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I asked this question on MSE, but I haven't received any comments or responses (also, it has a very low view count), so I thought I would also ask it here.

In this paper, Knight, Pillay, and Steinhorn prove that for any O-minimal structure $\mathfrak{A}$, in which the underlying order types is dense, and if $\mathfrak{B} \equiv \mathfrak{A}$, then $\mathfrak{B}$ is also O-minimal. Is there a counterexample for when the underlying order type is not dense? I haven't been able to construct a counterexample.

Thanks

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The restriction that the ordering be dense was removed by Pillay and Steinhorn in Definable Sets in Ordered Structures III, Trans. Amer. Math. Soc. 309 (1988) 469-476. The abstract of that paper reads: "We show that any o-minimal structure has a strongly o-minimal theory."

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