Strictly join irreducible elements in the lattice of varieties of BL-algebras

The logic BL and its corresponding semantics, BL-algebras, were introduced by Petr Hajek in 1998. From then, many scholars investigated the structure of the lattice of the subvarieties of BL. In this work we will show that every variety of BL-algebras and, more in general, every variety of MTL-algebras, is the join of a set of strictly join irreducible varieties, in the lattice of subvarieties of BL-algebras (MTL-algebras). We will study some general properties of the varieties of MTL-algebras which are strictly join irreducible (SJI), as elements of the lattice of the subvarieties of MTL. Finally, we will provide a partial classification for the SJI varieties of BL-algebras.