Amalgamation Property for Some Varieties of BL-Algebras Generated by One Finite Set of BL-Chains with Finitely Many Components

BL-algebras are the algebraic semantics of Basic logic BL, the logic of all continuous t-norms and their residua. In a previous work, we provided the classification of the amalgamation property (AP) for the varieties of BL-algebras generated by one BL-chain with finitely many components. As an open problem, we left the analysis of the AP for varieties of BL-algebras generated by one finite set of BL-chains with finitely many components. In this paper we provide a partial solution to this problem. We provide a classification of the AP for the varieties of BL-algebras generated by one finite set of BL-chains with finitely many components, which are either cancellative hoops or finite Wajsberg hoops. We also discuss the difficulties to generalize this approach to the more general case.