# Frankl's Conjecture Is True for Lower Semimodular Lattices

@article{Reinhold2000FranklsCI, title={Frankl's Conjecture Is True for Lower Semimodular Lattices}, author={J. Reinhold}, journal={Graphs and Combinatorics}, year={2000}, volume={16}, pages={115-116} }

Abstract. It is shown that every finite lower semimodular lattice L with |L|≥2 contains a join-irreducible element x such that at most |L|/2 elements y∈L satisfy y≥x.

#### Topics from this paper

#### 21 Citations

Strong semimodular lattices and Frankl's conjecture

- Mathematics
- 2000

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.

Frankl's Conjecture for Subgroup Lattices

- Computer Science, Mathematics
- Electron. J. Comb.
- 2017

We show that the subgroup lattice of any finite group satisfies Frankl's Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable… 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.

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

The union-closed sets conjecture almost holds for almost all random bipartite graphs

- Computer Science, Mathematics
- Eur. J. Comb.
- 2017

It is proved that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankls conjecture. Expand

Monoidal networks

- Mathematics
- 2019

In this paper we define and study the notion of a monoidal network, which consists of a commutative ring $R$ and a collection of groups $\Gamma_I$, indexed by the ideals of $R$, with $\Gamma_I$… 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

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

#### References

SHOWING 1-2 OF 2 REFERENCES

Frankl's Conjecture is True for Modular Lattices

- Computer Science, Mathematics
- Graphs Comb.
- 1998

Abstract. It is shown that every finite modular lattice L with |L|≥2 contains a join-irreducible element x∈L such that at most |L|/2 elements y∈L satisfy y≥x.