O-minimal

Let $L$ be a first-order language containing a binary relation symbol $<$ and let $M$ be an $L$-structure (cf. Structure) in which $<$ is interpreted as a total order (cf. Order (on a set)). Then $M$ is called $o$-minimal if every parametrically definable subset of $M$ is a finite union of intervals of $M$. An interval of $M$ is a subset of the form $\{ x \in M : a <_1 x <_2 b \}$ for some $a,b \in M \cup \{\pm\infty \}$, where $<_1,\,<_2$ stand for $<$ or $\le$. For $n \ge 1$, a subset $A$ of the Cartesian product $M^n$ is called parametrically definable if there are an $L$-formula $\phi(x_1,\ldots,x_n,y_1,\ldots,y_k)$ and $b_1,\ldots,b_k \in M$ such that $$ A = \{ (a_1,\ldots,a_n) \in M^n : \phi(a_1,\ldots,a_n,b_1,\ldots,b_k)\ \text{is true in}\ M \} \ . $$

An elementary theory is called $o$-minimal if every model of it is $o$-minimal.

This notion was introduced by L. van den Dries in [a2], while studying the expansion $(\mathbf{R},\exp)$ of the ordered field $\mathbf{R}$ of the real numbers by the real exponential function. He observed that the sets parametrically definable in Cartesian products $M^n$ for an $o$-minimal expansion $M$ of $\mathbf{R}$ share many of the geometric properties of semi-algebraic sets. For example, a semi-algebraic set has only finitely many connected components, each of them semi-algebraic (cf. [a1]), and van den Dries showed that this result remains true if one replaces "semi-algebraic" by "parametrically definable in an $o$-minimal expansion of $\mathbf{R}$" . This is a finiteness theorem, and van den Dries aims to explain the other finiteness phenomena in real algebraic and real analytic geometry as consequences of $o$-minimality (cf. [a3]).

In [a6], J.F. Knight, A. Pillay and C. Steinhorn prove the following results.

1) $o$-minimality is preserved under elementary equivalence.

2) An ordered group is $o$-minimal if and only if it is divisible Abelian.

3) An ordered ring is $o$-minimal if and only if it is a real closed field.

4) Any parametrically definable unary function in an $o$-minimal structure is piecewise strictly monotone or constant, and continuous. The real closed field $\mathbf{R}$ is $o$-minimal. The expansion of $\mathbf{R}$ by restricted analytic functions (cf. Model theory of the real exponential function) is $o$-minimal (cf. [a4]), as a consequence of Gabrielov's theorem of the complement that the complement of a subanalytic set is subanalytic (cf. [a5]). It follows from work of A. Wilkie [a7] that $(\mathbf{R},\exp)$ is $o$-minimal. His recent generalization of Gabrielov's theorem establishes the much stronger result that the expansion of $\mathbf{R}$ by Pfaffian chains of total functions is $o$-minimal, see [a8]. A. Macintyre, van den Dries and D. Marker establish in [a4] the $o$-minimality of $\mathbf{R}$ expanded by the restricted analytic functions and the exponential function. For a research account on $o$-minimal structures, see [a3].

References

[a1]	G.E. Collins, "Quantifier elimination for real closed fields by cylindrical algebraic decomposition, automata theory and formal language" , 2nd G.I. Conf. Kaiserslautern , Springer (1975) pp. 134–183
[a2]	L. van den Dries, "Remarks on Tarski's problem concerning $(\mathbf{R},{+},{\cdot},\exp)$" G. Lolli (ed.) G. Longo (ed.) A. Marcja (ed.) , Logic Colloquium '82 , North-Holland (1984) pp. 97–121
[a3]	L. van den Dries, "$o$-minimal structures" W. Hodges (ed.) M. Hyland (ed.) C. Steinhorn (ed.) J. Truss (ed.) , Logic: From Foundations to Applications, European Logic Colloquium , Oxford (1996) pp. 137–185
[a4]	L. van den Dries, A.J. Macintyre, D. Marker, "The elementary theory of restricted analytic fields with exponentiation" Ann. of Math. , 140 (1994) pp. 183–205
[a5]	A. Gabrielov, "Projections of semi-analytic sets" Funct. Anal. Appl. , 2 (1968) pp. 282–291
[a6]	J.F. Knight, A. Pillay, C. Steinhorn, "Definable sets in ordered structures I, II" Trans. Amer. Math. Soc. , 295 (1986) pp. 565–605
[a7]	A.J. Wilkie, "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function" J. Amer. Math. Soc. , 9 : 4 (1996)
[a8]	A.J. Wilkie, "A general theorem of the complement and new $o$-minimal expansions of the reals" , manuscript (1996)