O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable theories. This is the first in an anticipated series of papers whose aim is the development of model theory for ordered structures of rank greater than one. A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here. These include all (...) ordered structures definable in o-minimal structures. The principal result in this paper, Theorem 5.1, asserts roughly that a decomposable structure [Formula: see text] can be partitioned into finitely many definable subsets such that on each set the restriction of < is a "twisted lexicographic" order. As a consequence, for all n and linear orders ≺ definable on a subset X ⊆ Mn in an o-minimal structure [Formula: see text], there is a definable partition of X such that the restriction of ≺ to each set in the partition is "lexicographic". (shrink)

We give a short proof of the Marker–Steinhorn theorem for o-minimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures.

For an o-minimal expansion R of a real closed field and a set $\fancyscript{V}$ of Th(R)-convex valuation rings, we construct a “pseudo completion” with respect to $\fancyscript{V}$ . This is an elementary extension S of R generated by all completions of all the residue fields of the $V \in \fancyscript{V}$ , when these completions are embedded into a big elementary extension of R. It is shown that S does not depend on the various embeddings up to an R-isomorphism. For polynomially (...) bounded R we can iterate the construction of the pseudo completion in order to get a “completion in stages” S of R with respect to $\fancyscript{V} $ . S is the “smallest” extension of R such that all residue fields of the unique extensions of all $V \in \fancyscript{V}$ to S are complete. (shrink)

A box type is an n-type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M. From this, we deduce various structure theorems for subsets of $M^k $ , definable in the expansion M of M by all convex subsets of the line. We show that M after naming constants, is model complete provided M is model complete.

We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve has a definable extension to a closed set. This situation is translated into a question about types: What are the conditions on an n-type such that, for any bounded definable function, the germ of the function on the type has a definable continuous global extension? Certain categories of (...) definable types have this property, and we give the precise conditions that are equivalent to existence of the global extension. (shrink)

Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.

Assume$G\prec H$are groups and${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of theG-flow$S({\cal A})$and theH-flow$S({\cal B})$. We apply these results in the model theoretic context. Namely, assumeGis a group definable in a modelMand$M\prec ^* N$. Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups$S_{ext,G}(M)$and$S_{ext,G}(N)$. Assuming every minimal left ideal in$S_{ext,G}(N)$is a group we prove (...) that the Ellis groups of$S_{ext,G}(M)$are isomorphic to closed subgroups of the Ellis groups of$S_{ext,G}(N)$. (shrink)

We exhibit a theory where definable types lack the amalgamation property.

In, Marker and Steinhorn characterized models of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models for which all 1‐types over M realized in N are definable has already the desired property. Although it is true that if M is an algebraically closed valued field such that all 1‐types over M are definable then (...) all types over M are definable, we build a counterexample for the relative statement, i.e., we show for any that there is a pair of algebraically closed valued fields such that all n‐types over M realized in N are definable but there is an ‐type over M realized in N which is not definable. (shrink)

This paper was motivated by the following two questions which arise in the theory of complexity for computation over ordered rings in the now famous computational model introduced by Blum, Shub and Smale: 1. is the answer to the question P = ?NP the same in every real-closed field?2. if P ≠ NP for , does there exist a problem of which is NP but neither P nor NP-complete ?Some unclassical complexity classes arise naturally in the study of these questions. (...) They are still open, but we could obtain unconditional results of independent interest.Michaux introduced/const complexity classes in an effort to attack question . We show that , answering a question of his. Here A is the class of real problems which are algorithmic in bounded time. We also prove the stronger result: , where PAR stands for parallel polynomial time. In our terminology, we say that is A-saturated and PAR-saturated. We also prove, at the nonuniform level, the above results for every real-closed field. It is not known whether is P-saturated. In the case of the reals with addition and order we obtain P-saturation ). More generally, we show that an ordered -vector space is P-saturated at the nonuniform level ).We also study a stronger notion that we call P-stability. Blum, Cucker, Shub and Smale have shown that fields of characteristic 0 are P-stable. We show that the reals with addition and order are P-stable, but real-closed fields are not.Questions and and the/const complexity classes have some model theoretic flavor. This leads to the theory of complexity over “arbitrary” structures as introduced by Poizat. We give a summary of this theory with a special emphasis on the connections with model theory and we study/const complexity classes from this point of view. Note also that our proof of the PAR-saturation of shows that an o-minimal structure which admits quantifier elimination is A-saturated at the nonuniform level. (shrink)