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.