On the lattice formulation of the union-closed sets conjecture

Abstract

The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every finite lattice $L$ with more than one element contains a join-irreducible element that is less than or equal to at most half of the elements in $L$. In this work, we obtain several necessary conditions for any counterexample $\tilde{L}$ of minimum size.

Figures (3)

• Figure : Figure 1.1: Example corresponding structures in $L$ and (non-lattice) $L \setminus \{x\}$.
• Figure : Figure 2.1: Example corresponding structures in $\tilde{L}$ and $\hat{L}=\tilde{L}\setminus\{j\}$.
• Figure : Figure 2.2:$\hat{L}=({\uparrow}x)_{\tilde{L}}$ must also be a counterexample to Conjecture 1.1.