O-minimal

Let be a first-order language containing a binary relation symbol and let be an -structure (cf. Structure) in which is interpreted as a total order (cf. Order (on a set)). Then is called -minimal if every parametrically definable subset of is a finite union of intervals of . An interval of is a subset of the form for some , where stand for or . For , a subset of the Cartesian product is called parametrically definable if there are an -formula and such that

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

This notion was introduced by L. van den Dries in [a2], while studying the expansion of the ordered field of the real numbers by the real exponential function (cf. Exponential function, real). He observed that the sets parametrically definable in Cartesian products for an -minimal expansion of share many of the geometric properties of semi-algebraic sets (cf. Semi-algebraic set). 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 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 -minimality (cf. [a3]).

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

1) -minimality is preserved under elementary equivalence.

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

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

4) Any parametrically definable unary function in an -minimal structure is piecewise strictly monotone or constant, and continuous. The real closed field is -minimal. The expansion of by restricted analytic functions (cf. Model theory of the real exponential function) is -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 is -minimal. His recent generalization of Gabrielov's theorem establishes the much stronger result that the expension of by Pfaffian chains of total functions is -minimal, see [a8]. A. Macintyre, van den Dries and D. Marker establish in [a4] the -minimality of expanded by the restricted analytic functions and the exponential function. For a research account on -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 " 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 -minimal expansions of the reals" , manuscript (1996)