O-minimal Theory - Set-theoretic Definition

Set-theoretic Definition

O-minimal structures can be defined without recourse to model theory. Here we define a structure on a nonempty set M in a set-theoretic manner, as a sequence S = (S_n), n = 0,1,2,... such that

1. S_n is a boolean algebra of subsets of Mn
2. if A ∈ S_n then M × A and A ×M are in S_n+1
3. the set {(x₁,...,x_n) ∈ Mn : x₁ = x_n} is in S_n
4. if A ∈ S_n+1 and π : Mn+1 → Mn is the projection map on the first n coordinates, then π(A) ∈ Mn.

If M has a dense linear order without endpoints on it, say <, then a structure S on M is called o-minimal if it satisfies the extra axioms

5. the set {(x,y) ∈ M2 : x < y} is in S₂
6. the sets in S₁ are precisely the finite unions of intervals and points.

The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set.