The Non-Cancelling Intersections Conjecture

TL;DR

The paper investigates whether the union of a family of sets, as computed by inclusion-exclusion from non-cancelling intersections, can itself be constructed using only disjoint union and subset-complement from those same non-cancelling intersections. It develops two reformulations: (i) an intersection-lattice formulation (NCI) and (ii) a Boolean-lattice, dot-algebra formulation (NCPD), proving their equivalence. A key conceptual advance is the reduction to tight intersection lattices and the introduction of the lifting lemma to transfer local constructions to the full Boolean lattice. A partial result shows that if a non-trivial Möbius-zero occurs, one can still realize the target downset by excluding that zero and lifting back, using adjacent-pair moves and Euler-characteristic arguments. The work lays groundwork for further extensions, including decomposability conditions and potential counterexample searches, and connects these abstract structures to practical questions in probabilistic query evaluation and database theory.

Abstract

In this note, we present a conjecture on intersections of set families, and a rephrasing of the conjecture in terms of principal downsets of Boolean lattices. The conjecture informally states that, whenever we can express the measure of a union of sets in terms of the measure of some of their intersections using the inclusion-exclusion formula, then we can express the union as a set from these same intersections via the set operations of disjoint union and subset complement. We also present a partial result towards establishing the conjecture.

Figures (5)

• Figure 1: Hasse diagrams of the intersection lattices from Example \ref{['expl:intersection-lattices']}. The integer value besides each node $n$ is $\mu_L(n,\hat{1})$ and is computed top-down following Definition \ref{['def:mobius']}. The orange nodes are the non-cancelling non-trivial intersections.
• Figure 2: A tight intersection lattice of seven sets, and a witnessing tree showing that it does not violate the conjecture. For brevity we omit curly braces and commas when writing sets, i.e., $ab$ stands for $\{a, b\}$.
• Figure 3: Various witnessing trees. The sets that internal nodes correspond to are shown in orange.
• Figure 4: Visual representation of the configuration $\mathcal{C}$ from Example \ref{['expl:0111001001110010']} and of its associated Möbius function $\hat{\mu}_{\mathbbm{B}_S,\mathcal{C}}$ beside each node. Here, $\mathcal{C}$ consists of all the colored nodes.
• Figure 5: Imagine that the path at the top occurs in $G_S$ for some configuration $\mathcal{C}$ of $\mathbbm{B}_S$ (in orange), and let $P = \{X_0,\ldots,X_4\}$. The consecutive orange configurations of $\mathbbm{B}_S$ are obtained by single steps of the transformation. The total transformation illustrates what is called teleporting in Lemma \ref{['lem:chaining']}: we go from $\mathcal{C}$ to $\mathcal{C}'$ in the bottom path by teleporting $X_1$ to $X_4$.

Theorems & Definitions (55)

• definition 1
• definition 2: stanley2011enumerative, Section 3.7.2
• Remark 1
• definition 3
• proposition 1: Möbius inversion formula; see, e.g., Prop. 3.7.1 of stanley2011enumerative
• proposition 2
• proof
• definition 4
• proposition 3
• definition 5