Abstract
Much work has been done on specific instances of residuated lattices with modal operators (either nuclei or conuclei). In this paper, we develop a general framework that subsumes three important classes of modal residuated lattices: interior algebras, Abelian l-groups with conuclei, and negative cones of l-groups with nuclei. We then use this framework to obtain results about these three cases simultaneously. In particular, we show that a categorical equivalence exists in each of these cases. The approach used here emphasizes the role played by reducts in the proofs of these categorical equivalences. Lastly, we develop a connection between translations of logics and images of modal operators.
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