  We investigate inertial frames in the absence of Lorentz invariance,
reconsidering the usual group structure implied by the relativity principle. We
abandon the relativity principle, discarding the group structure for the
transformations between inertial frames, while requiring these transformations
to be at least linear (to preserve homogeneity). In theories with a preferred
frame (aether), the set of transformations between inertial frames forms a
groupoid/pseudogroup instead of a group, a characteristic essential to evading
the von Ignatowsky theorems. In order to understand the dynamics, we also
demonstrate that the transformation rules for energy and momentum are in
general affine. We finally focus on one specific and compelling model
implementing a minimalist violation of Lorentz invariance.
