Isomorphism in Union-Closed Sets

Abstract

We prove that for any isomorphism $h:{\mathcal{K}}_1\to {\mathcal{K}}_2$ between pure union-closed families, there exists a hyperisomorphism $H:\bigcup {\mathcal{K}}_1\to \bigcup {\mathcal{K}}_2$ such that $h(A)=\{H(a)\mid a\in A\}$, for all $A\in {\mathcal{K}}_1$. Since every union-closed family forms a lattice under inclusion, this result establishes a strong connection between the two frameworks. More precisely, any such family can be uniquely reconstructed from its lattice up to isomorphism. Hence, the lattice representation provides a faithful encoding, offering a perspective that may yield new insights into problems on union-closed families, including Frankl's union-closed sets conjecture.

12 pages, 1 figures