Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701Frankl's Conjecture for a subclass of semimodular lattices1972068573010.29252/cgasa.11.1.197ENVinayakJoshiDepartment of Mathematics, Savitribai Phule Pune University (Formerly, University of Pune) Ganeshkhind Road, Pune - 4110070000-0001-9105-4634BalooWaphareDepartment of Mathematics, Savitribai Phule Pune University,
Pune-411007, India.Journal Article20180701 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.https://cgasa.sbu.ac.ir/article_85730_335445da865e1a5e147830cee5b78a6e.pdf