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Let M = (M, <, . . .) be a weakly o-minimal structure with the strong cell decomposition property. In this note, we show that the canonical o-minimal extension M of M is the unique prime model of the full first order theory of M over any set A ⊆ M. We also show that if two weakly o-minimal structures with the strong cell decomposition property are isomorphic then, their canonical o-minimal extensions are isomorphic too. Finally, we show the uniqueness of the prime models in a complete weakly o-minimal theory with prime models.

Continuous extension (CE) cell decomposition in o-minimal structures was introduced by Simon Andrews to establish the open cell property (OCP) in those structures. Here, we define strong CE-cells in weakly o-minimal structures, and prove that every weakly o-minimal structure with strong cell decomposition has SCE-cell decomposition if and only if its canonical o-minimal extension has CE-cell decomposition. Then, we show that every weakly ominimal structure with SCE-cell decomposition satisfies OCP. Our last result implies that every o-minimal structure in which every definable open set is a union of finitely many open CE-cells, has CE-cell decomposition.