The setup is that we have a finite poset $P$, with a multiplicative rank function $r_{xy}:P\times P\rightarrow \mathbb{N}$, and a symmetric pairing $\langle\ ,\ \rangle:P\times P\rightarrow\mathbb{N}$. By multiplicative rank function we mean $r_{xy}r_{yz}=r_{xz}$ for $x\leq y \leq z$, but no other conditions on $r$, it need not vanish when $x$ and $y$ are incomparable, so I dont think $P$ is ranked in the usual sense. Our poset has a unique minimal element $\hat{0}$, and a distinguished maximal element $1$, but $1$ doesn't necessarily cover every element of $P$. From this, we get an associated automorphism $L:\mathbb{Q}[P]\rightarrow \mathbb{Q}[P]$ given by $$L(y)=\sum_{x\leq y}\frac{r_{\hat{0}x}}{r_{\hat{0}1}}\mu_P(x,y)x.$$

Extending our form to $\mathbb{Q}[P]$ by linearity, we are interested in the functions $f:P\rightarrow \mathbb{Z}$ that satisfy the functional equation: $$f(x)=\sum_{y\in P}f(y)\langle x,L(y)\rangle.$$

My question is, have you seen this bunch of structure before in any other posets? I was told that this might resemble Khazdan Lustzig type recurrences, but I couldn't see how to relate this to the general Khazdan-Luztig-Stanley polynomial of a poset. If you've seen this kind of thing in a different context, that would also be very useful to hear, I don't have much familiarity with these things, and any references that exhibit this setup would be helpful for me.