Algebra universalis

, 80:48 | Cite as

A completion for distributive nearlattices

  • Luciano J. GonzálezEmail author
  • Ismael Calomino


The aim of this article is to propose an adequate completion for distributive nearlattices. We give a proof of the existence of such a completion through a representation theorem, which allows us to prove that this completion is a completely distributive algebraic lattice. We show several properties about this completion, and we present a connection with the free distributive lattice extension of a distributive nearlattice. Finally, we consider how can be extended n-ary operations on distributive nearlattices, and we study the basic properties of these extensions.


Nearlattices Completion Extensions of operations Free lattice extension 

Mathematics Subject Classification

06A12 06B23 03G10 06A15 



  1. 1.
    Abbott, J.: Semi-boolean algebra. Matematički Vesnik 4(19), 177–198 (1967)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Araújo, J., Kinyon, M.: Independent axiom systems for nearlattices. Czech. Math. J. 61(4), 975–992 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Celani, S., Calomino, I.: Stone style duality for distributive nearlattices. Algebra Univ. 71(2), 127–153 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Celani, S., Calomino, I.: On homomorphic images and the free distributive lattice extension of a distributive nearlattice. Rep. Math. Log. 51, 57–73 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures. Heldermann Verlag, Lemgo (2007)zbMATHGoogle Scholar
  6. 6.
    Chajda, I., Halaš, R.: An example of a congruence distributive variety having no near-unanimity term. Acta Univ. M. Belii Ser. Math. 13, 29–31 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chajda, I., Kolařík, M.: A decomposition of homomorphic images of nearlattices. Acta Univ. Palacki. Olomuc. Fac. rer. nat. Mathematica 45(1), 43–51 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chajda, I., Kolařík, M.: Ideals, congruences and annihilators on nearlattices. Acta Univ. Palacki. Olomuc. Fac. rer. nat. Mathematica 46(1), 25–33 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chajda, I., Kolařík, M.: Nearlattices. Discrete Math. 308(21), 4906–4913 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cornish, W., Hickman, R.: Weakly distributive semilattices. Acta Math. Hung. 32(1), 5–16 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Davey, B., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)zbMATHCrossRefGoogle Scholar
  12. 12.
    Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Log. 70, 713–740 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238(1), 345–371 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gehrke, M., Jansana, R., Palmigiano, A.: \(\Delta _1\)-completions of a poset. Order 30(1), 39–64 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Jpn. 40(2), 207–215 (1994)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gehrke, M., Jónsson, B.: Monotone bounded distributive lattice expansions. Math. Jpn. 52(2), 197–213 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94(1), 13–45 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    González, L.J.: The logic of distributive nearlattices. Soft Comput. 22(9), 2797–2807 (2018)zbMATHCrossRefGoogle Scholar
  19. 19.
    Halaš, R.: Subdirectly irreducible distributive nearlattices. Miskolc Math. Notes 7, 141–146 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hickman, R.: Join algebras. Commun. Algebra 8(17), 1653–1685 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Jónnson, B., Tarski, A.: Boolean algebras with operators. Part II. Am. J. Math. 74(1), 127–162 (1952)CrossRefGoogle Scholar
  22. 22.
    Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I. Am. J. Math. 73(4), 891–939 (1951)zbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Facultad de Ciencias Exactas y NaturalesUniversidad Nacional de La PampaSanta RosaArgentina
  2. 2.CIC and Facultad de Ciencias ExactasUniversidad Nacional del CentroTandilArgentina

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