Let $\mathfrak L$ be a lattice and $f:\mathfrak L\to \mathbb R$ be a function. We say that $f$ obeys $\mathfrak L$ if for any $A,B\in \mathfrak L$ we have
$$f(A\vee B)=f(A)+f(B)-f(A\wedge B)$$
I expect that such definition already exists. Any reference or comment is welcome.