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Let I be a dense linear order with a left endpoint but no right endpoint. Weconsider the lattice L(I) of finite unions of closed intervals of I. Thislattice arises naturally in the setting of o-minimality, as these are preciselythe closed definable sets in any o-minimal expansion of I. Our main result saysthat L(I), the expansion of the lattice by constants for the empty set and thesmallest element of I (viewed as a singleton subset) as well as four unaryfunctions, is model-complete. The proof of the main result makes use ofprevious results regarding the weak monadic second order theory of I from theauthors PhD thesis.

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