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arXiv · 2501.02637

Isomorphism in Union-Closed Sets

M. J. Moghaddas Mehr
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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.

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M. J. Moghaddas Mehr. 2025-01-05. Isomorphism in Union-Closed Sets. https://arxiv.org/abs/2501.02637

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