半环上线性代数理论及其应用

线性方程的求解是数学中最基本的问题之一。国内外研究者虽然在一些特殊半环上有关线性方程的求解问题研究方面取得了丰硕的成果，如模糊关系方程的求解、max-plus代数上线性方程的求解以及经典线性方程的求解等，但对统一在半环上的线性方程的求解研究没有取得满意进展，许多问题有待解决。本项目就是结合目前半环上线性方程的研究现状和申请人及其团队的研究积累，开展半环上半线性空间基的向量个数惟一性、系数矩阵的秩与线性方程解的关系及判定线性方程是否有解的Kronecker-Capelli定理等的研究，以期把模糊关系方程、max-plus代数上线性方程及经典线性方程等特殊半环上线性方程统一于半环上，建立求解半环上线性方程的线性代数理论，然后应用到模糊关系方程的求解、max-plus代数上线性方程及优化问题的的求解，为线性代数理论的发展做贡献。

The resolution of linear equations is one of the fundamental problems in mathematics. Although reseachers all over the world have made a great developments on solving linear equations on certain special semirings, for example, fuzzy relational equations, linear equations over max-plus algebras and classical linear equations, etc.. However, the investigation of linear algebras over semirings is still at the beginning steps. Based on the previous studies in linear algebras over semirings, this project will focus on the explorations of the unique problem of number of vectors in a basis over semilinear spaces, the relationships between the rank of coefficient matrices and the solutions of linear equations and the Kronecker-Capelli theorem which is used to determine whether a linear system is solvable over semirings, and so on. The accomplishment of this project will unify all the linear algebras over all kinds of special semirings, establish the theory of linear algebras to solve the linear equations over semirings. After that, we apply the theroy to solve fuzzy relation equations, linear equations over max-plus algebra and optimization problem, and make a controbution to the theory of linear algebras.