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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

REFERENCES

Belluce, Lawrence Peter and Chang, Chen Chun, A weak completeness theorem for infinite valued predicate logic. The Journal of Symbolic Logic, vol. 28 (1963), pp. 43–50.CrossRefGoogle Scholar
Belluce, Lawrence Peter and Di Nola, Antonio, Yosida type representation for perfect MV-algebras. Mathematical Logic Quarterly, vol. 42 (1996), pp. 551–563.Google Scholar
Belluce, Lawrence Peter, Di Nola, Antonio, and Gerla, Brunella, Abelian ℓ-groups with strong unit and perfect MV-algebras. Order, vol. 25 (2008), pp. 387–401.Google Scholar
Belluce, Lawrence Peter, Di Nola, Antonio, and Gerla, Giangiacomo, Perfect MV-algebras and their logic. Applied Categorical Structures, vol. 15 (2007), pp. 135–151.CrossRefGoogle Scholar
Bigard, Alain, Keimel, Klaus, and Wolfenstein, Samuel, Groupes et Anneaux Réticulés, vol. 608, Lecture Notes in Mathematics, 1977.Google Scholar
Caramello, Olivia, The unification of Mathematics via Topos Theory, arXiv:math.CT/1006.3930, 2010.Google Scholar
Caramello, Olivia, Syntactic characterizations of properties of classifying toposes. Theory and Applications of Categories, vol. 26 (2012), pp. 176–193.Google Scholar
Caramello, Olivia, Universal models and definability. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 152 (2012), pp. 279–302.Google Scholar
Caramello, Olivia, Yoneda representations of flat functors and classifying toposes. Theory and Applications of Categories, vol. 26 (2012), pp. 538–553.Google Scholar
Caramello, Olivia, Extensions of flat functors and theories of presheaf type, Theories, Sites, Toposes: Relating and Studying Mathematical Theories through Topos-Theoretic ‘Bridges’, Oxford University Press, forthcoming.Google Scholar
Caramello, Olivia, Lattices of theories, Theories, Sites, Toposes: Relating and Studying Mathematical Theories through Topos-Theoretic ‘Bridges’, Oxford University Press, forthcoming.Google Scholar
Caramello, Olivia, Topos-theoretic background, Theories, Sites, Toposes: Relating and Studying Mathematical Theories through Topos-Theoretic ‘Bridges’, Oxford University Press, forthcoming.Google Scholar
Caramello, Olivia and Russo, Anna Carla, The Morita-equivalence between MV-algebras and abelian ℓ-groups with strong unit. Journal of Algebra, vol. 422 (2015), pp. 752–787.Google Scholar
Chang, Chen Chung, Algebraic analysis of many valued logics. Transactions of the American Mathematical Society, vol. 88 (1958), pp. 467–490.Google Scholar
Cignoli, Roberto, D’Ottaviano, Itala M. L., and Mundici, Daniele, Algebraic Foundations of Many-Valued Reasoning, vol. 7, Trends in Logic, 2000.Google Scholar
Dubuc, Eduardo J. and Poveda, Yuri, Representation theory of MV-algebras, Annals of Pure and Applied Logic. vol. 161 (2010), no. 8, pp. 1024–1046.Google Scholar
Gabriel, Peter and Ulmer, Friedrich, Lokal präsentierbare Kategorien, vol. 221, Lecture Notes in Mathematics, Springer, 1971.Google Scholar
Jacubík, Ján, Direct product decomposition of MV-algebras. Czechoslovak Mathematical Journal, vol. 44 (1994), pp. 725–739.Google Scholar
Johnstone, Peter T., Sketches of an Elephant: A Topos Theory Compendium. Vols. 1–2, vol. 43–44, Oxford University Press, 2002.Google Scholar
McLane, Saunders and Moerdijk, Ieke, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer, New York, 1992.Google Scholar
Mundici, Daniele, Interpretation of af C*-algebras in Ł ukasiewicz sentential calculus. Journal of Functional Analysis, vol. 65 (2002), pp. 15–63.Google Scholar
Di Nola, Antonio, Esposito, Ivano, and Gerla, Brunella, Local algebras in the representation of MV-algebras. Algebra Universalis, vol. 56 (2007), pp. 133–164.CrossRefGoogle Scholar
Di Nola, Antonio, Georgescu, George, and Leuçteanc, Laurenţiu, Boolean products of BL-algebras. Journal of Mathematical Analysis and Applications, vol. 251 (2000), pp. 106–131.Google Scholar
Di Nola, Antonio and Lettieri, Ada, Perfect MV-algebras are categorically equivalent to abelian ℓ-groups. Studia Logica, vol. 53 (1994), pp. 417–432.Google Scholar
Rodríguez Salas, Antonio Jesús, Un estudio algebraico de los cálculos proposicionales de Lukasiewicz, Ph.D. Thesis, Universidad de Barcelona, 1980.Google Scholar