DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications

Article in press


Authors:

H. Gaitan

Hernando Gaitan

Universidad Nacional de Colombia, Facultad de Ciencias, Departmanto de Matematicas

email: hgaitano@unal.edu.co

0000-0001-5691-8134

Title:

Duality for Stonean Hilbert algebras

PDF

Source:

Discussiones Mathematicae - General Algebra and Applications

Received: 2023-07-01 , Revised: 2024-04-09 , Accepted: 2024-04-11 , Available online: 2025-02-19 , https://doi.org/10.7151/dmgaa.1473

Abstract:

In this paper we characterize the dual space of Stonean Hilbeert algebras, a special kind of Hilbert algebras and we study the relationship between their structura and the monoid of their endomorphisms

Keywords:

Hilbert algebras, deductve system, Stone identity.

References:

  1. D. Bu\c{sneag}, On the maximal deductive systems of a bounded Hilbert algebra, Bull. Math. Soc. Math R.S. de Roumanie 31 (79) No. 1 (1987).
  2. D. Buşneag, Categories of Algebraic Logic, Ed. Academiei Române, 2006.
  3. S.A Celani, $\alpha$-ideals and $\alpha$-deductive systems in bounded Hilbert algebras, J.Mult.-Valued Logic & Soft Computing 21 (2013) 493–510.
  4. S.A. Celani and D. Montangie, Hilbert algebras with supremum, Algebra Univ. 67(3) (2012) 237–255.
    https://doi.org/10.1007/s00012-012-0178-z
  5. S.A. Celani, L.M. Cabrer and D. Montangie, Representation and duality for Hilbert algebras, Central Eur. J. Math. 7(3) (2009) 463–478.
    https://doi.org/10.2478/s11533-009-0032-5
  6. S.A. Celani, A note on homomorphisms of Hilbert algebras, Int. J. Math. and Math. Sci. 29(1) (2002) 55–61.
    https://doi.org/10.1155/S0161171202011134
  7. S.A. Celani, Notes on bounded Hilbert with supremum, Acta Sci. Math. 80 (2014) 3–19.
  8. A. Diego, Sur les algebres de Hilbert, Collection de Logique Mathematique, Ser. A (Ed. Hermann, Paris) 21 (1966) 1–52.
  9. I. Chajda, R. Halas and J. Kuhr, Semilattice Structures, Research and Exposition in Mathematics 30 (Heldermann Verlag, 2007).
  10. Ch.T. Dan, Hilbert Algebras of Fractions, Int. J. Math. and Math. Sci. Volume 2009.
    https://doi.org/10.1155/2009/589830
  11. A. Figallo, E. Pick, S. Saad and M. Figallo, Free algebras in varieties of Hilbert algebras with supremum generated by finite chains, arXiv:1307.8184v1 [math.LO] 2013.
  12. H. Gaitán, Congruences and closure endomorphisms of Hilbert algebras, Commun. Algebra 43 (2015) 1135–1145.
    https://doi.org/10.1080/00927872.2013.865039
  13. H. Gaitán, Duality for Hilbert algebras with supremum: an application, Math. Bohem. 142(3) (2017) 263–276.
    https://doi.org/10.21136/MB.2017.0056-15
  14. H. Gaitán, Hilbert algebras with supremum generated by finite chains, Math. Slovaca 69(4) (2019) 953–963.
    https://doi.org/10.1515/ms-2017-0262
  15. F. Guzman and C. Lynch, Varieties of positive implicative BCK-algebras subdirectly irreducible and free algebras, Math. Japonica 37 (1992) 27–39.
  16. P.M. Idziak, Lattice operations in BCK-algebras, Math. Japonica 29(6) (1984) 839–846.
  17. P.M. Idziak, Filters and congruences relations in BCK-algebras, Math. Japonica 29(6) (1984) 975–980.
  18. M. Kondo, Hilbert algebras are dual isomorphic to positive implicative BCK-algebras, Math. Japonica 49(2) (1999) 265–268.
  19. V. Koubek and H. Radovanská, Algebras determined by their endomorphism monoids, Cahiers de Topologie et Géometrie Deffréntielle Catégoriques 35(3) (1994) 187–225.
  20. A.S. Nasab and A.B. Saeid, Stonean Hilbert algebra, J. Intelligent & Fuzzy Systems 30 (2016) 485–492.
    https://doi.org/10.3233/ifs-151773
  21. A.S. Nasab and A.B. Saeid, Some results in local Hilbert algebras, Math. Slovaca 67 (2017) 541–552.
    https://doi.org/10.1515/ms-2016-0288
  22. R. Rasiowa, An Algebraic Approach to Non-classical Logics, Studies in Logic and the Foundations of Mathematics 8 (North Holland, Amsterdam Elsevier, New York, 1974).

Close