Linear functions preserving Green's relations over fields

TL;DR

This paper characterizes linear functions on matrix spaces over fields that preserve Green's relations, providing complete descriptions over algebraically closed fields and revealing complex behaviors over more general fields.

Contribution

It offers a comprehensive classification of linear preservers of Green's relations on matrices, including new results over various types of fields.

Findings

Non-zero $ ext{J}$-preservers are bijective and rank-1 preservers.

Non-zero $ ext{H}$-preservers are exactly the invertibility preservers.

Preservers for $ ext{L}$ and $ ext{R}$ relations can be quite complex over fields with few roots.

Abstract

We study linear functions on the space of $n \times n$ matrices over a field which preserve or strongly preserve each of Green's equivalence relations ($\mathcal{L}$, $\mathcal{R}$, $\mathcal{H}$ and $\mathcal{J}$) 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 $\mathcal{J}$-preservers are all bijective and coincide with the bijective rank-$1$ preservers, while the non-zero $\mathcal{H}$-preservers turn out to be exactly the invertibility preservers, which are known. The $\mathcal{L}$- and $\mathcal{R}$-preservers over a field with "few roots" seem harder to describe: we give a family of examples showing that they can be quite wild.