Morita-equivalences for MV-algebras

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Morita-equivalences for MV-algebras
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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. In and we generalize to a topos-theoretic setting two classical equivalences arising in the context of MV-algebras: Mundici's equivalence 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 between the category of perfect MV-algebras and the category of` -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 `-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 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 `-groups. Furthermore, we generalize to this setting the representation results obtained in.
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