Additive functions on a lattice Consider a lattice $L$. Can one classify all functions $f:L\rightarrow \mathbb{R}$, that satisfy
$f(a \wedge b)+f(a\vee b) = f(a)+f(b)$.
Some examples are the the set of all finite subsets of a given set $S$. Then every such function is uniquely determined by the element $(f(\{s\}))_{s\in S}\in \prod_S\mathbb{R}$ plus the value on the empty set. Indeed this gives a vector space isomorphism from the set of all such functions to $\mathbb{R}^{|S|+1}$.
For other lattices there are also other additive functions arising naturally. For example if one considers the set of all natural numbers (without $0$) ordered by divisibility and assigns to a natural number (for a fixed prime $p$) the biggest $n$, so that $p^n$ divides this number. The logarithm is another example for such a function.
More generally a ultrafilter on the underlying poset can also be viewed as such a function. So there are a lot of interesting examples. There are even more interesting examples like lattices of measurable sets and their measures, Euler-characteristic of subcomplexes and so on. So my question is: Can one classify the set of all those functions, probably in terms of filters / ultrafilters on the underlying poset?
 A: An additive function with the additional property that $f(a) < f(b)$ whenever $a < b$ (i.e., strict monotonicity) can exist only when the lattice is modular.  A partial converse was mentioned in a comment above: If the lattice is modular and has finite height, so that it has an integer-valued rank function, then this rank function is additive.  But modular lattices of infinite height need not support any strictly monotone additive function. Any additive function on a non-modular lattice factors through the projection to a modular quotient lattice.  I believe all this information is in Birkhoff's classic book "Lattice Theory".
A: This was done by Birkhoff [1] for modular lattices and extended by Fleischer and Traynor [2] to all lattices, see https://www.findstat.org/StatisticsDatabase/St001617.
[1] Birkhoff, G. Lattice theory. (available here)
[2] Fleischer, I., Traynor, T. Group-valued modular functions.  Algebra Universalis 14 (1982), no. 3, 287–291. (DOI link)
A: Update: My answer is incorrect.  See comments below.
This answer may be obvious, and it probably isn't what you were looking for, but I realized that for any lattice $L$, there is a subset $M$, in which any $f:M\rightarrow \mathbb{R}$ extends uniquely to an additive $f$ on $L$.
I can't think of any great characterization of this set, except to just pick it using the axiom of choice, i.e. well-order $L$ and build both the set $M$ of generators and $N$ of determined points.  For each $x \in L$, add it to $M$ if $M\cup N$ doesn't contain a three elements of $a,b,a\wedge b,a \vee b$ where $x$ is the fourth.  Else the value $f(x)$ of any additive $f$ is determined by the values of $f$ on the elements in $M\cup N$, which are in turn determined by the values on $M$. So add $x$ to $N$.
In the linear case, $M$ is just all of $L$.  In the case of a finite Boolean algebra with atoms, then $M$ can be set of the atoms plus the bottom element.
