Subvarieties of BL-algebras generated by single-component chains

In this paper we study and equationally characterize the subvarieties of BL, the variety of BL-algebras, which are generated by families of single-component BL-chains, i.e. MV-chains, Product-chain or Gödel-chains. Moreover, it is proved that they form a segment of the lattice of subvarieties of BL which is bounded by the Boolean variety and the variety generated by all single-component chains, called Ł G. 1. The variety of BL algebras BL-algebras have been introduced by Hájek [Haj98a] as the algebraic counterpart of Basic Fuzzy logic BL, the logic of continuous t-norms and their residua in the sense that a formula is a theorem in BL if and only if it is a tautology in any BL-algebra on the unit real interval [0,1] induced by a continuous t-norm and its residuum (see [CEGT00]). In [Haj98a] Hájek defines a BL-algebra as an algebraic structure