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Preprint Number 1958
1958. Masato Fujita Almost o-minimal structures and X-structures E-mail: Submission date: 3 April 2021 Abstract:
We propose new structures called almost o-minimal structures and
X-structures. The former is a first-order expansion of a dense linear order
without endpoints such that the intersection of a definable set with a bounded
open interval is a finite union of points and open intervals. The latter is a
variant of van den Dries and Miller's analytic geometric categories and
Shiota's X-sets and Y-sets. In them, the family of
definable sets are closed only under proper projections unlike first-order
structures. We demonstrate that an X-expansion of an ordered
divisible abelian group always contains an o-minimal expansion of an ordered
group such that all bounded X-definable sets are definable in the
structure. Mathematics Subject Classification: Primary 03C64, Secondary 14P99 Keywords and phrases: |

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