The lattice of subvarieties of the variety of BL-algebras generated by [0, 1]MV ⊕ [0, 1]G.

JOSÉ PATRICIO DÍAZ VARELA; NOEMÍ LUBOMIRSKY

Concepción

Congreso; XVIII SLALM; 2019

Universidad de Concepción

BL-algebras were introduced by H´ajek to formalize fuzzy logics in whichthe 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 Godel 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 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 these functions we need to decompose the domain An = ([0, 1]MV ⊕ [0, 1]G)^nin a finite number of pieces. In each piece a function F ∈ F reeMG(n) coincides either with McNaughton functions or functions of FreeG(n).We give a description of the elements in the lattice of subvarieties of the variety MG and the equational characterization of them.