In [P. E. Eleftheriou et al., J. Symb. Log. 82, No. 4, 1482–1495 (2017; Zbl 1385.03041)] several aspects of non-valuational weakly o-minimal expansions of ordered groups are studied. The canonical example of such a structure is obtained by considering a dense pair \(\langle \mathcal M, \mathcal N\rangle\) of o-minimal expansions of groups and considering the structure induced on the smaller model, \(\mathcal M\). Such a structure is dubbed an o-minimal trace. Eleftheriou et. al. [loc. cit.] show, among others, that o-minimal traces expanding a group do not admit definable Skolem functions.
In the paper under review, it is shown that several of the results of Eleftheriou et. al. on o-minimal traces can be obtained without the underlying group structure. In particular, o-minimal traces have the strong cell decomposition (Corollary 3.5), the notion of an irrational non-valuational cut is extended to this context (Definition 3.7) and it is shown that any expansion of an o-minimal structure by irrational non-valuational cuts is an o-minimal trace (Theorem 3.14). The paper concludes by showing, that – unlike the above mentioned result of Eleftheriou et. al. [loc. cit.] – any expansion of DLO by convex sets is an o-minimal trace with definable Skolem fubctions.