In [MPP] it was shown that in every reduct of = ‹ ℝ, +, ·, <› that properly expands ℳ = ‹ℝ, +, <, λa›a∈ℝ, all the bounded semi-algebraic (that is, -definable) sets are definable. Said differently, every such is an expansion of = ‹ℝ, +, <, λa, Bi›a∈ℝ, i∈I where {Bi}i∈I is the collection of all bounded semialgebraic sets and the λa's are scalar multiplication by a. In [PSS] (see Theorem 1.2 below) it was shown that the structure is a proper reduct of ; that is, one cannot define in it all the semialgebraic sets. In [Pe] we show that is the only reduct properly between ℳ and . As a first step towards this result, we investigate in this paper the definable sets in reducts such as . (We point out that ‘definable’ will always mean ‘definable with parameters’.)
Definition 1.1. Let X ⊆ ℝn. X is called semi-bounded if it is definable in the structure ‹ℝ, +, <, λa, B1, …, Bk›a∈ℝ, where the Bi's are bounded subsets of ℝn.
The main result of this paper (see Theorem 3.1) shows roughly that, in Ominimal expansions of that satisfy the partition condition (see Definition 2.3), every semibounded set can be partitioned into finitely many sets, each of which is of a form similar to a cylinder. Namely, these sets are obtained through the “stretching” of a bounded cell by finitely many linear vectors. As a corollary (see Theorem 1.4), we get different characterizations of semibounded sets, either in terms of their structure or in terms of their definability power.
The following result, by A. Pillay, P. Scowcroft and C. Steinhorn, was the main motivation for this paper. The theorem is formulated here in a slightly stronger form than originally, but the proof itself is essentially the original one. A short version of the proof is included in §4.