Abstract

We compute the model-theoretic Grothendieck ring, \\), of a dense linear order with or without end points, \\), as a structure of the signature \, and show that it is a quotient of the polynomial ring over \ generated by \\) by an ideal that encodes multiplicative relations of pairs of generators. This ring can be embedded in the polynomial ring over \ generated by \. As a corollary we obtain that a DLO satisfies the pigeon hole principle for definable subsets and definable bijections between them—a property that is too strong for many structures.