We introduce an equivalence relation ρ on matrices over a class of semirings, which is relevant to the Kleene star operator on matrices. We characterize those surjective linear transformations for matrices preserving ρ. Also, we show that a surjective linear transformation preserves Kleene stars of matrices if and only if it commutes with the Kleene star operator if and only if it preserves ρ. Finally, we illustrate that there exist non-surjective linear transformations that preserve ρ or preserve Kleene stars of matrices or commute with the Kleene star operator.

中文翻译：

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一类半环上矩阵的线性保持器

我们在一类半环上的矩阵上引入等价关系ρ，这与矩阵上的 Kleene 星型算子有关。我们描述了保留ρ的矩阵的满射线性变换。此外，我们证明满射线性变换保留矩阵的 Kleene 星当且仅当它与 Kleene 星算子交换当且仅当它保留ρ。最后，我们说明存在保留ρ或保留矩阵的 Kleene 星或与 Kleene 星算子交换的非满射线性变换。