Operators that preserve semiring matrix functions (Q1097325)

Let S be a commutative, antinegative (i.e. none of its nonzero elements have additive inverses) semiring without zero divisors, \(\epsilon_ r(A)\) be the sum of all the \(r\times r\) principal minor permanents of A. It is shown that the only linear operators on the \(n\times n\) matrices over S that preserve \(\epsilon_ r(n>r\geq 3)\) are compositions of (a) multiplication by an rth root of unity, (b) transposition, and (c) similarity transformations. Let \(\Phi_ r\) denote the coefficient of x r in the rook polynomial of the \(m\times n\) matrix A. The authors prove that the linear operators T that preserve \(\Phi_ r(2\leq r\leq m\leq n)\) are compositions of (a) multiplication by a permutation matrix, (b) multiplication by an rth root of unity, (c) transposition (if \(m=n)\), and (d) \(T(A)=D_ 1AD_ 2\) where \(D_ 1\) and \(D_ 2\) are diagonal matrices with \(per(D_ 1D_ 2)=1\) (if \(r=m)\). The authors also characterize the preservers of the term rank of matrices with term rank r.