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We characterize all residuated lattices that have height equal to $3$ andshow that the variety they generate has continuum-many subvarieties. Moregenerally, we study unilinear residuated lattices: their lattice is a union ofdisjoint incomparable chains, with bounds added. We we give two generalconstructions of unilinear residuated lattices, provide an axiomatization and aproof-theoretic calculus for the variety they generate, and prove the finitemodel property for various subvarieties.

Unilinear residuated lattices: axiomatization, varieties and FEP