Epimorphisms in varieties of semilinear residuated lattices

We provide sufficient conditions for a variety of residuated lattices to have surjective epimorphisms. The lattices under consideration are square-increasing [involutive] commutative residuated lattices (S[I]RLs) that are semilinear, i.e., that embed into a direct product of totally ordered algebras.

We say that an S[I]RL is negatively generated when it is generated by the elements beneath its monoid identity. We show that epimorphisms are surjective in all varieties of negatively generated semilinear S[I]RLs.

These results settle natural questions about Beth-style definability of a range of relevance logics.