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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.
Aglianò et al. (Mon,) studied this question.
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