The bideterminants of matrices over semirings

Strong linearly independent vectors in semilinear spaces and their applications

“…And on the other hand, it is clear that a standard orthogonal vectors are strong linearly independent, but the converse is not true (see [17,19]). We know that in L-semilinear space over commutative semirings which are generated by standard orthogonal, the basis is standard orthogonal, which is different with the conclusion in classical algebra.…”

“…Note that if det 1 (A) = det 2 (A), then det(A) ≡ 0 (see [19]). Otherwise, we use symbols det(A) ≡ 0.…”