the category of Godel algebras, is transferred by a contravariant functor H into the category M V ( C ) G of MV-algebras generated by perfect MV-chains via the operators of direct products, subalgebras and direct limits. Conversely, the category M V ( C ) G is transferred into the category GS by means of a contravariant functor P . Moreover, it is shown that the functor H is faithful, the functor P is full and the both functors are dense. The description of finite coproduct of algebras, which are isomorphic to Chang algebra, is given. Using duality a characterization of projective algebras in M V ( C ) G is given.

Local MV-Algebras

Local MV-algebras are MV-algebras with only one maximal ideal that, hence, contains all infinitesimal elements.

Pro-finite MV-spaces

Abstract In this work it is shown that every MV-space, i.e. the prime ideal space of an MV-algebra, is pro-finite if and only if it is a completely normal dual Heyting space. An example is given showing that MV-spaces and completely normal spectral spaces are not pro-finite.

Formulas of one propositional variable in intuitionistic logic with the Solovay modality

A description of the free cyclic algebra over the variety of Solovay algebras, as well as over its pyramid locally finite subvarieties is given.

Projective MV-algebras

A characterization of finitely generated projective MV-algebras is given, introducing the new notions of correct partitions and e-relatively complete subalgebras.

Finiteness and Duality in MV-algebras Theory

Some results about finiteness properties of MV-algebras and some dualities between categories of MV-algebras and categories of certain ordered structures are presented. Actually, finite MV-algebras are presented as algebras of words. Moreover, it is presented a duality between the category of MV-algebras which are finitely generated, having finite spectrum, and the category of finite linear dual Heyting algebras.