Abstract
In this paper, the inner products on the semimodules over a commutative semiring are investigated. Some characterizations for orthogonal sets and standard orthogonal basis in the semimodules are given. In particular, an equivalent description is obtained for a semifield S satisfying the property that every standard orthogonal set in a finitely generated semimodule over S can be extended to a standard orthogonal basis for . Also, the adjoint homomorphisms of the semimodules are discussed and some properties of the adjoint homomorphisms are obtained. Partial results obtained in this paper generalize and develop corresponding results for semilinear spaces of n-dimensional vectors over commutative zerosumfree semirings and for unitary spaces.
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