Extending orthogonal subsets of semimodules

Let R be a commutative semiring and S be an R  -semimodule with inner product operation. A subset B⊂SB⊂S is called standard orthogonal if, for any different b,c∈Bb,c∈B, 〈b,b〉〈b,b〉 is a unit and 〈b,c〉=0〈b,c〉=0. Tan posed the following problem: Characterize semirings R such that, for every R  -semimodule with standard orthogonal basis and every standard orthogonal set B⊂SB⊂S, there is a standard orthogonal basis of S containing B. In this paper, we present a negative result on this problem. Namely, we show that such a description is not possible in terms of first-order logic. Also, we provide several interesting examples related to the problem.