Soft Computing

, Volume 22, Issue 9, pp 2797–2807 | Cite as

The logic of distributive nearlattices

  • Luciano J. González


In this paper, we propose a sentential logic naturally associated, in the sense of Abstract Algebraic Logic, with the variety of distributive nearlattices. We show that the class of algebras canonically associated (in the sense of Abstract Algebraic Logic) with this logic is the variety of distributive nearlattices. We also present several properties of this sentential logic.


Distributive nearlattices Sentential logic Gentzen system 



I greatly appreciate the comments and suggestions of the referees that helped to improve the paper. This paper was partially supported by Universidad Nacional de La Pampa (Fac. de Cs. Exactas y Naturales) under the Grant P.I. 64 M, Res. No. 432/14 CD and also by CONICET (Argentina) under the Grant PIP 112-20150-100412CO.

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interests regarding the publishing of this paper.


  1. Abbott JC (1967) Semi-boolean algebra. Matematički Vesnik 4(40):177–198MathSciNetzbMATHGoogle Scholar
  2. Abbott JC (1976) Orthoimplication algebras. Stud Logica 35(2):173–177MathSciNetCrossRefzbMATHGoogle Scholar
  3. Araújo J, Kinyon M (2011) Independent axiom systems for nearlattices. Czech Math J 61(4):975–992MathSciNetCrossRefzbMATHGoogle Scholar
  4. Blok WJ, Pigozzi D (1989) Algebraizable logics. 396. Memoirs of the American Mathematical SocietyGoogle Scholar
  5. Calomino I (2015) Supremo álgebra distributivas: una generalización de las álgebras de tarski. Ph.D. thesis, Universidad Nacional del SurGoogle Scholar
  6. Celani S, Calomino I (2014) Stone style duality for distributive nearlattices. Algebra Universalis 71(2):127–153MathSciNetCrossRefzbMATHGoogle Scholar
  7. Celani S, Calomino I (2016) On homomorphic images and the free distributive lattice extension of a distributive nearlattice. Rep Math Log 51:57–73MathSciNetzbMATHGoogle Scholar
  8. Chajda I, Halaš R (2005) An implication in orthologic. Int J Theoret Phys 44(7):735–744MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chajda I, Halaš R, Kühr J (2007) Semilattice structures, vol 30. Heldermann, LemgozbMATHGoogle Scholar
  10. Chajda I, Halas R, Länger H (2001) Orthomodular implication algebras. Int J Theoret Phys 40(11):1875–1884MathSciNetCrossRefzbMATHGoogle Scholar
  11. Chajda I, Kolařík M (2008) Nearlattices. Discrete Math 308(21):4906–4913MathSciNetCrossRefzbMATHGoogle Scholar
  12. Cornish W, Hickman RC (1978) Weakly distributive semilattices. Acta Mathematica Hungarica 32(1):5–16MathSciNetCrossRefzbMATHGoogle Scholar
  13. Czelakowski J (2001) Protoalgebraic logics. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  14. Davey BA, Priestley HA (2002) Introduction to lattices and order. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  15. Font JM, Jansana R (2009) A general algebraic semantics for sentential logics, Lecture Notes in Logic, vol. 7, 2 edn. The Association for Symbolic LogicGoogle Scholar
  16. Font JM, Jansana R, Pigozzi D (2003) A survey of abstract algebraic logic. Stud Logica 74(1–2):13–97MathSciNetCrossRefzbMATHGoogle Scholar
  17. Font JM, Verdú V (1991) Algebraic logic for classical conjunction and disjunction. Stud Logica 50(3–4):391–419MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hickman R (1980) Join algebras. Commun Algebra 8(17):1653–1685MathSciNetCrossRefzbMATHGoogle Scholar
  19. Monteiro A (1980) Sur les algèbres de heyting symétriques. Portugaliae Mathematica 39(1–4):1–237MathSciNetzbMATHGoogle Scholar
  20. Rebagliato J, Verdú V (1993) On the algebraization of some gentzen systems. Fundam Inform 18:319–338MathSciNetzbMATHGoogle Scholar
  21. Rebagliato J, Verdú V (1995) Algebraizable gentzen systems and the deduction theorem for gentzen systems. Mathematics Preprint Series 175Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Fac. de Cs. Exactas y NaturalesUniversidad Nacional de La PampaSanta RosaArgentina

Personalised recommendations