2
$\begingroup$

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.

$\endgroup$
11
  • $\begingroup$ I am curious, what is Lattice ? I did a problem like this before, but I don't understand if this and that same. $\endgroup$
    – MAN-MADE
    Jul 4, 2017 at 10:44
  • 1
    $\begingroup$ This seems to be related to the concept of additive function in measure theory. Notice among their basic properties: $\mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B)$. $\endgroup$
    – amrsa
    Jul 4, 2017 at 11:30
  • 1
    $\begingroup$ @MANMAID Lattice (order) $\endgroup$
    – amrsa
    Jul 4, 2017 at 11:34
  • $\begingroup$ @amrsa so it is not what you are saying. I don't see measure theory in that link. $\endgroup$
    – MAN-MADE
    Jul 4, 2017 at 11:37
  • 1
    $\begingroup$ This post on MathOverflow seems to be (to some extent) related: Additive functions on a lattice. $\endgroup$
    – Martin Sleziak
    Jul 4, 2017 at 14:19

1 Answer 1

Reset to default
2
$\begingroup$

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

$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .