交换半环上半线性空间的线性变换

Linear Transformations of Semilinear Spaces over Commutative Semirings

介绍半线性空间上线性变换、幂等变换、可逆变换的概念,定义线性变换的运算,讨论幂等变换、可逆变换的一些基本性质,并得到线性变换的值域与核的一些关系.特别地,证明在单变换下,值域的基的原像构成半线性空间的基.

This paper mainly introduces the concepts of a linear transformation,an idempotent transformation and an invertible transformation,and defines the operations of transformations in semilinear spaces over commutative semirings.It then discusses some properties of the idempotent transformations and the invertible transformations,respectively.Some relationships between images and kernels of the linear transformation are showed.In particular,it proves that the inverses of vectors of a basis in images form a basis of a semilinear space under the condition that the linear transformation is injective.