Strictly join irreducible varieties of BL-algebras: The missing pieces

Basic Logic BL, introduced by P. Hájek in 1998, is the logic of all continuous t-norms and their residua. The variety of BL-algebras forms the algebraic semantics of BL. Let L be a variety of BL-algebras, and let L(L) be its lattice of subvarieties, ordered by inclusion. L is called strictly join irreducible (SJI) if, whenever L is the join of a set S of varieties of BL-algebras, then L∈S. Every variety in L(L) is obtained as join of SJI varieties, which may be considered as the building blocks of all the varieties in L(L). In a previous work by the second author, a partial classification for the SJI varieties of BL-algebras has been found. In this paper we provide a full classification of the SJI varieties of BL-algebras. Our main result is that a variety of BL-algebras is SJI iff it is generated by a BL-chain with finitely many components, each of them being a cancellative hoop or a Wajsberg hoop with finite rank. As an application, we provide a characterization for the varieties of BL-algebras having only finitely many subvarieties, and we study some additional topics.