Unilinear residuated lattices: axiomatization, varieties and FEP

We characterize all residuated lattices that have height equal to 3 and show that the variety they generate has continuum-many subvarieties. More generally, we study unilinear residuated lattices: their lattice is a union of disjoint incomparable chains, with bounds added. We we give two general constructions of unilinear residuated lattices, provide an axiomatization and a proof-theoretic calculus for the variety they generate, and prove the finite model property for various subvarieties.