Frankl's Conjecture for a subclass of semimodular lattices | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 12، دوره 11، Special Issue Dedicated to Prof. George A. Grätzer، مهر 2019، صفحه 197-206 اصل مقاله (585.7 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.29252/cgasa.11.1.197 | ||
| نویسندگان | ||
| Vinayak Joshi1؛ Baloo Waphare2 | ||
| 1Department of Mathematics, Savitribai Phule Pune University (Formerly, University of Pune) Ganeshkhind Road, Pune - 411007 | ||
| 2Department of Mathematics, Savitribai Phule Pune University, Pune-411007, India. | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Union-Closed Sets Conjecture؛ Frankl's Conjecture؛ semimodular lattice؛ adjunct operation | ||
| مراجع | ||
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[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. | ||
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