A biresiduation algebra is a 〈/,\,1〉-subreduct of an integral residuated lattice. These algebras arise as algebraic models of the implicational fragment of the Full Lambek Calculus with weakening. We axiomatize the quasi-variety B of biresiduation algebras using a construction for integral residuated lattices. We define a filter of a biresiduation algebra and show that the lattice of filters is isomorphic to the lattice of B-congruences and that these lattices are distributive. We give a finite basis of terms for generating filters (...) and use this to characterize the subvarieties of B with EDPC and also the discriminator varieties. A variety generated by a finite biresiduation algebra is shown to be a subvariety of B. The lattice of subvarieties of B is investigated; we show that there are precisely three finitely generated covers of the atom. (shrink)

In this paper we extend to various classes of subreducts of hoops some results about splitting algebras. In particular we prove that every finite chain in the purely implicational fragment of basic hoops is splitting and that every finite chain in the \ fragment of hoops is splitting. We also produce explicitly the splitting equations in most cases.

In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework.