# Enriched Grothendieck ring

The Grothendieck ring of a structure is a ring whose certain "positive" elements represent the definable parts of the structure, and the sum and the product correspond to the disjoint union and the Cartesian product. In order to keep track of these data, as well as the structure itself, we consider the notion of "enriched ring": a ring (commutative unitary associative) endowed with a subset $P$ of "positive elements" and an element $a$ satisfying the axioms:

- $a \in P$,

- $P + P$,

- $P.P \subset P$,

- $P-P = A.$

To any structure $M$, we can associate an enriched Grothendieck ring $(K_0(M), [M], P)$ where

- $K_0(M)$ is the Grothendieck ring of $M$,

- $[M]$ is the class of $M$ in $K_0(M)$,

- $P$ is the set of "positive elements of $K_0(M)$" that is the set of elements $x\in K_0(M)$ such that there exists $A$, a definable set of $M$, with $x=[A]$.

One check that any enriched ring with underlying ring $\mathbb{Z}$ has the form $(\mathbb{Z}, k, \mathbb{Z})$ where $k\in \mathbb{Z}$ or of the form $ (\mathbb{Z}, k, \mathbb{N})$ with $k>0$.

We prove that any enriched ring with underlying ring $\mathbb{Z}$ is indeed isomorphic to the enriched Grothendieck ring of a structure.