论文标题
线性功能保留了格林在字段上的关系
Linear functions preserving Green's relations over fields
论文作者
论文摘要
我们在$ n \ times n $矩阵的空间上研究线性功能，该领域在维护或强烈保留格林的等效关系的每个字段（$ \ nathcal {l} $，$ \ MATHCAL {r} $，$ \ MATHCAL {h} $ {h} $ and $ \ Mathcal {j} $ {J} $）和相应的Pre-pre-orders。对于这些关系中的每一个，我们都可以在代数封闭的字段上完全描述所有保存器（或更一般而言，每个$ n $的多项式都有一个词根），以及所有强大的保留者和在任何字段上都有强大的保留剂和三个领域。在一般字段上，非零$ \ MATHCAL {J} $ - 保存器都是两种物物，并且与Bioxtive Cark-$ 1 $ preservers相吻合，而非零$ \ MATHCAL {H} $ - 前列器 - 前者的预言是确切的可逆性预备者。 $ \ mathcal {l} $ - 和$ \ Mathcal {r} $ - 在一个领域上的保存者，“很少的根”似乎很难描述：我们给出了一个例子，表明它们可以很疯狂。
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.
