A non-associative generalization of Hájek's BL-algebras

Hájek introduced basic logic BL as the logic of continuous t-norms and their residua. Basic logic is a fuzzy logic, i.e. it is complete with respect to linearly ordered models. Algebraic semantics of BL is the variety of BL algebras. It was proved by Cignoli, Esteva, Godo and Torrens that the variety of BL algebras is generated just by the continuous t-norms on the interval [0,1] of reals.The main goal of the paper is to present a non-associative generalization of Hájek's BL logic which has a class naBL of non-associative BL algebras as its algebraic semantics. Moreover, it is shown that naBL forms a variety generated just by non-associative t-norms. Consequently, the non-associative BL logic is the logic of non-associative t-norms and their residua.