Almost minimal varieties of commutative residuated lattices

Abstract

We study minimal and almost minimal varieties of commutative integral residuated lattices. In particular, we prove that the variety [Formula: see text] of generalized Boolean algebras and the variety [Formula: see text] generated by the negative integers are the only two semilinear atoms. Furthermore, we give a characterization of all the finitely generated covers of [Formula: see text], showing that there are infinitely many, we axiomatize the unique common cover of [Formula: see text] and [Formula: see text], and we construct continuum-many semilinear covers of [Formula: see text] and infinitely many non-semilinear ones.