Value groups and residue fields of models of real exponentiation

Abstract

Let F be an archimedean field, G a divisible ordered abelian group and h a group exponential on G. A triple (F, G, h) is realised in a non-archimedean exponential field (K, exp) if the residue field of K under the natural valuation is F and the induced exponential group of (K, exp) is (G, h). We give a full characterisation of all triples (F, G, h) which can be realised in a model of real exponentiation in the following two cases: i) G is countable. ii) G is of cardinality κ and κ-saturated for an uncountable regular cardinal κ with κ <κ = κ. Moreover, we show that for any o-minimal exponential field (K, exp) satisfying the differential equation exp ′ = exp, its residue exponential field is a model of real exponentiation.

Embedding the prime model of real exponentiation into o‐minimal exponential fields

“…We denote the corresponding first-order language  or ∪ {𝐸}, where 𝐸 is a unary function symbol, by  exp . Following the terminology of Krapp [6], we call an ordered exponential field  an EXP-field if its exponential satisfies the first-order  exp -sentence expressing the differential equation 𝐸 ′ = 𝐸 with initial condition 𝐸(0) = 1. Although ℝ exp and, more generally, any model of 𝑇 exp is an o-minimal EXP-field, the following is still open:…”

“…We denote the corresponding first‐order language $$\mathcal {L}_{\mathrm{or}}\cup \lbrace E\rbrace$$, where $$E$$ is a unary function symbol, by $$\mathcal {L}_{\exp}$$. Following the terminology of Krapp [6], we call an ordered exponential field $$\mathcal {K}$$ an $$\operatorname{EXP}$$‐field if its exponential satisfies the first‐order $$\mathcal {L}_{\exp}$$‐sentence expressing the differential equation $$E^{\prime}=E$$ with initial condition $$E(0)=1$$. Although $$\mathbb {R}_{\exp}$$ and, more generally, any model of $$T_{\exp}$$ is an o‐minimal $$\operatorname{EXP}$$‐field, the following is still open:Transfer Conjecture Any o‐minimal $$\operatorname{EXP}$$‐field is elementarily equivalent to $$\mathbb {R}_{\exp}$$.…”