# A homological algebra approach to the Union-closed sets conjecture

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.