I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just be a simple reformulation but it looks cool and might give a new perspective. Since I made no progress for some time and my experience with lattices is very little, I thought to share this here to see how far one can get and maybe prove the conjecture in some special cases.

Let $L$ be a finite lattice with minimum $m$ and maximum $M$ and $m \neq M$. The incidence algebra $A=A_L$ of $L$ is by definition the $K$-algebra with basis $p_y^x$ for each $x \leq y$ (where we write $e_i :=p_i^i$) with multiplication $p_{y_1}^{x_1} p_{y_2}^{x_2}= \delta_{x_2,y_1} p_{y_2}^{x_1}$ where $\delta_{i,j}$ is the Kronecker delta. $A$ is isomorphic to the quiver algebra $KQ/I$ where $Q$ is the Hasse quiver of $L$ and the relations $I$ are such that any two paths with the same start and ending points get identified.

Then the Union-closed sets conjecture is equivalent to the statement that there exists a join-irreducible element $x \in L$ with $|[x,M]| \leq \frac{|L|}{2}$ (see conjecture 1 in https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.081/Henning/UCSurvey.pdf ).

Now let $A$ be the incidence algebra of $L$ and note that to every point $x \in L$ we have a unique indecomposable injective module (we use right modules) $e_x A$, injective indecomposable module $D(Ae_x)$ and simple module $S_x=top(e_x A)=socle(D(Ae_x))$. For example, as a vector space $e_x A$ is just the span of the elements $p_y^x$ of paths starting at $x$.

Then the conjecture is equivalent to the following statement (in case I made no mistake):

There exists an indecomposable projective module $e_x A$ with $injdim(S_x)=1$ (this is equivalent to $x$ being join-irreducible) that has a minimal injective coresolution $I_i$ such that $\dim I_0 \leq 2 (\dim I_1 - \dim I_2 + \dim I_3 - ....)$.

(To prove the equivalence, note that $|[x,M]| \leq \frac{|L|}{2}$ is equivalent to $dim(e_x A) \leq dim ( \Omega^{-1}(e_x A))$ and then replace $e_x A$ and $ \Omega^{-1}(e_x A)$ by the terms $dim(I_i)$ using the euler characteristic of the minimal injective coresolution, which is finite since $A$ has finite global dimension).

Now this seems to be a nice formulation, but calculating the minimal injective coresolution seems to be extremely complicated for lattices. This was done in https://arxiv.org/pdf/2009.07170.pdf in the distributive case and the same method just gives a non-minimal injective coresolution in general for a non-distributive lattice, which does not help much.

Main question: Can this homological approach work to give a proof of the Union-closed sets conjecture in some special cases?

It might be interesting to also study the terms $I_i$ for other properties. For example I wonder whether the following is true:

Question: Let $x$ be join-irreducible (equivalently $injdim(S_x)=1$) with a minimal injective coresolution $I_i$ of $e_x A$. Do we have that the sequence $dim(I_i)$ is weakly decreasing, that is $dim(I_i) \geq dim(I_{i+1})$ for all $i$?

Some random tests and small cases found no counterexample to this yet.


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