We develop a first-order theory of ordered transexponential fields in the language { + , ⋅ , 0 , 1 , < , e , T } , where e and T stand for unary function symbols. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural valuation. We establish necessary and sufficient conditions on the value group of an ordered exponential field ( K , e ) to admit a transexponential function T compatible with e . Moreover, we give a full characterisation of all countable ordered transexponential fields in terms of their valuation theoretic invariants.

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Abstract Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for a t-Henselian non-Henselian ordered field elementarily equivalent to a Henselian field with a specified value group.

Abstract We give a valuation theoretic characterization for a real closed field to be recursively saturated. This builds on work in [9], where the authors gave such a characterization forκ-saturation, for a cardinal $\kappa \ge \aleph _0 $ . Our result extends the characterization of Harnik and Ressayre [7] for a divisible ordered abelian group to be recursively saturated.

Abstract The lexicographic power ΔΓ of chains Δ and Γ is, roughly, the Cartesian power ΠγЄΓΔ totally ordered lexicographically from the left. Here the focus is on certain powers in which either Δ = ℝ or ℚ = ℝ, with emphasis on when two such powers are isomorphic and on when ΔΓ is 2-homogeneous. The main results are: (1) For a countably infinite ordinal (2) ℝℝ ≄ ℝℚ. (3) For Δ a countable ordinal ≥ 2, Δℝ with its smallest element deleted, is 2-homogeneous.

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over (the prime field of characteristicp) with values in the ordinals (cf. [F]).