# Strong semimodular lattices and Frankl's conjecture

@article{Abe2000StrongSL, title={Strong semimodular lattices and Frankl's conjecture}, author={Tetsuya Abe}, journal={algebra universalis}, year={2000}, volume={44}, pages={379-382} }

Abstract. In this paper, we show that Frankl's conjecture holds for strong semimodular lattices. The result is the first step to deal with the case of upper semimodular lattices.

#### 19 Citations

A proof of Frankl’s union-closed sets conjecture for dismantlable lattices

- Mathematics
- 2016

In this paper, we prove Frankl’s Union-Closed Sets Conjecture for the class of dismantlable lattices, a more general class than the class of planar lattices. As a consequence of this result, we also… Expand

Frankl’s Conjecture and the Dual Covering Property

- Mathematics, Computer Science
- Graphs Comb.
- 2009

We have proved that the Frankl’s Conjecture is true for the class of finite posets satisfying the dual covering property.

AN ATTEMPT AT FRANKL'S CONJECTURE

- Mathematics
- 2007

In 1979 Frankl conjectured that in a finite union-closed family F of finite sets, F � {∅} there has to be an element that belongs to at least half of the sets inF .W e prove this when| F| 10.

Frankl's Conjecture for a subclass of semimodular lattices

- Mathematics
- Categories and General Algebraic Structures with Application
- 2019

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… Expand

The Journey of the Union-Closed Sets Conjecture

- Mathematics, Computer Science
- Graphs Comb.
- 2015

The state of the union-closed sets conjecture is surveyed and the proposed solution to this conjecture is shown to be correct on the basis of the inequality of the Following inequality. Expand

FRANKL'S CONJECTURE FOR LARGE SEMIMODULAR AND

- Mathematics
- 2008

A lattice L is said to satisfy (the lattice theoretic version of) Frankl's conjecture if there is a join-irreducible element f 2 L such that at most half of the elements x of L satisfy f x. Frankl's… Expand

Frankl's Conjecture for Large Semimodular and Planar Semimodular

- Mathematics
- 2008

A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x.… Expand

FRANKL ’ S CONJECTURE AND SEMIMODULARITY

A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x.… Expand

A new generic class of Frankl’s families

- Mathematics
- 2013

Frankl’s conjecture states that in a family of sets closed by union F such that F 6= {∅}, there is an element that belongs to at least half of the sets of F. There are several partial results of this… Expand

On averaging Frankl's conjecture for large union-closed-sets

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2009

The sum of the n-2s(a), for all a@?A, is shown to be non-positive and this stronger version does not hold for all union-closed families; however, it is conjecture that it holds for a much wider class of families than considered here. Expand

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