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Frankl’s conjecture for a subclass of semimodular lattices. (English) Zbl 07122741
Summary: 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.
06A07 Combinatorics of partially ordered sets
05D05 Extremal set theory
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