Average-Rare Order Ideals in Functional Preorders
Masahiro Hachimori, Kenji Kashiwabara
TL;DR
The paper addresses whether the order-ideal family $\mathcal{I}(V, ≤)$ of a functional preorder is average-rare, formalized as $NDS(\mathcal{I}(V, ≤)) ≤ 0$, within the context of Frankl-type conjectures. The authors encode preorders via a function $f: V \to V$, reduce to rooted forests through a trace-based reduction, and prove nonpositivity by induction, with a complete Lean 4 formalization. They prove a Main Theorem for all functional preorders and a Secondary Main Theorem for rooted forests, connect functional preorders to rooted-set representations with singleton stems, and offer a conjecture extending to larger rooted-set stems to push toward broader Frankl-type results. The work provides a rigorously verified instance of average-rare for a natural class of closure systems and lays out a concrete path for extending Frankl-type results to more general intersection-closed families.
Abstract
We prove that for the preorder induced by a function f: V -> V, the family of all order ideals is average-rare, that is, its normalized degree sum (nds) is nonpositive. As a base case in our reduction, we establish the same result for functional partial orders (or rooted forests). We also propose a conjecture related to Frankl's Conjecture. All proofs have been formally verified in the proof assistant Lean 4.