Mathematica Bohemica, online first, 14 pp.

Duality for Hilbert algebras with supremum: An application

Hernando Gaitán

Received September 24, 2015.   First published January 2, 2017.

Hernando Gaitán, Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia, Ciudad Universitaria, Bogotá, Colombia, e-mail:

Abstract: We modify slightly the definition of $H$-partial functions given by Celani and Montangie (2012); these partial functions are the morphisms in the category of $H^\vee$-space and this category is the dual category of the category with objects the Hilbert algebras with supremum and morphisms, the algebraic homomorphisms. As an application we show that finite pure Hilbert algebras with supremum are determined by the monoid of their endomorphisms.

Keywords: Hilbert algebra; duality; monoid of endomorphisms; BCK-algebra

Classification (MSC 2010): 06A12, 03G25

DOI: 10.21136/MB.2017.0056-15

Full text available as PDF.

  [1] J. Berman, W. J. Blok: Algebras defined from ordered sets and the varieties they generate. Order 23 (2006), 65-88. DOI 10.1007/s11083-006-9032-2 | MR 2258461 | Zbl 1096.08002
  [2] S. A. Celani: A note on homomorphisms of Hilbert algebras. Int. J. Math. Math. Sci. 29 (2002), 55-61. DOI 10.1155/S0161171202011134 | MR 1892332 | Zbl 0993.03089
  [3] S. A. Celani, L. M. Cabrer: Duality for finite Hilbert algebras. Discrete Math. 305 (2005), 74-99. DOI 10.1016/j.disc.2005.09.002 | MR 2186683 | Zbl 1084.03050
  [4] S. A. Celani, L. M. Cabrer, D. Montangie: Representation and duality for Hilbert algebras. Cent. Eur. J. Math. 7 (2009), 463-478. DOI 10.2478/s11533-009-0032-5 | MR 2534466 | Zbl 1184.03064
  [5] S. A. Celani, D. Montangie: Hilbert algebras with supremum. Algebra Univers. 67 (2012), 237-255. DOI 10.1007/s00012-012-0178-z | MR 2910125 | Zbl 1254.03117
  [6] A. Diego: Sur les algèbres de Hilbert. Collection de logique mathématique. Ser. A, vol. 21. Gauthier-Villars, Paris; E. Nauwelaerts, Louvain (1966). MR 0199086 | Zbl 0144.00105
  [7] H. Gaitán: Congruences and closure endomorphisms of Hilbert algebras. Commun. Algebra 43 (2015), 1135-1145. DOI 10.1080/00927872.2013.865039 | MR 3298124 | Zbl 1320.03090
  [8] P. M. Idziak: Lattice operations in BCK-algebras. Math. Jap. 29 (1984), 839-846. MR 0803438 | Zbl 0555.03030
  [9] K. Iseki, S. Tanaka: An introduction to the theory of BCK-algebras. Math. Jap. 23 (1978), 1-26. MR 0500283 | Zbl 0385.03051
  [10] M. Kondo: Hilbert algebras are dual isomorphic to positive implicative BCK-algebras. Math. Jap. 49 (1999), 265-268. MR 1687626 | Zbl 0930.06017

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