WebO-minimality can be regarded as a weak form of quantifier elimination. A structure M is o-minimal if and only if every formula with one free variable and parameters in M is equivalent to a quantifier-free formula involving only the ordering, also with parameters in M. 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 = (Sn), n = 0,1,2,... such that 1. Sn is a boolean algebra of subsets of M 2. if A ∈ Sn then M × A and A ×M are in Sn+1 3. the set {(x1,...,xn) ∈ M : x1 = xn} is in Sn
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