Dimensions of ℒ-semilinear spaces over zerosumfree semirings

This paper investigates the dimensions of ℒ-semilinear subspaces of V_n. It first obtains some necessary and sufficient conditions that each basis has the same number of elements. Then it presents that over some zerosumfree semirings, each basis also has the same number of elements. In the end, it discusses the relationships between the direct sums and the dimensions of semilinear subspaces of n-dimensional vectors and shows a necessary and sufficient condition that a sum of semilinear subspaces is a direct sum over some zerosumfree semirings.