Isomorphism in Union-Closed Sets

TL;DR

This paper demonstrates that isomorphisms between pure union-closed families correspond to hyperisomorphisms of their unions, linking lattice structures to set families and enabling potential new approaches to longstanding conjectures.

Contribution

It establishes a correspondence between isomorphisms of union-closed families and hyperisomorphisms of their unions, connecting lattice and set family frameworks.

Findings

Isomorphisms induce hyperisomorphisms of unions

Union-closed families can be reconstructed from their lattice structures

Lattice representation preserves the structure of union-closed families

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.