Inner products on semimodules over a commutative semiring

In this paper, the inner products on the semimodules over a commutative semiring are investigated. Some characterizations for inner products and for orthogonal sets in the semimodules are given, and standard orthogonal basis and orthogonal subsemimodule are discussed. In particular, an equivalent description is obtained for a semifield S   satisfying the property that every standard orthogonal set in a finitely generated semimodule MM over S   can be extended to a standard orthogonal basis for MM. Also, the analogue of the Kronecker–Capelli theorem is proved to be valid for a matrix equation over a commutative semiring. Partial results obtained in this paper generalize and develop corresponding results for semilinear spaces of n-dimensional vectors over commutative zerosumfree semirings.