Symmetries of the Partial Order of Traces

This paper deals with the automorphism group of the partial order of finite traces. We show that any group can arise as such an automorphism group if we allow arbitrary large dependence alphabets. Restricting to finite dependence alphabets, the automorphism groups are profinite and possess only finitely many simple decomposition factors. Finally, we show that the partial order associated with the Rado graph as dependence alphabet does not give rise to a homogeneous domain thereby answering an open question from Boldi, P., Cardone, F. and Sabadini, N. (1993).