Model-completeness for the lattice of finite unions of closed intervals of a dense linear order

TL;DR

This paper proves that the lattice of finite unions of closed intervals in a dense linear order with endpoints is model-complete when expanded with specific constants and functions, advancing understanding in o-minimal structures.

Contribution

It establishes the model-completeness of the lattice of finite unions of closed intervals in a dense linear order with endpoints, with specific expansions.

Findings

L(I) is model-complete with added constants and functions.

Uses weak monadic second order theory of I for proof.

Results relate to o-minimality and definable sets.

Abstract

Let I be a dense linear order with a left endpoint but no right endpoint. We consider the lattice L(I) of finite unions of closed intervals of I. This lattice arises naturally in the setting of o-minimality, as these are precisely the closed definable sets in any o-minimal expansion of I. Our main result says that L(I), the expansion of the lattice by constants for the empty set and the smallest element of I (viewed as a singleton subset) as well as four unary functions, is model-complete. The proof of the main result makes use of previous results regarding the weak monadic second order theory of I from the authors PhD thesis.