Linear functions preserving Green's relations over fields

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.

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