Tropical矩阵半群和Tropical矩阵群

Many problems arising from application areas are naturally expressed as tropical matrix algebra problems, and much of the theory of tropical algebra is concerned with matrices. An important aspect is the algebraic structure of tropical matrices under multiplication; many authors have proved a number of interesting ad hoc results，but until recently there has been no systematic study in this area. This surprising omission is due largely to the difficulty, both conceptual and technical, of the subject. We believe that the development of a coherent and comprehensive theory of tropical matrix semigroups of arbitrary finite dimension is a major challenge. Based on our works in early days, combining with the trend of international, we shall further study tropical matrix semigroups and tropical matrix groups. From an algebraic and geometric perspective，we shall introduce and study some new equivalences on a tropical matrix semigroup which are similar to Green equivalences; study Tropical matrix semigroups and its some important subsemigroups；study the representations of groups, Clifford semigroups and some special semigroups by n ×n tropical matrices; find some new identities satisfied by tropical matrix semigroups or its some special subsemigroups; by the representations of groups, Clifford semigroups and some special semigroups by n ×n tropical matrices, we shall study some open classic problems in theory of smigroup variety; find some new applications of tropical algebra theory in theoretical computer science or other application discipline.

缘自于应用领域的许多问题可表示为Tropical矩阵代数问题和许多跟Tropical矩阵有关的Tropical代数理论问题。这样系统地和全面地研究开发Tropical矩阵半群和Tropical矩阵群的代数理论是十分必要的，也充满了魅力和挑战。基于前期工作, 结合国际趋势，我们将进一步研究Tropical矩阵半群和Tropical矩阵群。从几何和代数的角度出发，引入和研究Tropical矩阵半群上一些类似于Green关系的新的等价关系；研究Tropical矩阵半群及其若干重要的子结构；给出群、Clifford半群和一些特殊半群在Tropical矩阵半群上的表示；寻求Tropical矩阵半群及其一些特殊子半群所满足的恒等式，借此来进一步研究半群簇理论中的若干经典问题；探索Tropical代数理论在理论计算机科学或其它应用学科中的应用。

Tropical矩阵的代数理论和许多应用领域的问题有着密切联系。我们预定的主要研究的内容是：Tropical矩阵群；Tropical矩阵半群上的格林关系，子结构；半群在Tropical矩阵半群上的表示；Tropical矩阵半群生成簇的一些问题。该项目大致上按预定计划进行，未作大的调整。部分预定研究目标已取得较好的结果，譬如，在Tropical矩阵半群的极大子群的刻画，Tropical矩阵半环上的同余，幂半群，半环簇等方面，我们取得了较好的研究成果，拓宽了前人的研究工作。该项目的实施推动了相关课题的发展。