Linear functions preserving Green's relations over fields

We study linear functions on the space of n × matrices over a field which preserve or strongly each Green's equivalence relations ( L , R H and J ) corresponding pre-orders. For these we are able to completely describe all preservers an algebraically closed (or more generally, in every polynomial degree has root), strong bijective any field. Over general field, non-zero -preservers coincide with rank-1 preservers, while turn out be exactly invertibility known. The - “few roots” seem harder describe: give family examples showing that they can quite wild.