In a combinatorial computation, I came across the following quantity:

Consider a finite meet semilattice $L$, that is, a finite poset which is closed under $\min$. Denote the least element of $L$ by $0$. Now, define $Z := \{ S \subset L : \min S = 0 \}$. I want to compute the quantity

$$ \chi := \sum_{S \in Z} (-1)^{|S|+1}. $$

I noticed that the complement $Z^c$ of $Z$ in $2^{L}$ (the collection of subsets of $L$ whose meet is *not* $0$) is a simplicial complex on $L$. Moreover, since the Euler characteristic of $2^L$ is $0$, the above quantity $\chi$ is actually just equal to the Euler characteristic of $Z^c$:

$$ \chi = \sum_{S \in Z^c} (-1)^{|S|}. $$

My question is:

- Is there any way to efficiently compute $\chi$ or bounds on it/approximations to it using some properties of $L$? (I already know that in many of the cases I'm interested in $\chi \ne 0$).
- Since (1) seems likely quite difficult to answer, especially in this general context, are there any references that can tell me more about the relationship between $L$ and $Z$ or $Z^c$, $H^{\cdot}(Z^c), \chi$, etc? These seem like natural enough objects to study, but I wasn't able to find a reference on them. When I look up simplicial complexes associated to a semilattice, mostly I find references to the order complex, which seems very different. I could find something called "the ideal zero divisor complex" for rings which seems similar, but I am working with a semilattice, not a ring, and elements, not ideals. I could also find information about zero divisor graphs of semigroups, which is related, but still rather far from the idea I'm looking for.

(And of course, I am actually only concerned about a very particular family of semilattices $L$, which happen to consist of certain partitions of {1,...,N}; the meet here is given by common refinement. I don't want to say too much about them here to avoid making the question too specific, but suffice to say I don't necessarily need a completely general answer to this question.)