View source for O-minimal

Let $L$ be a first-order language containing a binary [[relation symbol]] $<$ and let $M$ be an $L$-structure (cf. [[Structure(2)|Structure]]) in which $<$ is interpreted as a [[total order]] (cf. [[Order (on a set)|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 [[#References|[a2]]], while studying the expansion $(\mathbf{R},\exp)$ of the ordered field $\mathbf{R}$ of the real numbers by the [[Exponential function, real|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 set]]s.  For example, a semi-algebraic set has only finitely many connected components, each of them semi-algebraic (cf. [[#References|[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. [[#References|[a3]]]).

In [[#References|[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. [[#References|[a4]]]), as a consequence of Gabrielov's theorem of the complement that the complement of a subanalytic set is subanalytic (cf. [[#References|[a5]]]). It follows from work of A. Wilkie [[#References|[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 [[#References|[a8]]]. A. Macintyre, van den Dries and D. Marker establish in [[#References|[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 [[#References|[a3]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Gabrielov,   "Projections of semi-analytic sets"  ''Funct. Anal. Appl.'' , '''2'''  (1968)  pp. 282–291</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  J.F. Knight,   A. Pillay,   C. Steinhorn,   "Definable sets in ordered structures I, II"  ''Trans. Amer. Math. Soc.'' , '''295'''  (1986)  pp. 565–605</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  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)</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  A.J. Wilkie,   "A general theorem of the complement and new $o$-minimal expansions of the reals" , manuscript  (1996)</TD></TR>
</table>

{{TEX|done}}