A note on ${\aleph }_{\alpha }$-saturated o-minimal expansions of real closed fields

Abstract

We give necessary and sufficient conditions for a polynomially bounded o-minimal expansion of a real closed field (in a language of arbitrary cardinality) to be ${\aleph }_{\alpha }$-saturated. The conditions are in terms of the value group, residue field, and pseudo- Cauchy sequences of the natural valuation on the real closed field. This is achieved by an analysis of types, leading to the trichotomy. Our characterization provides a construction method for saturated models, using fields of generalized power series.

Key words and phrases. natural valuation, value group, residue field, pseudo- Cauchy sequences, polynomially bounded o-minimal expansion of a real closed field, definable closure, dimension, saturation. To appear in Algebra and Logic volume 54 Issue 5 November 2015