The cardinality of bases in semilinear spaces over commutative semirings

This paper investigates the cardinality of a basis in semilinear spaces of n-dimensional vectors over commutative semirings. It first discusses the cardinality of a basis and gives a necessary and sufficient condition that each basis has the same number of elements, which is then used to present the characterizations of bases, by the way, it obtains an equivalent description of an invertible matrix. It finally shows a necessary and sufficient condition that each basis has the same number of elements.