Abstract

The category $\mathbb {DRDL}{'}$, whose objects are c-differential residuated distributive lattices satisfying the condition $\textbf{CK}$, is the image of the category $\mathbb {RDL}$, whose objects are residuated distributive lattices, under the categorical equivalence $\textbf{K}$ that is constructed in Castiglioni et al. (Stud Log 90:93–124, 2008). In this paper, we introduce weak monadic residuated lattices and study some of their subvarieties. In particular, we use the functor $\textbf{K}$ to relate the category $\mathbb {WMRDL}$, whose objects are weak monadic residuated distributive lattices, and the category $\mathbb {WMDRDL}{'}$, whose objects are pairs formed by an object of $\mathbb {DRDL}{'}$ and a center weak universal quantifier.