Aldo V. Figallo

DISCRETE DUALITY FOR TSH-ALGEBRAS

In this article, we continue the study of tense symmetric Heyting algebras (or TSH-algebras). These algebras constitute a generalization of tense algebras. In particular, we describe a discrete duality for TSH-algebras bearing in mind the results indicated by Or lowska and Rewitzky in [E. Or lowska and I. Rewitzky, Discrete Dualities for Heyting Algebras with Operators, Fund. Inform. 81 (2007), no. 1-3, 275–295] for Heyting algebras. In addition, we introduce a propositional calculus and prove this calculus has TSH-algebras as algebraic counterpart. Finally, the duality mentioned above allowed us to show the completeness theorem for this calculus.

LUKASIEWICZ RESIDUATION ALGEBRAS WITH INFIMUM

Lukasiewicz residuation algebras with an underlying ordered structure of meet semilattice (or iLR-algebras) are studied. These algebras are the algebraic counterpart of the {—>, A}-fragment of Lukasiewiczs many-valued logic. An equational basis for this class of algebras is shown. In addition, the subvariety of ( n + 1)—valued iLR-algebras for 0 < n < u> is considered. In particular, the structure of the free finitely generated (n + 1)—valued iLR-algebra is described. Moreover, a formula to compute its cardinal number in terms of n and the number of free generators is obtained. 1. Preliminares B. Bosbach ([5, 6]) undertook the investigation of a class of residuated structures that were related to but considerably more general than Brouwerian semilattices and the algebras associated with {—A}-fragment of Lukasiewiczs many valued logic. In a manuscript by J . Büchi and T. Owens ([8]) devoted to a study of Bosbachs algebras, written in the mid-seventies, the commutative members of this equational class were given the name hoops. More precisely, they are algebras (A,—•, 1) of type (2, 2,0) that satisfy: (HI) (A, •, 1) is a commutative monoid, (H2) x ^ x = 1, (H3) x-y(y^>z) = (x • (H4) x • (x^>y) = y • (y^x). An important subclass of the variety of hoops is the variety of Wajsberg hoops, so named and studied by W. Blok and I. Ferreirim in [3]. These

(n+1)-valued modal implicative semilattices

The equational class of (n+1)-valued modal implicative semilattices is defined and investigated. These algebras are the natural generalization of three-valued modal implicative semilattices. A characterization of (n+1)-valued Post algebras is given.<<ETX>>

On Tarski algebras with a finite set of free generators

In this paper, we describe a method to determine the structure of the Tarski algebra with a finite set of free generators which is different to that given by Iturrioz and Monteiro in [Les algebres de Tarski avec un nombre fini de generateurs libres, in Informe Tecnico, Vol. 37 (Instituto de Matematica de la Universidad Nacional del Sur, Bahia Blanca, 1994)].

Localization of tetravalent modal algebras

The main aim of this paper is to define the localization of a tetravalent modal algebra A with respect to a topology ℱ on A. In Sec. 5, we prove that the tetravalent modal algebra of fractions relative to a ∧-closed system (defined in Definition 3.1) is a tetravalent modal algebra of localization.

Free three-valued Lukasiewicz, Post and Moisil algebras over a poset

A construction of the free three-valued Lukasiewicz algebra L(L) over a poset L is given. The number of elements of L(L) for some particular cases of finite posets L is determined. The free (three-valued) Post and Moisil algebras over a poset are determined.<<ETX>>