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Moritaequivalences for MValgebras
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Title  Moritaequivalences for MValgebras 
Title of Series  Topos à l'IHES 
Part Number  23 
Number of Parts  28 
Author 
Russo, Anna Carla

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/20738 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
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Subject Area  Mathematics 
Abstract  We shall make a survey of the most recent results obtained in connection with the programme of investigating notable categorical equivalences for MValgebras from a topostheoretic perspective commenced in. In and we generalize to a topostheoretic setting two classical equivalences arising in the context of MValgebras: Mundici's equivalence between the category of MValgebras and the category of`u groups (i.e., latticeordered abelian groups with strong unit) and Di NolaLettieri's equivalence between the category of perfect MValgebras and the category of` groups (i.e., latticeordered abelian groups, not necessarily with strong unit). These generalizations yield respectively a Moritaequivalence between the theory MV of MValgebras and the theory L u of `u groups and one between the theory P of perfect MValgebras and the theory L of `groups. These Moritaequivalences 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 biinterpretabilitity 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 MValgebras. Given the fact that perfect MValgebras are exactly the local MValgebras in the variety generated by Chang's algebra, it is natural to wonder whether analogues of Di NolaLettieri's equivalence exist for local MValgebras in a given proper subvariety of MValgebras. In a forthcoming paper, we prove that the theory of local MValgebras in any subvariety V of MValgebras is of presheaf type (i.e., classied by a presheaf topos) and establish a Moritaequivalence with a theory that extends that of `groups. Furthermore, we generalize to this setting the representation results obtained in. 
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