ON INNER PRODUCT SPACES OVER ZEROSUMFREE SEMIRINGS

A semiring A is called a zerosumfree semiring if for all a, b ∈  A, a+b = 0 implies that a = 0 and b = 0. Here we have endowed every semimodule over a zerosumfree semiring A with a natural inner product of values in A. A dimension theorem for orthonormal bases of these inner product spaces over zerosumfree semiring is proved. The main results in this paper generalize the corresponding results on the Boolean inner product spaces [5].