# Valuations, simplicial complexes and arithmetics

I am currently working on an Introduction to geometric probability (Klein, Rota, 1997). This book is very stimulating, and I find myself toying with the subject, and a note in particular (pp 95-97 for the lucky ones who keep this book close).

So, before I state my questions, here is some context. One of the central subject of this book is the study of valuations over the set of polyconvex subsets of $\mathbb{R}^n$. A valuation is a finitely additive measure (that is, $\mu$ is a valuation if $\mu (A \cup B) = \mu (A) + \mu (B) - \mu (A \cap B)$ whenever that makes sense), and the polyconvex sets are the finite union of compact convex sets. Hence, to define a valuation on the polyconvexes one only need to define its value for compact convex sets. Moreover, one may want additional properties, for instance the invaraince of the valuation under rigid motions, or some form of continuity. Such valuations include the volume, the surface, or the Euler characteristic.

Now, for a more abstract point of view, one may consider a (finite, to keep it simple) partially ordered set $(X, \le)$, and develop the same theory. A simplex is a set $\{y \in X : y \leq x\}$ for some $x$ (or the empty set), and a simplicial complex is a finite union of simplexes. One can then define a valuation over the set of simplexes, and extend it to the simplicial complexes. An invariant valuation is then a valuation which gives the same value to any two order-isomorphic simplicial complexes (or simplexes).

Here come my questions. I was trying to translate some arithmetical objects in terms of simplicial complexes and valuations. For instance, let me take $X = \mathbb{N}^*$, and define the order $x \preceq y \iff x|y$. A simplex is a set $\hat{x} = \{y \in \mathbb{N}^* : y|x\}$, and two simplexes $\hat{x}$ and $\hat{y}$ are isomorphic if the multi-sets $\{v_p (x) : p \in \mathcal{P}\}$ and $\{v_p (y) : p \in \mathcal{P}\}$ (where $v_p$ is the p-adic valuation) are identical. Hence, invariant valuations can be generated by the number of divisors ($\mu (\hat{x}) = d (x)$),and I think the Liouville and Möbius functions.

First question : which arithmetic functions can one write as valuations, i.e. as $f(x) = \mu (x)$ or $f(x) = \mu (\{1, \cdots, x\})$ for some invariant valuation $x$ ? I am interested, for instance, by the Euler function, for which I have seen no evident translation of this kind. Conversely, do geometric quantities such as the Euler characteristic generate interesting arithmetic functions, and do they share more than a formal link ?

Second question : is looking at $\mathbb{N}^*$ as some kind of infinite-dimensional simplicial complex some kind of abstract nonsense, or does it have some value (i.e. can you learn something about it, or discover easily some properties, with this kind of interpretation) ?

Third question : I was trying to make the same kind of construction to interpret the p-adic valuations as valuations, thanks to identities such as $v_p (m) + v_p (n) = v_p (gcd (m,n)) + v_p (lcm (m,n))$ and defining valuations over the ideals of $\mathbb{Z}$, but I failed. Is there a way to do it ?

I am aware that my questions are not precise at all. That's what you get when you play with concept you have just discovering. I would be very thankful for references.

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