Abstract
A near-Heyting algebra is a join-semilattice with a top element such that every principal upset is a Heyting algebra. We establish a one-to-one correspondence between the lattices of filters and congruences of a near-Heyting algebra. To attain this aim, we first show an embedding from the lattice of filters to the lattice of congruences of a distributive nearlattice. Then, we describe the subdirectly irreducible and simple near-Heyting algebras. Finally, we fully characterize the principal congruences of distributive nearlattices and near-Heyting algebras. We conclude that the varieties of distributive nearlattices and near-Heyting algebras have equationally definable principal congruences.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abbott, J.C.: Semi-Boolean algebra. Matematički Vesnik 4(19), 177–198 (1967)
Araújo, J., Kinyon, M.: Independent axiom systems for nearlattices. Czech. Math. J. 61(4), 975–992 (2011)
Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)
Calomino, I.: Supremo álgebra distributivas: una generalización de las álgebra de Tarski. Ph.D. thesis, Universidad Nacional del Sur (2015)
Calomino, I., Celani, S.: A note on annihilators in distributive nearlattices. Miskolc Math. Notes 16(1), 65–78 (2015)
Celani, S., Cabrer, L., Montangie, D.: Representation and duality for Hilbert algebras. Cent. Eur. J. Math. 7(3), 463–478 (2009)
Celani, S., Calomino, I.: Stone style duality for distributive nearlattices. Algebra Universalis 71(2), 127–153 (2014)
Celani, S., Calomino, I.: On homomorphic images and the free distributive lattice extension of a distributive nearlattice. Rep. Math. Logic 51, 57–73 (2016)
Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures, vol. 30. Heldermann, Lemgo (2007)
Chajda, I., Kolařík, M.: Ideals, congruences and annihilators on nearlattices. Acta Univ. Palacki. Olomuc. Fac. Rer. Nat. Math. 46(1), 25–33 (2007)
Chajda, I., Kolařík, M.: Nearlattices. Discret. Math. 308(21), 4906–4913 (2008)
Cornish, W., Hickman, R.C.: Weakly distributive semilattices. Acta Math. Hungar. 32(1), 5–16 (1978)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)
Day, R.A.: A note on the congruence extension property. Algebra Universalis 1, 234–235 (1971)
González, L.: The logic of distributive nearlattices. Soft Comput. 22(9), 2797–2807 (2018)
Grätzer, G.: Universal Algebra, 2nd edn. Springer, Berlin (2008)
Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Boston (2011)
Halaš, R.: Subdirectly irreducible distributive nearlattices. Miskolc Math. Notes 7, 141–146 (2006)
Hickman, R.: Join algebras. Comm. Algebra 8(17), 1653–1685 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by M. Ploščica
This work was partially supported by Universidad Nacional de La Pampa (Facultad de Ciencias Exactas y Naturales) under the Grant P.I. 64 M, Res. 432/14 CD. The first author was also partially supported by CONICET under the Grand PIP 112-20150-100412CO.
Rights and permissions
About this article
Cite this article
González, L.J., Lattanzi, M.B. Congruences on near-Heyting algebras. Algebra Univers. 79, 78 (2018). https://doi.org/10.1007/s00012-018-0560-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-018-0560-6