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

Ramakrishnan, Janak

doi:10.1090/S0002-9939-2012-11424-8

Definable linear orders definably embed into lexicographic orders in o-minimal structures
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by Janak Ramakrishnan PDF

Proc. Amer. Math. Soc. 141 (2013), 1809-1819

Abstract:

We classify definable linear orders in o-minimal structures expanding groups. For example, let $(P,\prec )$ be a linear order definable in the real field. Then $(P,\prec )$ embeds definably in $(\mathbb {R}^{n+1},<_{\text {lex}})$, where $<_{\text {lex}}$ 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.

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