Linear functions preserving Green's relations over fields

Abstract

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