Morita-equivalences for Mv-algebras

We shall make a survey of the most recent results obtained in connection with the programme of investigating notable categorical equivalences for MV-algebras from a topos-theoretic perspective commenced in [3]. In [3] and [2] we generalize to a topos-theoretic setting two classical equivalences arising in the context of MV-algebras: Mundici’s equivalence [4] between the category of MV-algebras and the category of `-u groups (i.e., lattice-ordered abelian groups with strong unit) and Di Nola-Lettieri’s equivalence [5] between the category of perfect MV-algebras and the category of `-groups (i.e., latticeordered abelian groups, not necessarily with strong unit). These generalizations yield respectively a Morita-equivalence between the theory MV of MV-algebras and the theory Lu of `-u groups and one between the theory P of perfect MV-algebras and the theory L of `-groups. These Morita-equivalences allow us to apply the ‘bridge technique’ of [1] to transfer properties and results from one theory to the other, obtaining new insights on the theories which are not visible by using classical techniques. Among these results, we mention a bijective correspondence between the geometric theory extensions of the theory MV and those of the theory Lu, a form of completeness and compactness for the infinitary theory Lu, the identification of three different levels of bi-interpretabilitity between the theory P and the theory L and a representation theorem for the finitely presentable objects of Chang’s variety as finite products of perfect MV-algebras. Given the fact that perfect MV-algebras are exactly the local MV-algebras in the variety generated by Chang’s algebra, it is natural to wonder whether analogues of Di Nola-Lettieri’s equivalence exist for local MV-algebras in a given proper subvariety of MV-algebras. In a forthcoming paper, we prove that the theory of local MV-algebras in any subvariety V of MV-algebras is of presheaf type (i.e., classified by a presheaf topos) and establish a Morita-equivalence with a theory that extends that of `-groups. Furthermore, we generalize to this setting the representation results obtained in [2].