Soft Computing

, Volume 23, Issue 2, pp 375–381 | Cite as

Topological spaces of monadic MV-algebras

  • Antonio Di Nola
  • Revaz Grigolia
  • Giacomo LenziEmail author


We construct a covariant functor \(\gamma \) from the category of monadic MV-algebras into the category of Q-distributive lattices, i.e., distributive lattices with quantifier introduced by R. Cignoli. For every monadic MV-algebra, we construct a dual object named QM-space; these objects form a special subcategory of spectral spaces and of Q-spaces developed by R. Cignoli for Q-distributive lattices.


Topological duality Monadic MV-algebra 


Compliance with Ethical Standards

Conflict of interest

All authors declare that they have no conflicts of interest.


  1. Belluce LP (1986) Semisimple algebras of infinite-valued logic and bold fuzzy set theory. Can J Math 38:1356–1379MathSciNetCrossRefzbMATHGoogle Scholar
  2. Belluce LP, Chang CC (1963) A weak completeness theorem for infinite valued 2rst-order logic. J Symb Logic 28:43–50CrossRefzbMATHGoogle Scholar
  3. Belluce LP, Grigolia R, Lettieri A (2005) Representations of monadic MV- algebras. Stud Logica 81:125–144MathSciNetzbMATHGoogle Scholar
  4. Cecilia C, Diaz Varela JP (2014) Monadic MV-algebras I: a study of subvarieties. Algebra Universalis 71(1):71–100MathSciNetCrossRefzbMATHGoogle Scholar
  5. Chang CC (1958) Algebraic analysis of many-valued logics. Trans Am Math Soc 88:467–490MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cignoli R (1978) The lattice of global section of sheaves of chains over Boolean spaces. Algebra Universalis 8:357–373MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cignoli R (1991) Quantifiers on distributive lattices. Discrete Math 96:183–197MathSciNetCrossRefzbMATHGoogle Scholar
  8. Davey BA, Priestley HA (2002) Introduction to lattices and order, 2nd edn. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  9. Di Nola A, Grigolia R (2002) MV-algebras in duality with labeled root systems. Discrete Math 243:79–90MathSciNetCrossRefzbMATHGoogle Scholar
  10. Di Nola A, Grigolia R (2004a) On monadic MV-algebras. Ann Pure Appl Logic 128:125–139MathSciNetCrossRefzbMATHGoogle Scholar
  11. Di Nola A, Grigolia R (2004b) Profinite MV-spaces. Discrete Math 283:61–69MathSciNetCrossRefzbMATHGoogle Scholar
  12. Diego C, Cecilia C, Diaz Varela JP, Laura R (2017) Monadic BL-algebras: the equivalent algebraic semantics of Hajek’s monadic fuzzy logic. Fuzzy Sets Syst 320:40–59CrossRefzbMATHGoogle Scholar
  13. Hay LS (1958) An axiomatization of the infinitely many-valued calculus, M.S. Thesis, Cornell UniversityGoogle Scholar
  14. Horn A (1969) Logic with truth values in a linearly ordered Heyting algebra. J Symb Logic 34:395–408MathSciNetCrossRefzbMATHGoogle Scholar
  15. Łukasiewicz J, Tarski A (1930) Untersuchungen über den Aussagenkalkül. Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie 23(cl iii):30–50zbMATHGoogle Scholar
  16. Muresan C (2008) The reticulation of a residuated lattice. Bull Math Soc Sci Math Roumanie (N.S.) 51(99), no. 1, 4765Google Scholar
  17. Priestley HA (1972) Representation of distributive lattices by means of ordered Stone spaces. Bull Lond Math Soc 2:186–190MathSciNetCrossRefzbMATHGoogle Scholar
  18. Priestley HA (1984) Ordered sets and duality for distributive lattices. Ann Discret Math 23:39–60MathSciNetzbMATHGoogle Scholar
  19. Rutledge JD (1959) A preliminary investigation of the infinitely many-valued predicate calculus, Ph.D. Thesis, Cornell UniversityGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of SalernoFiscianoItaly
  2. 2.I.I.A.S.S. “E. R. Caianiello”Vietri sul MareItaly
  3. 3.University of TbilisiTbilisiGeorgia

Personalised recommendations