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LATTICE-ORDERED ABELIAN GROUPS AND PERFECT MV-ALGEBRAS: A TOPOS-THEORETIC PERSPECTIVE

Published online by Cambridge University Press:  05 July 2016

OLIVIA CARAMELLO  and
ANNA CARLA RUSSO
OLIVIA CARAMELLO
Affiliation:
UNIVERSITÉ PARIS DIDEROT UFR DE MATHÉMATIQUES, BÂTIMENT SOPHIE GERMAIN 5 RUE THOMAS MANN, 75205PARIS CEDEX 13, FRANCEE-mail: [email protected]
ANNA CARLA RUSSO
Affiliation:
DIPARTIMENTO DI MATEMATICA E INFORMATICA UNIVERSITÁ DI SALERNO FISCIANO (SA), VIA GIOVANNI PAOLO II, 132 - 84084, ITALYE-mail: [email protected]
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Abstract

We establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.

Keywords

06F2006F3518B2503G30Lattice-ordered abelian groupordered monoidpositive coneperfectMV-algebraChang’s varietyMorita-equivalencegeometric theoryclassifying toposDi Nola-Lettieri’s equivalencebi-interpretability
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 22 , Issue 2 , June 2016 , pp. 170 - 214
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

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