×

Finite distributive nearlattices. (English) Zbl 07369640

Summary: Our main goal is to develop a representation for finite distributive nearlattices through certain ordered structures. This representation generalizes the well-known representation given by Birkhoff for finite distributive lattices through finite posets. We also study finite distributive nearlattices through the concepts of dual atoms, boolean elements, complemented elements and irreducible elements. We prove that the sets of boolean elements and complemented elements form semi-boolean algebras. We show that the set of boolean elements of a finite distributive lattice is a boolean lattice.

MSC:

06-XX Order, lattices, ordered algebraic structures
06Dxx Distributive lattices
03Gxx Algebraic logic
06Axx Ordered sets
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abbott, J., Semi-boolean algebra, Mat. Vesn., 4, 19, 177-198 (1967) · Zbl 0153.02704
[2] Araújo, J.; Kinyon, M., Independent axiom systems for nearlattices, Czechoslov. Math. J., 61, 4, 975-992 (2011) · Zbl 1249.06003
[3] Balbes, R.; Dwinger, P., Distributive Lattices (1974), University of Missouri Press · Zbl 0321.06012
[4] Birkhoff, G., Rings of sets, Duke Math. J., 3, 3, 443-454 (1937) · JFM 63.0832.02
[5] Calomino, I., Supremo álgebra distributivas: una generalización de las álgebra de Tarski (2015), Universidad Nacional del Sur, PhD thesis
[6] Calomino, I., Note on α-filters in distributive nearlattices, Math. Bohem., 144, 3, 241-250 (2019) · Zbl 1474.06044
[7] Calomino, I.; Celani, S., A note on annihilators in distributive nearlattices, Miskolc Math. Notes, 16, 1, 65-78 (2015) · Zbl 1340.06007
[8] Calomino, I.; Celani, S.; González, L. J., Quasi-modal operators on distributive nearlattices, Rev. Unión Mat. Argent., 61, 2, 339-352 (2020) · Zbl 1484.06046
[9] Calomino, I.; González, L. J., Remarks on normal distributive nearlattices, Quaest. Math., 44, 4, 513-524 (2021) · Zbl 1493.06002
[10] Celani, S.; Calomino, I., Stone style duality for distributive nearlattices, Algebra Univers., 71, 2, 127-153 (2014) · Zbl 1301.06030
[11] Celani, S.; Calomino, I., On homomorphic images and the free distributive lattice extension of a distributive nearlattice, Rep. Math. Log., 51, 57-73 (2016) · Zbl 1390.06008
[12] Celani, S.; Calomino, I., Distributive nearlattices with a necessity modal operator, Math. Slovaca, 69, 35-52 (2019) · Zbl 1496.06015
[13] Chajda, I.; Halaš, R.; Kühr, J., Semilattice Structures, Research and Exposition in Mathematics, vol. 30 (2007), Heldermann Verlag: Heldermann Verlag Lemgo · Zbl 1117.06001
[14] Chajda, I.; Kolařík, M., Ideals, congruences and annihilators on nearlattices, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math., 46, 1, 25-33 (2007) · Zbl 1147.06002
[15] Chajda, I.; Kolařík, M., Nearlattices, Discrete Math., 308, 21, 4906-4913 (2008) · Zbl 1151.06004
[16] Davey, B.; Priestley, H., Introduction to Lattices and Order (2002), Cambridge University Press · Zbl 1002.06001
[17] González, L. J., The logic of distributive nearlattices, Soft Comput., 22, 9, 2797-2807 (2018) · Zbl 1398.06019
[18] González, L. J., Selfextensional logics with a distributive nearlattice term, Arch. Math. Log., 58, 219-243 (2019) · Zbl 1506.03129
[19] González, L. J.; Calomino, I., A completion for distributive nearlattices, Algebra Univers., 80, 48 (2019) · Zbl 1455.06004
[20] Grätzer, G., Lattice Theory: Foundation (2011), Springer Science & Business Media · Zbl 1233.06001
[21] Halaš, R., Subdirectly irreducible distributive nearlattices, Miskolc Math. Notes, 7, 141-146 (2006) · Zbl 1120.06003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.