Definable linear orders definably embed into lexicographic orders in o-minimal structures

TL;DR

This paper characterizes all definable linear orders in o-minimal structures expanding groups, showing they can be embedded into lexicographic orders, thus generalizing previous results in the real field context.

Contribution

It provides a complete characterization of definable linear orders in o-minimal structures expanding groups, demonstrating their embeddability into lexicographic orders based on o-minimal dimension.

Findings

Definable linear orders embed into lexicographic orders in o-minimal structures.

The embedding dimension equals the o-minimal dimension of the order.

Generalizes previous results from the real field to broader o-minimal structures.

Abstract

We completely characterize definable linear orders in o-minimal structures expanding groups. For example, let (P,<_p) be a linear order definable in the real field R. Then (P,<_p) embeds definably in (R^{n+1},<_l), where <_l is the lexicographic order and n is the o-minimal dimension of P. This improves a result of Onshuus and Steinhorn in the o-minimal group context.