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)

It is well known that there is a categorical equivalence between lattice-ordered Abelian groups (or l-groups) and conical BCK-algebras (see [COR 80]). The aim of this paper is to study this equivalence from the perspective of logic, in particular, to study the relationship between two deductive systems: conical logic Co and a logic of l-groups, Balo. In [GAL 04] the authors introduce a system Bal which models the logic of balance of opposing forces with a single distinguished truth value, that (...) represents equilibrium. Its equivalent algebraic semantics BAL (via Blok-Pigozzi's construction) is definitionally equivalent to the variety of l-groups. In this paper we define the system Balo which is equivalent to Bal and whose Lindenbaum-Tarski algebra is an l-group. On the other hand, we define the conic logic Co and we prove that it can be naturally merged in the system Balo. Also, we prove that every formula of Balo has a normal form that depends on translations of formulas of Co. (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)