Homomorphisms And Subspaces Of Semilinear Spaces Over Commutative Semirings

This paper investigates the homomorphisms and subspaces of finitely gener-ated semilinear spaces over commutative semirings. It first introduces the homomorphism of semilinear spaces over commutative semirings and the notion of Ker(?), and discusses some relevant properties. It gives some sufficient and necessary conditions that two semilinear s-paces are isomorphic and proves the invariance of bases under isomorphisms. Next, it shows some sufficient and necessary conditions that any finitely generated semilinear space is free and obtains the range of the cardinalities of bases over any finitely generated free semilinear space. Then, it obtains some necessary and sufficient conditions that a set of n-dimensional vectors in semilinear spaces can be extended to a basis. This paper finally presents some necessary and sufficient conditions that a sum of subspaces in a semilinear space is a direct sum.