Abstract
In this paper, we prove Frankl’s Union-Closed Sets Conjecture for the class of dismantlable lattices, a more general class than the class of planar lattices. As a consequence of this result, we also prove that an upper semimodular lattice with breadth at most two satisfies the Conjecture.
Similar content being viewed by others
References
Abe T.: Strong semimodular lattices and Frankl’s Conjecture. Algebra Universalis 44, 379–382 (2000)
Abe T., Nakano B.: Lower semimodular types of lattices: Frankl’s Conjecture holds for lower quasi-modular lattices. Graphs Combin. 16, 1–16 (2000)
Ajtai M.: On a class of finite lattices. Period. Math. Hungar. 4, 217–220 (1973)
Baker K.A., Fishburn P.C., Roberts F.S.: Partial orders of dimension 2. Networks 2, 11–28 (1972)
Czédli G., Schmidt E.T.: Frankl’s conjecture for large semimodular and planar semimodular lattices. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 47, 47–53 (2008)
Grätzer G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)
Hunh A.P.: Schwach distributive Verbdnde-I. Acta Sci. Math. (Szeged) 33, 297–305 (1972)
Kelly D., Rival I.: Crowns, fences, and dismantlable lattices. Canad. J. Math. 26, 1257–1271 (1974)
Poonen B.: Union-closed families. J. Combin. Theory Ser. A 59, 253–268 (1992)
Rival I.: Lattices with doubly irreducible elements. Canad. Math. Bull. 17, 91–95 (1974)
Rival I.: Combinatorial inequalities for semimodular lattices of breadth two. Algebra Universalis 6, 303–311 (1976)
Shewale R.S., Joshi V., Kharat V.S.: Frankl’s conjecture and the dual covering property. Graphs Combin. 25, 115–121 (2009)
Stanley R.P.: Enumerative Combinatorics, vol I. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, (1986)
Thakare N.K., Pawar M.M., Waphare B.N.: A structure theorem for dismantlable lattices and enumeration. Period. Math. Hungar. 45, 147–160 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by R. Quackenbush.
Rights and permissions
About this article
Cite this article
Joshi, V., Waphare, B.N. & Kavishwar, S.P. A proof of Frankl’s union-closed sets conjecture for dismantlable lattices. Algebra Univers. 76, 351–354 (2016). https://doi.org/10.1007/s00012-016-0405-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-016-0405-0