# Is there any definition of such functions in lattice theory?

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.

• I am curious, what is Lattice ? I did a problem like this before, but I don't understand if this and that same. Jul 4, 2017 at 10:44
• This seems to be related to the concept of additive function in measure theory. Notice among their basic properties: $\mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B)$. Jul 4, 2017 at 11:30
• @MANMAID Lattice (order) Jul 4, 2017 at 11:34
• @amrsa so it is not what you are saying. I don't see measure theory in that link. Jul 4, 2017 at 11:37
• This post on MathOverflow seems to be (to some extent) related: Additive functions on a lattice. Jul 4, 2017 at 14:19

This is called a valuation on $\mathfrak L$, see Birkhoff's book for more on that.