Definable combinatorics with dense linear orders

TL;DR

This paper computes the model-theoretic Grothendieck ring of dense linear orders, revealing algebraic structure and properties like the pigeon hole principle for definable sets, which are not common in many structures.

Contribution

It provides an explicit description of the Grothendieck ring of dense linear orders, showing it as a quotient of a polynomial ring with relations, and establishes the PHP for definable sets.

Findings

Grothendieck ring of DLO is a quotient of a polynomial ring.

DLO satisfies the pigeon hole principle for definable sets.

The algebraic structure encodes multiplicative relations of generators.

Abstract

We compute the model-theoretic Grothendieck ring, $K_0(\mathcal{Q})$, of a dense linear order (DLO) with or without end points, $\mathcal{Q}=(Q,<)$, as a structure of the signature $\{<\}$, and show that it is a quotient of the polynomial ring over $\mathbb{Z}$ generated by $\mathbb N_+\times(Q\sqcup\{-\infty\})$ by an ideal that encodes multiplicative relations of pairs of generators. As a corollary we obtain that a DLO satisfies the pigeon hole principle (PHP) for definable subsets and definable bijections between them--a property that is too strong for many structures.