极大代数上有限生成模的凸性

THE CONVEXITY OF FINITELY GENERATED MODULE OVER MAX-PLUS ALGEBRA

研究极大代数上有限生成模的凸性.基于极大代数上有限生成模的几何形态,运用代数与几何方法,分析空间维数n≤3和生成向量数m≥1的有限生成模的凸性.证明n=1,2的有限生成模是凸集.对于n=3,给出m=2的有限生成模为凸集的一个充分必要条件,以及m≥3的有限生成模为凸集的一个充分条件.此外,对于极大代数上有限生成模的几何形态,发现n=3,m≥3的形态有三种情形.

The convexity of finitely generated module over max-plus algebra is stud- ied in this paper. By applying the geometric shape of finitely generated module and using algebraic method combined with geometric method, the convexity of finitely generated module with dimension n ≤ 3 and the number of generating vector rn ≥ 1 are analyzed. It is proved that the finitely generated module with n = 1, 2 is a convex set. For n = 3, the necessary and sufficient condition for the convexity with m = 2 and the sufficient condition for the convexity with rn ≥ 3 are obtained, respectively. Besides, with regard to the geometric shape of finitely generated module over max-plus algebra, it is found out that there are three types of geometric shapes with n = 3 and rn≥ 3.