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. (shrink)

An equivalence between the category of MV-algebras and the category $${{\rm MV^{\bullet}}}$$ MV ∙ is given in Castiglioni et al. :67–92, 2014). An integral residuated lattice with bottom is an MV-algebra if and only if it satisfies the equations $${a = \neg \neg a, \vee = 1}$$ a = ¬ ¬ a, ∨ = 1 and $${a \odot = a \wedge b}$$ a ⊙ = a ∧ b. An object of $${{\rm MV^{\bullet}}}$$ MV ∙ is a residuated lattice which in (...) particular satisfies some equations which correspond to the previous equations. In this paper we extend the equivalence to the category whose objects are pairs, where A is an MV-algebra and I is an ideal of A. (shrink)

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. (...) Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL ew. (shrink)

An equivalence between the category of MV-algebras and the category \ is given in Castiglioni et al. :67–92, 2014). An integral residuated lattice with bottom is an MV-algebra if and only if it satisfies the equations \ \vee = 1}\) and \ = a \wedge b}\). An object of \ is a residuated lattice which in particular satisfies some equations which correspond to the previous equations. In this paper we extend the equivalence to the category whose objects are pairs, where (...) A is an MV-algebra and I is an ideal of A. (shrink)