Nonnegative group-monotone matrices and the minus partial order

Bapat et al. previously described a class of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus partial order. In this paper, we improve upon that description by first presenting a more general result that gives the precise structure of nonnegative matrices dominated by a group-monotone matrix under the minus partial order. As a special case we derive the complete class of nonnegative matrices dominated by a nonnegative idempotent matrix that includes the class obtained by Bapat et al.