Study On Some Preserver Problems

Linear Preserver Problems is a very active topic in the field of matrix theory research. The problems to characterize the linear operators which preserve certain functions, subsets, relations or transformations invariants between matrix sets are called"Linear Preserver Problems". Problems about preserving determinant and minimal rank are discussed in this paper by changing the fundamental field of determinant preserving into nonnegative commutative semiring without zero divisors, and replacing the linear condition of minimal rank preserving by preserving a nonsingular bilinear function.Firstly, the linear transformation over some special nonnegative commutative semirings without zero divisors is characterized in detail.Secondly, the conclusion, when n≥4 the surjective linear transformation which preserves the positive (negative) determinant on M_n(R) over the nonnegative commutative semiring without zero divisors is a ( P , Q, B ) operator, is proved; and when n≥2 the surjective linear transformation which preserves permanent on M_n(R) is also a ( P , Q, B ) operator. Besides, when the semiring is R = Z~+∪{0 }, the surjective additive transformation which preserves permanent on M_n(R) is characterized in detail. This work extends the results of Beasley et al on bideterminant preserver.Finally, the transformations which preserve both minimal rank and a nonsingular bilinear function on full matrix spaces M_n(F) over the field F of characteristic not 2 are characterized when n≥3 . And then the linear transformation on 2×2 full matrix spaces which preserves minimal rank is proved.