Grothendieck Semirings and Definable Endofunctions

Abstract

We define the Grothendieck semiring of a category in a way suitable for the categories of definable sets and functions which occur in Model Theory, and also emphasize Euler characteristics, that is, invariant measures simultaneously additive and multiplicative on the classes of objects of such categories. The semiring of any strongly minimal group, as well as the one of any infinite vector space over any division ring, is isomorphic to the semiring of nonnegative polynomials over the integers. A natural ordering on the semiring is related to an interesting property that definable endofunctions can have, namely, being surjective if and only if being injective. This property holds, in particular, if the structure in question is pseudofinite, but it can be established for vector spaces independently of pseudofiniteness. The class of division rings all whose vector spaces are pseudofinite is axiomatizable, and we use an explicit axiomatization to build a non-pseudofinite vector space. Other results include: extension of an Euler characteristic to interpretable sets; construction of translation-invariant measures on lattices generated by cosets of subgroups; said property of endofunctions for vector spaces and omega-categorical structures; and presentation of the semiring of the category of bounded semilinear sets in an ordered field, and their volume-preserving functions, "up to null sets".