Let T be the theory of divisible ordered Abelian groups in a language L
where T has quantifier elimination. Let f be a new unary function symbol. We
would like to consider the L(f)-theory T(a) expanding T together with axioms
for “f is an automorphism”. Unfortunately it is well known that T(a) does not
have a model companion and generally is not easy to analyze. Rather we look at
a weaker theory T(l) once again expanding T but with axioms for “l is a linear
bijection”. T(l) has a model companion and we provide a detailed analysis of
this theory.