Call a connected (finite) poset $P$ lattice-like in case an element $x \in P$ is join-irreducible (recall that an element a of a poset is join-irreducible if there is no subset $X \subseteq P$ and $a=\lor X$) iff $injdim(S_x)=1$ where $S_x$ is the simple module corresponding to $x$ in the incidence algebra $A$ of $P$. Every lattice is lattice-like but not every poset is lattice-like.

All posets with $\leq 4$ points are lattice like and of all posets with 5 points (there are 44 connected posets with 5 points) only 2 are not lattice-like, namely those two with the following Hasse quiver:

$[ [ "'x1'", "'x2'" ], [ "'x1'", "'x3'" ], [ "'x2'", "'x4'" ], [ "'x2'", "'x5'" ], [ "'x3'", "'x4'" ], 
              [ "'x3'", "'x5'" ] ]$

$[  [ "'x1'", "'x5'" ], [ "'x2'", "'x3'" ], [ "'x2'", "'x4'" ], [ "'x3'", "'x5'" ], [ "'x4'", "'x5'" ] ]$

The (right) UC-condition (we only look at the right condition and not the two-sided here) for a poset $P$ with incidence algebra $A$ says:

>>There exists an indecomposable projective module $e_x A$ with $injdim(S_x)=1$ that has a minimal injective coresolution $I_i$ such that $\dim I_0 \leq 2 (\dim I_1 - \dim I_2 + \dim I_3 - ....)$.

For lattices this condition is equivalent to the truth of the Union-closed sets conjecture (see https://mathoverflow.net/questions/372033/a-homological-algebra-approach-to-the-union-closed-sets-conjecture).

I wonder whether one can generalize the Union-closed sets conjecture from lattices to lattice-like posets, hence the question:

>Question 1: Does every (bounded) lattice-like poset satisfy the UC-condition?

The question has a positive answer for all lattice-like posets with at most 6 points and for all lattice-like bounded posets with at most 8 points. Unfortunately, I can not test the conjecture for more points at the moment, but some random examples also found no counterexmaple. Of course, I can only really expect to see a counterexample or some evidence that the question might be true.

>Question 2: Are there attempts in the literature already to generalise the Union-closed sets conjecture from lattices to a larger class of posets?