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Joshi, V., Waphare, B. (2019). Frankl's Conjecture for a subclass of semimodular lattices. Categories and General Algebraic Structures with Applications, 11(Special Issue Dedicated to Prof. George A. Grätzer), 197-206.
Vinayak Joshi; Baloo Waphare. "Frankl's Conjecture for a subclass of semimodular lattices". Categories and General Algebraic Structures with Applications, 11, Special Issue Dedicated to Prof. George A. Grätzer, 2019, 197-206.
Joshi, V., Waphare, B. (2019). 'Frankl's Conjecture for a subclass of semimodular lattices', Categories and General Algebraic Structures with Applications, 11(Special Issue Dedicated to Prof. George A. Grätzer), pp. 197-206.
Joshi, V., Waphare, B. Frankl's Conjecture for a subclass of semimodular lattices. Categories and General Algebraic Structures with Applications, 2019; 11(Special Issue Dedicated to Prof. George A. Grätzer): 197-206.

Frankl's Conjecture for a subclass of semimodular lattices

Article 12, Volume 11, Special Issue Dedicated to Prof. George A. Grätzer, Summer and Autumn 2019, Page 197-206  XML PDF (585.7 K)
Document Type: Research Paper
Authors
Vinayak Joshiorcid 1; 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.
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
Union-Closed Sets Conjecture; Frankl's Conjecture; semimodular lattice; adjunct operation
References
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