Defining new linear functions in tame expansions of the real ordered additive group

We explore semibounded expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. For R = 〈R,<,+, . . .〉, a semibounded o-minimal structure and P ⊆ R a set satisfying certain tameness conditions, we discuss under which conditions 〈R, P 〉 defines total linear functions that are not definable in R. Examples of such structures that does define new total linear functions include the cases when R is a reduct of 〈R, <,+, ·↾(0,1)2 , (x 7→ λx)λ∈I⊆R〉, and P = 2 Z, or P is an iteration sequence (for any I) or P = Z, for I = Q.

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