On semibounded expansions of ordered groups

We explore semibounded expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if R=⟨R,<,+,…⟩ is a semibounded o-minimal structure and P⊆R is a set satisfying certain tameness conditions, then ⟨R,P⟩ remains semibounded. Examples include the cases when R=⟨R,<,+,(x↦λx)λ∈R,⋅↾[0,1]2⟩, and P=2Z or P is an iteration sequence. As an application, we show that smooth functions definable in such ⟨R,P⟩ are definable in R.