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

Abstract

We classify definable linear orders in o-minimal structures expanding groups. For example, let ( P , ≺ ) (P,\prec ) be a linear order definable in the real field. Then ( P , ≺ ) (P,\prec ) embeds definably in ( R n + 1 , > lex ) (\mathbb {R}^{n+1},>_{\text {lex}}) , where > lex >_{\text {lex}} is the lexicographic order and n n is the o-minimal dimension of P P . This improves a result of Onshuus and Steinhorn in the o-minimal group context.