Advertisement

Order

, Volume 35, Issue 1, pp 23–45 | Cite as

Semi-Nelson Algebras

  • Juan Manuel Cornejo
  • Ignacio Viglizzo
Article

Abstract

Generalizing the well known and exploited relation between Heyting and Nelson algebras to semi-Heyting algebras, we introduce the variety of semi-Nelson algebras. The main tool for its study is the construction given by Vakarelov. Using it, we characterize the lattice of congruences of a semi-Nelson algebra through some of its deductive systems, use this to find the subdirectly irreducible algebras, prove that the variety is arithmetical, has equationally definable principal congruences, has the congruence extension property and describe the semisimple subvarieties.

Keywords

Semi-Heyting algebras Semi-Nelson algebras Twist structures Heyting algebras Nelson algebras 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abad, M., Cornejo, J.M., Díaz Varela, J.P.: The variety generated by semi-Heyting chains. Soft Comput. 15(4), 721–728 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Abad, M., Cornejo, J.M., Díaz Varela, J.P.: The variety of semi-Heyting algebras satisfying the equation (0→1)∨(0→1)∗∗≈. Rep. Math. Logic 46, 75–90 (2011)Google Scholar
  3. 3.
    Abad, M., Cornejo, J.M., Díaz Varela, J.P.: Semi-Heyting algebras term-equivalent to Gödel algebras. Order 2, 625–642 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Abad, M., Cornejo, J.M., Díaz Varela, P.: Free-decomposability in varieties of semi-Heyting algebras. Math. Log. Q. 58(3), 168–176 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blok, W.J., Köhler, P., Pigozzi, D.: On the structure of varieties with equationally definable principal congruences. II. Algebra Universalis 18(3), 334–379 (1984)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burris, S., Sankappanavar, H.P.: A course in universal algebra. Springer-Verlag, New York (1981). volume 78 of Graduate Texts in MathematicsGoogle Scholar
  7. 7.
    Cornejo, J.M.: Semi-intuitionistic logic. Stud. Logica. 98(1-2), 9–25 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cornejo, J.M.: The semi Heyting-Brouwer logic. Stud. Logica. 103(4), 853–875 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cornejo, J.M., Viglizzo, I.D.: On some semi-intuitionistic logics. Stud. Logica. 103(2), 303–344 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Day, A.: A note on the congruence extension property. Algebra Universalis 1, 234–235 (1971/72)Google Scholar
  11. 11.
    Fried, E., Grätzer, G., Quackenbush, R.: Uniform congruence schemes. Algebra Universalis 10(2), 176–188 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kalman, J.A.: Lattices with involution. Trans. Amer. Math. Soc. 87, 485–491 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Köhler, P., Pigozzi, D.: Varieties with equationally definable principal congruences. Algebra Universalis 11(2), 213–219 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kracht, M.: On extensions of intermediate logics by strong negation. J. Philos. Logic 27(1), 49–73 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nelson, D.: Constructible falsity. J. Symbolic Logic 14, 16–26 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Odintsov, S.P.: On the representation of N4-lattices. Stud. Logica. 76(3), 385–405 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rasiowa, H.: N-lattices and constructive logic with strong negation. Fund. Math. 46, 61–80 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rasiowa, H.: An algebraic approach to non-classical logics, vol. 78. North-Holland Publishing Co., Amsterdam (1974)Google Scholar
  19. 19.
    Rivieccio, U.: Implicative twist-structures. Algebra Universalis 71(2), 155–186 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sankappanavar, H.P.: Semi-Heyting algebras: an abstraction from Heyting algebras. In: Proceedings of the 9th Dr. Antonio A. R. Monteiro Congress (Spanish), Actas Congr. Dr. Antonio A. R. Monteiro. Bahía Blanca, 2008. Univ. Nac. del Sur., pp 33–66Google Scholar
  21. 21.
    Sankappanavar, H.P.: Expansions of semi-Heyting algebras I: Discriminator varieties. Stud. Logica. 98(1-2), 27–81 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sendlewski, A.: Some investigations of varieties of n-lattices. Stud. Logica. 43(3), 257–280 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sendlewski, A.: Nelson algebras through Heyting ones. I. Stud. Logica. 49(1), 105–126 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sholander, M.: Postulates for distributive lattices. Canadian J. Math. 3, 28–30 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Vakarelov, D.: Notes on N-lattices and constructive logic with strong negation. Stud. Logica. 36(1–2), 109–125 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Viglizzo, I. Álgebras de Nelson, Instituto De Matemática De Bahía Blanca, Universidad Nacional del Sur, 1999. Magister dissertation in Mathematics, Universidad Nacional del Sur, Bahía Blanca, available at https://sites.google.com/site/viglizzo/viglizzo99nelson (1999)

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Instituto de Matemática de Bahía BlancaUniversidad Nacional del Sur-CONICET, Departamento de Matemática -Bahía BlancaArgentina

Personalised recommendations