半线性空间的基与基数

Bases and the Cardinality of a Basis in Semilinear Spaces

半环上的线性代数技术在选择理论,分离事件网络模型,以及半环是除半环时在图论上都有重要的应用.然而不同于经典线性代数的是,半环上L-半线性空间vn中不同基可能有不同的数量的元素.主要讨论了交换的零和自由半环上L-半线性空间vn中的基数问题.首先给出每组基有相同基数的充要条件,回答了A.Di Nola等在其论文(Fuzzy Sets and Systems,2007,158:1-22)中提出的开问题.其次证明文中给出的充要条件和已有充要条件之间的关系.最后证明在一些交换零和自由半环上不同基也有相同的基数.

The techniques of linear algebra over a semiring have important applications in optimization theory, models of discrete e- vent networks, and graph theory, particularly when the semiring is in fact a semifiled. But different with classcal linear algebra, differ- ent basis may have different number elements in L-semilinear space Vn. In this paper, we discuss the cardinality of a basis in an B semilinear space Vn. First, we give a necessary and sufficient condition for all the bases to have the same cardinality and give an an- swer to the open question proposed in paper（ Fuzzy Sets and Systems ,2007,158：1-22）. Then we discuss the relationship between the necessary and sufficient condition in this paper and the others. In the end, we present that over some commutative zerosumfree semir- ings, all the bases also have the same cardinality.