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　semimoduleMover　S　can　be　extended　to　a　standard　orthogonal　basis　forM.　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.