TY - JOUR

T1 - Linear functions preserving green's relations over fields

AU - Guterman, Alexander

AU - Johnson, Marianne

AU - Kambites, Mark

AU - Maksaev, Artem

PY - 2020/10/29

Y1 - 2020/10/29

N2 - We study linear functions on the space of n x 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.

AB - We study linear functions on the space of n x 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.

U2 - 10.1016/j.laa.2020.10.033

DO - 10.1016/j.laa.2020.10.033

M3 - Article

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

ER -