Nonnegative rank-preserving operators

AbstractAnalogues of characterizations of rank-preserving operators on field-valued matrices are determined for matrices witheentries in certain structures S contained in the nonnegative reals. For example, if S is the set of nonnegative members of a real unique factorization domain (e.g. the nonnegative reals or the nonnegative integers), M is the set of m×n matrices with entries in S, and min(m,n)⩾4, then a “linear” operator on M preserves the “rank” of each matrix in M if and only if it preserves the ranks of those matrices in M of ranks 1, 2, and 4. Notions of rank and linearity are defined analogously to the field-valued concepts. Other characterizations of rank-preserving operators for matrices over these and other structures S are also given