Axiomatic derivation of the Doppler factor and related relativistic laws

In the context of the relativistic Doppler effect [DE] and the Lorentz-Fitzgerald contraction [LF], we investigate the consequences of two abstract axioms [R] and [M] expressed in terms of an operation circle plus generalizing the addition of velocities and a function L : (lambda, nu) bar right arrow L(lambda, nu). The latter can represent either the Doppler effect or the Lorentz-Fitzgerald Contraction. In words, these axioms state the following: [R] iterating the function L has the same effect as adding velocities; [M] adding a velocity via the operation. preserves the order of the function L. We prove that these axioms are equivalent to each other, and also to generalized forms of the Doppler effect and the standard expression [AV] for the relativistic addition of velocities, taken jointly. We also show that if [AV] holds, then the axioms [R] and [M] are inconsistent with [LF].