Alicia Ziliani

Monadic Distributive Lattices

The purpose of this paper is to investigate the variety of algebras, which we call monadic distributive lattices, as a natural generalization of monadic Heyting algebras [16]. It is worth mentioning that the latter is a proper subvariety of the first one, as it is shown in a simple example. Our main interest is the characterization of simple and subdirectly irreducible monadic distributive lattices. In order to do this, a duality theory for these algebras is developed. The duality enables us to describe the lattice of congruences on monadic distributive lattices. Finally, our attention is focused upon the relationship between the category of dual spaces associatted with these algebras and the category of perfect Ono frames considered by Bezhanishvili in order to represent monadic Heyting algebras.

FREE ALGEBRAS OVER A POSET IN VARIETIES

In 1945, the notion of free lattice over a poset was introduced by R. Dilworth (Trans. Am. Math. Soc. 57 (1945), 123{154). In this note, a construction of the free algebra over a poset in varieties finitely generated is shown. Finally, this result is applied to different classes of algebras.

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

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>>