非负交换整半环上矩阵的正/负行列式保持问题

Positive/Negative Determinant Preservers for Matrices over Nonnegative Commutative Semiring without Zero Divisors

设R为非负交换整半环,用M_n(R)表示R上所有n×n矩阵构成的矩阵半环.令T是M_n(R)到其自身的线性变换,若T满足|T(X)|^+=|X|^+,■X∈M_n(R)(或|T(X)|^-=|X|^-,(?)X∈Mn(R)),称T为M_n(R)上保持正行列式(负行列式)的线性变换.刻画了n≥4时,M_n(R)上保持正行列式/负行列式的线性满射形式.

Suppose R is a nonnegative commutative semiring without zero divisors, and let M~ （R） be the matrix semiring of all n x n matrices over R. A linear transformation T from Mn（R） to itself, is said to positive determinant or negative determinant preserver if ｜T（X）｜＋=｜X｜＋,VX∈Mn（R）（orT（X）｜-=｜X｜=,VX∈Mn（R）） forms of the surjective linear transformation on Mn （R） which preserve positive determinant / negative determinant are characterized when n ≥4 in this paper.