Surjective linear transformations of tropical matrices preserving transitive closures

Abstract–

In this paper we characterize those surjective linear transformations of tropical matrices preserving the set of all weak transitive closures $M_n^W$ and the set of all strong transitive closures $M_n^S$, respectively. We illustrate that there exist non-surjective linear transformations that preserve $M_n^W$ or $M_n^S$. Also, we consider the strongly linear preservers of $M_n^W$ and $M_n^S$, respectively.

Keywords:

• Linear preserver
• transitive closure
• tropical matrix
• tropical semiring

2020MATHEMATICS SUBJECT CLASSIFICATION:

• 15A80
• 15A86

Acknowledgments

The authors thank Prof. Xianzhong Zhao from Northwest University for his supervision. The authors are grateful to the referee for his/her valuable comments and suggestions.

Additional information

Funding

The paper is supported by National Natural Science Foundation of China (11971383), Key Project of Henan Education Department (23B110012), Project for University Key Young Teacher by Education of Henan Province (2021GGJS158), the Natural Science Basic Research Plan in Shaanxi Province of China (2021JM-520), Science Foundation of Weinan city (2020ZDYF-JCYJ214) and Foundation of Weinan Normal University (20RC16).