Studia Logica

, Volume 103, Issue 2, pp 345–373 | Cite as

A Categorical Equivalence for Product Algebras

  • Franco MontagnaEmail author
  • Sara Ugolini


In this paper we provide a categorical equivalence for the category \({\mathcal{P}}\) of product algebras, with morphisms the homomorphisms. The equivalence is shown with respect to a category whose objects are triplets consisting of a Boolean algebra B, a cancellative hoop C and a map \({\vee_e}\) from B × C into C satisfying suitable properties. To every product algebra P, the equivalence associates the triplet consisting of the maximum boolean subalgebra B(P), the maximum cancellative subhoop C(P), of P, and the restriction of the join operation to B × C. Although several equivalences are known for special subcategories of \({\mathcal{P}}\), up to our knowledge, this is the first equivalence theorem which involves the whole category of product algebras. The syntactic counterpart of this equivalence is a syntactic reduction of classical logic CL and of cancellative hoop logic CHL to product logic, and viceversa.


Many-valued logics Product algebras Lattice ordered groups Boolean algebras Equivalences 

Mathematics Subject Classification

03B50 03G20 03G05 06F20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aglianó P., Montagna F.: Varieties of BL-algebras I: general properties. Journal of Pure and Applied Algebra 181, 105–129 (2003)CrossRefGoogle Scholar
  2. 2.
    Aglianó P., Ferreirim I.M.A., Montagna F.: Basic hoops: an algebriac study of continuous t-norms. Studia Logica 87(1), 73–98 (2007)CrossRefGoogle Scholar
  3. 3.
    Bigard, A., K. Keimel, and S. Wolfenstein, Groupes at anneaux reticulés, vol. 608 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1977.Google Scholar
  4. 4.
    Blok W.J., Ferreirim I.M.A.: On the structure of hoops. Algebra Universalis 43, 233–257 (2000)CrossRefGoogle Scholar
  5. 5.
    Blok, W., and D. Pigozzi, Algebraizable Logics, vol. 77 of Memoirs of the American Mathematical Society, American Mathematical Society, Providence, 1989, p. 396.Google Scholar
  6. 6.
    Burris, S., and H. P. Sankappanavar, A course in Universal Algebra, Graduate texts in Mathematics, Springer-Verlag, Berlin, 1981.Google Scholar
  7. 7.
    Busanice, M., and F. Montagna, Hájek’s Logic BL and BL-algebras, in P. Cintula, P. Hájek and C. Noguera, (eds.), Handbook of Mathematical Fuzzy Logic, Studies in Logic, vol. 38 of Mathematical Logic and Foundations, College Publications, London, 2011, pp. 355–447.Google Scholar
  8. 8.
    Cignoli, R., I. M. L. D’Ottaviano, and D. Mundici, Algebraic foundations of many-valued reasoning, Kluwer, 2000.Google Scholar
  9. 9.
    Cignoli R., Torrens A.: An algebraic analysis of product logic. Multiple-Valued Logic 5, 45–65 (2000)Google Scholar
  10. 10.
    Di Nola A., Lettieri A.: Perfect MV-algebras are categorical equivalent to abelian -groups. Studia Logica 53, 417–432 (1994)CrossRefGoogle Scholar
  11. 11.
    Dummett M.: A propositional logic with denumerable matrix. The Journal of Symbolic Logic 24, 96–107 (1959)CrossRefGoogle Scholar
  12. 12.
    Esteva F., Godo L., Hájek P.: A complete many-valued logic with product conjunction. Archive for Mathematical Logic 35, 191–208 (1996)CrossRefGoogle Scholar
  13. 13.
    Esteva F., Godo L., Hájek P., Montagna F.: Hoops and fuzzy logic. Journal of Logic and Computation 13(4), 531–555 (2003)CrossRefGoogle Scholar
  14. 14.
    Ferreirim, I. M. A., On varieties and quasi varieties of hoops and their reducts, PhD thesis, University of Illinois at Chicago, 1992.Google Scholar
  15. 15.
    Galatos, N., P. Jipsen, T. Kowalski and H. Ono, Residuated lattices: An algebraic glimpse at substructural logics, vol. 151 of Studies in Logic and the Foundations of Mathematics, Elsevier, 2007.Google Scholar
  16. 16.
    Galatos N., Tsinakis C.: Generalized MV-algebras. Journal of Algebra 283(1), 254–291 (2005)CrossRefGoogle Scholar
  17. 17.
    Hà àjek, P., Metamathematics of fuzzy logic, Kluwer, 1998.Google Scholar
  18. 18.
    Mac Lane, S., Categories for the Working Mathematician, second edition, Graduate Texts in Mathematics, Springer, 1997.Google Scholar
  19. 19.
    Montagna F., Tsinakis C.: Ordered groups with a conucleus. Journal of Pure and Applied Algebra 214(1), 71–88 (2010)CrossRefGoogle Scholar
  20. 20.
    Mundici D.: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. Journal of Functional Analysis 65, 15–63 (1986)CrossRefGoogle Scholar
  21. 21.
    Schmidt, J., Quasi-decompositions, exact sequences, and triple sums of semigroups I–II, vol.17 of Colloquia Mathematica Societatis János Bolyai, Contributions to Universal Algebra, Szeged, 1975.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Engineering and Mathematical SciencesSan NiccolòSienaItaly

Personalised recommendations