El reticulado de subvariedades de MG

Abstract

Contenido: - T-normas. - Hoops y BL-algebras. - Subvariedad MG. - Λ(MG). - Bases ecuacionales. - Álgebras libres. - Bibliografía.

BL-algebras were introduced by Hájek (see [5]) to formalize fuzzy logics in which the conjunction is interpreted by continuous t-norms over the real interval [0;1]. These algebras form a variety, usually called BL. In this work we will concentrate in the subvariety MG ⊆ BL generated by the ordinal sum of the algebra [0;1]MV and the Gödel hoop [0;1]G, that is, generated by A = [0;1]MV ⊕ [0;1]G. Though it is well-known that [0;1]G is decomposable as an infinite ordinal sum of two-elements Boolean algebra, the idea is to treat it as a whole block. The elements of this block are the dense elements of the generating chain and the elements in [0;1]MV are usually called regular elements of A: The main advantage of this approach, is that unlike the work done in [3] and [1], when the number n of generators of the free algebra increase the generating chain remains fixed. This provides a clear insight of the role of the two main blocks of the generating chain in the description of the functions in the free algebra: the role of the regular elements and the role of the dense elements. We have a functional representation for the free algebra FreeMG (n). To define this functions we need to decompose the domain Aⁿ = ([0;1]MV ⊕ [0;1]G)ⁿ in a finite number of pieces. In each piece a function F ∈ FreeMG (n) coincides either with McNaughton functions or functions of FreeG (n). Using [2] and [4] we give a description of the elements in the lattice of subvarieties of the variety MG and the equational characterization of them.