交换半环上半线性空间的结构及其应用

国内外广大研究者围绕交换半环上的线性代数问题做了大量的工作, 但至今为止交换半环上线性方程组有解的充要条件以及解的结构仍是公开问题. 本项目的研究内容主要包括: 交换半环上半线性空间的基的性质和线性无关的向量组是基的条件, 并应用于Cramer法则的推广; 半线性子空间的和是直和的等价条件及其在维数公式和半线性空间结构刻画上的应用; 最后讨论交换半环中线性方程系统的求解问题, 找到有解的充要条件并构造解集. 本项目旨在推动这些问题的深入研究和解决.

Many reseachers all over the world to linear algebra problems over semirings have done a lot of work, but so far, both the necessary and sufficient condition for the solvability and the structure of solution set of system of linear equations are still open. The content of this project mainly includes: Some characterizations of bases in semilinear spaces over commutative semirings and the condition that a set of linearly independent vectors is a basis of semilinear space, which is used to generalize Cramer's rule; Some equivalent conditions that a sum of semilinear spaces is a direct sum, which is used to study dimensional formulas and characterize the structure of semilinear spaces; At last, we will discuss the system of linear equations, investigate the condition for solvability and how to discribe the structure of solution sets. The main idea of this project is to promote the further study of the problems mentioned as above and solve them.