Ramaz Liparteliani

MV -algebras are the algebraic counterpart of the infinite valued Łukasiewicz sentential calculus, as Boolean algebras are with respect to the classical propositional logic. As it is well known, MV - algebras form a category which is equivalent to the category of abelian lattice ordered groups (ℓ-groups, for short) with strong unit. It is known that any subvariety of MV -algebras is generated by finitely many algebras. Notice that the free algebras over the subvarieties of MV -algebras have been described functionally. Finitely generated free MVn-algebras (that correspond to n-valued Lukasiewicz logic) was described algebraically. In this paper we give an algebraic description of finitely many generated free MV -algebras in the variety V(S₁^ω) generated by the algebra S₁^ω (denoted as C) that was introduced by Chang and later by Komori. Moreover, we give ordered spectral spaces of the free algebras. In this work it is represented the free algebras by means of subdirect product of the finite family of chains. 

In this work we deal with algebraic counterparts of expansions of Lukasiewicz logic enriched with finite number of truth constants. Denote these varieties by MVmSn. We show that these varieties contain non-trivial minimal subvariety generated by finite linearly ordered algebra which is functionally equivalent to Post algebra. The analysis and characterizations of appropriate varieties and corresponding logical systems are given. Free and Projective algebras are studied in these varieties as well as projective formulas and unification problems. We give two kinds of representations of MVmSn-functions. One is constructive algorithm for construction of conjunctive and disjunctive normal forms for these polynomials. Second is representation of these polynomials with McNaughtonstyle functions. Our main considerations are unification problems in the corresponding varieties. We study it through projectivity. Connections between projective algebras and projective formulas have been established. Unification types of these varieties are unitary. One of major parts of our work is the construction of unification algorithms for finding the most general unifiers for MVmSn-formulas.

In this work we deal with algebraic counterparts of finitely valued Lukasiewicz logic enriched with finite number of truth constants. Specifically we show that these varieties contain nontrivial minimal subvarieties generated by finite linearly ordered algebra which is functionally equivalent to Post algebra, give analysis and characterization of appropriate varieties and corresponding logical systems, Free and Projective algebras in these varieties as well as projective formulas. Unification problems and algorithm will be discussed.

In this work we deal with algebraic counterparts of Lukasiewicz logic enriched by finite number of truth constants. A propositional many-valued logical system which turned out to be equivalent to the expansion of Lukasiewicz Logic L by adding into the language a truth-constant r for each real r (0, 1), together with a number of additional axioms was proposed by Pavelk. Many authors have been studied many-valued logics enriched by truth constants with respect to their relationship to other parts of mathematics, as well as to various structures. We investigate the varieties of algebras, corresponding to of Lukasiewicz logic enriched by finite number of truth constants. Specifically we show that these varieties contain non-trivial minimal subvariety generated by finite linearly ordered algebra which is functionally equivalent to Post algebra.

A new logic - Lukasiewicz logic enriched with constant connective, and corresponding to it variety of algebras, is introduced. The unification problems are analyzed for the new logic. It is shown that the logic has finitary unification type.

Doctoral Thesis Referee

Master Theses Supervisor

Doctoral Thesis Supervisor/Co-supervisor

Scientific editor of monographs in foreign languages

Scientific editor of a monograph in Georgian

Editor-in-Chief of a peer-reviewed or professional journal / proceedings

Review of a scientific professional journal / proceedings

Member of the editorial board of a peer-reviewed scientific or professional journal / proceedings

Participation in a project / grant funded by an international organization

Participation in a project / grant funded from the state budget

Patent authorship

Membership of the Georgian National Academy of Science or Georgian Academy of Agricultural Sciences

Membership of an international professional organization

Membership of the Conference Organizing / Program Committee

National Award / Sectoral Award, Order, Medal, etc.

Honorary title

Monograph

Handbook

Research articles in high impact factor and local Scientific Journals

Publication in Scientific Conference Proceedings Indexed in Web of Science and Scopus