A preliminary study of MV-algebras with two quantifiers which commute. (English) Zbl 1357.06004
Summary: In this paper we investigate the class of MV-algebras equipped with two quantifiers which commute as a natural generalization of diagonal-free two-dimensional cylindric algebras (see [L. Henkin et al., Cylindric algebras. Part I. With an introductory chapter: General theory of algebras. 2nd printing. New York-Oxford: North-Holland (1985; Zbl 0576.03042); Cylindric algebras. Part II. Amsterdam-New York-Oxford: North-Holland (1985; Zbl 0576.03043)]). In the 40s, Tarski first introduced cylindric algebras in order to provide an algebraic apparatus for the study of classical predicate calculus. The diagonal-free two-dimensional cylindric algebras are special cylindric algebras. The treatment here of MV-algebras is done in terms of implication and negation. This allows us to simplify some results due to A. Di Nola and R. Grigolia [Ann. Pure Appl. Logic 128, No. 1–3, 125–139 (2004; Zbl 1052.06010)] related to the characterization of a quantifier in terms of some special sub-algebra associated to it. On the other hand, we present a topological duality for this class of algebras and we apply it to characterize the congruences of one algebra via certain closed sets. Finally, we study the subvariety of this class generated by a chain of
length \(n+1\) (\(n < \omega\)). We prove that the subvariety is semisimple and we characterize their simple algebras. Using a special functional algebra, we determine all the simple finite algebras of this subvariety.
MSC:
| 06D35 | MV-algebras |
| 03G25 | Other algebras related to logic |
| 08A30 | Subalgebras, congruence relations |
| 06D50 | Lattices and duality |
| 08C05 | Categories of algebras |
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