Studia Logica

, Volume 107, Issue 6, pp 1135–1158 | Cite as

Categorical Equivalence Between \(\varvec{PMV}_{\varvec{f}}\)-Product Algebras and Semi-Low \(\varvec{f}_{\varvec{u}}\)-Rings

  • Lilian J. CruzEmail author
  • Yuri A. Poveda


An explicit categorical equivalence is defined between a proper subvariety of the class of \({ PMV}\)-algebras, as defined by Di Nola and Dvurečenskij, to be called \({ PMV}_{f}\)-algebras, and the category of semi-low \(f_u\)-rings. This categorical representation is done using the prime spectrum of the \({ MV}\)-algebras, through the equivalence between \({ MV}\)-algebras and \(l_u\)-groups established by Mundici, from the perspective of the Dubuc–Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low \(f_u\)-rings associated to Boolean algebras are characterized.


\({ PMV}\)-algebra \({ PMV}_{f}\)-algebra \(l_u\)-ring Prime ideal Spectrum 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Birkhoff, G., Lattice Theory. 3rd ed., Colloquium Publications No. 25, American Mathatical Society, Providence, 1967.Google Scholar
  2. 2.
    Chang, C. C., A new proof of the completeness of the Łukasiewicz axioms. Transactions of the American Mathematical Society 93:74–90, 1959.Google Scholar
  3. 3.
    Cignoli, R. L, I. M. L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefGoogle Scholar
  4. 4.
    Di Nola, A., and A. Dvurečenskij, Product MV-algebras. Multiple-Valued Logics 193–215, 2001.Google Scholar
  5. 5.
    Dubuc, E. J., and Y. A. Poveda, On the Equivalence Between MV-Algebras and l-Groups whit Strong Unit. Studia Logica 103(4):807–814, 2015.CrossRefGoogle Scholar
  6. 6.
    Dubuc, E. J., and Y. A. Poveda, Representation Theory of MV-Algebras, Annals of Pure and Applied Logic, 161, 2010.CrossRefGoogle Scholar
  7. 7.
    Estrada, A., MVW-rigs. Master’s Thesis, Universidad Tecnológica de Pereira, 2016.Google Scholar
  8. 8.
    Montagna, F., An algebraic approach to propositional fuzzy logic, Journal of Logic, Language and Information 9:91–124, 2000.CrossRefGoogle Scholar
  9. 9.
    Montagna, F., Subreducts of MV-algebras with product and product residuation. Algebra Universalis 53(1):109–137, 2005.CrossRefGoogle Scholar
  10. 10.
    Mundici, D., Interpretation of AF C*-algebras in Lukasiewicz sentential calculus, Journal of Functional Analysis 65:15–63, 1986.CrossRefGoogle Scholar
  11. 11.
    Zuluaga, S., Los MVW-rigs provenientes de las MV-álgebras libres. Master’s Thesis, Universidad Tecnológica de Pereira, 2017.Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Universidad del ValleCaliColombia
  2. 2.Universidad Tecnológica de PereiraPereiraColombia

Personalised recommendations