Collection Topos à l'IHES
Organizer(s)
Date(s)
30/04/2024
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 l-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 l-groups (i.e., lattice-ordered abelian groups, not necessarily with strong unit). These generalizations yield respectively a Morita-equivalence between the theory MV of MV-algebras and the theory L u of l-u groups and one between the theory P of perfect MV-algebras and the theory L of l-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 L u, a form of completeness and compactness for the innitary theory L u, the identication of three dierent levels of bi-interpretabilitity between the theory P and the theory L and a representation theorem for the nitely presentable objects of Chang's variety as nite 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., classied by a presheaf topos) and establish a Morita-equivalence with a theory that extends that of l-groups. Furthermore, we generalize to this setting the representation results obtained in [2].
