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It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal). It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components." 0. Introduction. In this paper we study the… (More)

A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that… (More)

- Dugald Macpherson, Charles Steinhorn
- Ann. Pure Appl. Logic
- 1996

The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions… (More)

A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ∈ N and every formula φ(x, ȳ), where ȳ = (y1, . . . , ym): (i) There is a positive constant C and a finite set E ⊂ R>0 such that for every M ∈ C and ā ∈ Mm, either |φ(M, ā)| ≤ C, or for some μ ∈ E, ∣∣|φ(M, ā)| − μ|M |∣∣ ≤ C|M | 2 . (ii) For every μ ∈ E, there is an… (More)

- Ya’acov Peterzil, Charles Steinhorn
- 1999

We introduce the notion of definable compactness and within the context of o-minimal structures prove several topological properties of definably compact spaces. In particular a definable set in an ominimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. We then apply definable compactness to… (More)

- David Marker, Charles Steinhorn
- J. Symb. Log.
- 1994

- Michael C. Laskowski, Charles Steinhorn
- J. Symb. Log.
- 1995

Let T be a complete first order theory in a language L satisfying uniform finiteness and let TP be the model companion of T in an expanded language LP with a new unary predicate P . We show that: TP satisfies uniform finiteness; if T is an expansion of the theory of dense linear order then every cut definable in a model M of TP is definable in M L; and, if… (More)

- Saharon Shelah, Charles Steinhorn
- Notre Dame Journal of Formal Logic
- 1986