Tropical矩阵乘法半群的代数性质及应用

Tropical矩阵的代数理论被广泛的用于组合优化、排队问题、离散事件系统、控制理论和自动机理论中。Tropical矩阵乘法半群的代数性质及应用是目前人们关注的问题，但由于此半群在矩阵阶数大于2时并非正则半群，其研究还不完善。本项目主要研究tropical矩阵乘法半群上的代数性质及应用，研究内容为：1)对tropical幂等矩阵进行分类，进而对tropical矩阵乘法半群的极大子群进行分类刻画；2)将tropical幂等矩阵进行相似分解，在此基础上建立正则tropical矩阵的分解定理，进而分别给出具有自反广义逆、{1，3}-g-逆、{1，4}-g-逆和Moore-Penrose逆的tropical矩阵的刻画；3)探究tropical矩阵以及推广后的tropical矩阵在计数组合中的应用。本项目的研究结果将进一步丰富半环与半群代数理论。

The algebraic theory of tropical matrices has wide applications in combinatorial optimization, queuing problems, discrete event systems, control theory and automata theory. The research of the algebraic properties and applications for the multiplicative semigroup of tropical matrices is growing rapidly in recent years. Since the multiplicative semigroup of n×n (n≥3) tropical matrices is not a regular semigroup, the research is still not complete. In this project, we shall study the algebraic properties and applications for the multiplicative semigroups of n×n tropical matrices. The main aim of this project is as following. Firstly, we will classify the tropical idempotent matrices and characterize the maximal subgroups of the multiplicative semigroups of tropical matrices. Secondly, we will decompose the tropical idempotent matrix by similarity transformation, establish a decomposition theorem for regular tropical matrices, and give the characterization of the tropical matrices with reflexive generalized inverse, {1,3}-g-inverse, {1,4}-g-inverse and Moore-Penrose inverse. Finally, we will investigate the applications of tropical matrices and some extended tropical matrices in enumerative combinatorics. The results of this project will further enrich the content and research method of semiring and semigroup algebraic theory.