Let $\mathcal{L}$ be a meet-semilattice, and denote by $\Delta(\mathcal{L})$ the poset of chains in $\mathcal{L}\setminus\{\hat 0\}$, where $\hat 0$ is the minimum element of $\mathcal{L}$.
Let $\alpha:\mathcal{L}\to \Bbb F$ be any map into a field, such that for all $x\in \mathcal{L}$, if $[\hat 0, x]\cong \prod_{y\in I}[\hat 0, y]$, then $\alpha(x)=\sum_{y\in I}\alpha(y)$. An example of such a map is the rank function on a ranked poset. Another example is the map $x\mapsto |A(\mathcal{L})|_{\leq x}$; mapping $x$ to the number of atoms underneath $x$. Another example is the map $x\mapsto \ln(|\mathcal{L}_{\leq x}|)$.
Then consider the expression:
$$ \begin{align} F^\alpha_\mathcal{L}&=\sum_{c\in \Delta(\mathcal{L})}|\Delta(\mathcal{L})_{\geq c}|\frac{\prod_{x\in c}(\alpha(x)-1)}{\prod_{x\in c}\alpha(x)}\\ &=\sum_{c\in \Delta(\mathcal{L})}\left(\sum_{d\in \Delta(\mathcal{L})_{\geq c}}(-1)^{|d|-|c|}|\Delta(\mathcal{L})_{\geq d}|\right)\prod_{x\in c}\frac{1}{\alpha(x)} \end{align} $$
Just to be clear: $|\Delta(\mathcal{L})_{\geq c}|$ is the number of chains in which the chain $c$ is contained, i.e. the number of extensions of $c$.
The reason for asking is that this is a specific case of a class of poset invariants that I'm investigating. I'm not overly familiar with posets though, so maybe someone else recognises this as being related\similar to some known invariant?
