Quasianalytic algebras with weakly smooth germs generate o-minimal structures

In arXiv:1303.3724, the authors provide an axiomatic way of constructing new polynomially bounded o-minimal structures. However, all of the structures satisfying these axioms must also have smooth cell-decomposition. In this paper, we generalize their approach by allowing weakly smooth germs into the construction. In particular, we showed in arXiv:2501.17583 that the o-minimal structure constructed in [O. Le Gal, J.-P. Rolin. "An o-minimal structure which does not admit $C^\infty$ cellular decomposition" Ann. Inst. Fourier 59 (2009), pp 543-562] satisfies the assumptions of our theorem. As an application of our result, we show that there exists an o-minimal structure within which we can define a nowhere smooth function.