22/06/2016 – Aldo Figallo

A preliminary study of MV-algebras with two quantifiers which commute

Aldo Figallo Orellano ( Dep. de Mat.- Universidad Nacional del Sur and CLE-UNICAMP)

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 [2]). In the 40’s, 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 Di Nola and Grigolia in [1] related to the characterization of a quantifier in terms of some special subalgebra 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.

[1] A. Di Nola and R. Grigolia, On monadic $MV$–algebras, Ann. Pure Appl. Logic 128 (2004), no. 1-3, 125–-139.

[2] L. Henkin, D. Monk and A. Tarski, {\em Cylindric Algebras}, Parts I & II, North-Holland, 1971 & 1985

The paper of our talk can be consulted in:

http://www.cle.unicamp.br/e-prints/vol_15,n_3,2015.html

or

http://link.springer.com/article/10.1007%2Fs11225-016-9663-2

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