Some Studies About Linear Preserver Problems Over Antinegative Semirings

Linear preserver problem(LPP for short) concerns the characterization of linear operators on matrix spaces that leave certain functions, subsets, relations, etc., invariant. The first paper on LPP can be traced back to Frobenius's work in 1897. Since then, a number of works in this area were published. Among these works, the linear operators concerned are linear operators on matrix spaces over some fields or rings. In this paper, we shall characterize the linear operators that strongly preserve nilpotent matrices and that strongly preserve invertible matrices over Boolean algebras and antinegative semirings without zero divisors. And also we characterize the linear operators that strongly preserve commuting pairs of matrices over Boolean algebras.In order to characterize the linear operators that strongly preserve nilpotence and that strongly preserve invertibility, we first study the case of the binary Boolean algebra. Then, we study the case of finite Boolean algebra based on the fact that any finite Boolean algebra is the direct product of a finite number of binary Boolean algebra. By the means of the extension of linear operator, we characterize the linear operators that strongly preserve nilpotence and that strongly preserve invertibility over any Boolean algebra. Also, by the means of the pattern of matrix and the pattern of linear operator, we characterize the linear operators that strongly preserve nilpotence and that strongly preserve invertibility over antinegative commutative semirings without zero divisors. In the same way as above, we characterize linear operators that strongly preserve commuting pairs of matrices over general Boolean algebras.