Frankl's Conjecture for a subclass of semimodular lattices

Document Type : Research Paper

Authors

1 Department of Mathematics, Savitribai Phule Pune University (Formerly, University of Pune) Ganeshkhind Road, Pune - 411007

2 Department of Mathematics, Savitribai Phule Pune University, Pune-411007, India.

10.29252/cgasa.11.1.197

Abstract

 In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)\setminus A(L)| \leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices having breadth at most two.  We provide a very short proof of the Conjecture for the class of lattices having breadth at most two. This generalizes the results of Joshi, Waphare and Kavishwar as well as Czédli and Schmidt.

Keywords

References

[1] Abdollahi, A., Woodroofe, R., and Zaimi, G., Frankl’s Conjecture for subgroup lattices, Electron. J. Combin. 24(3) (2017), P3.25.
[2] Abe, T., Strong semimodular lattices and Frankl’s Conjecture, Algebra Universalis 44 (2000), 379-382.
[3] Abe, T. and Nakano, B., Lower semimodular types of lattices: Frankl’s Conjecture holds for lower quasi-modular lattices, Graphs Combin. 16 (2000), 1-16.
[4] Baker, K.A., Fishburn, P.C., and Roberts, F.S., Partial orders of dimension 2, Networks 2 (1972), 11-28.
[5] Bruhn, H. and Schaudt, O., The journey of the Union-Closed Sets Conjecture, Graphs Combin. 31 (2015), 2043-2074.
[6] Czédli, G. and Schmidt, E.T., Frankl’s conjecture for large semimodular and planar semimodular lattices, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 47 (2008), 47-53.
[7] Grätzer, G., “General Lattice Theory”, Birkhäuser, 1998.
[8] Hunh, A.P., Schwach distributive Verbdnde-I, Acta Sci. Math. (Szeged) 33 (1972), 297-305.
[9] Joshi, V., Waphare, B.N., and Kavishwar, S.P., A proof of Frankl’s Union-Closed Sets Conjecture for dismantlable lattices, Algebra Universalis 76 (2016), 351-354.
[10] Poonen, B., Union-closed families, J. Combin. Theory Ser. A. 59 (1992), 253-268.
[11] Rival, I., Combinatorial inequalities for semimodular lattices of breadth two, Algebra Universalis 6 (1976), 303-311.
[12] Shewale, R.S., Joshi, V., and Kharat, V.S., Frankl’s conjecture and the dual covering property, Graphs Combin. 25(1) (2009), 115-121.
[13] Stanley, R.P., “Enumerative Combinatorics”, Vol I. Wadsworth & Brooks/Cole Advanced Books & Software, 1986.
[14] Stern, M., “Semimodular Lattices”, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1999.
[15] Thakare, N.K., Pawar, M.M., and Waphare, B.N., A structure theorem for dismantlable lattices and enumeration, Period. Math. Hungar. 45(1-2) (2002), 147-160.

Files

Share

How to cite

Statistics