Preservers of Upper Ideals of Matrices: Tournaments; Primitivity

Abstract

Let \( \mathcal{M} \) denote the set of matrices over some semiring. An upper ideal of matrices in \( \mathcal{M} \) is a set \( \mathcal{U} \) such that if \( A \in \mathcal{U} \) and \( B \) is any matrix in \( \mathcal{M} \), then \( A + B \in \mathcal{U} \). We investigate linear operators that strongly preserve certain upper ideals (that is, linear operators on \( \mathcal{M} \) with the property that \( X \in \mathcal{U} \) if and only if \( T(X) \in \mathcal{U} \)). We then characterize linear operators that strongly preserve sets of tournament matrices and sets of primitive matrices. Specifically, we show that if \( T \) strongly preserves the set of regular tournaments when \( n \) is odd or nearly regular tournaments when \( n \) is even, then for some permutation matrix \( P \), \( T(X) = P^{t}XP \) for all matrices \( X \) with zero main diagonal, or \( T(X) = P^{t}X^{t}P \) for all matrices \( X \) with zero main diagonal. Similar results are shown for linear operators that strongly preserve the set of primitive matrices whose exponent is \( k \) for some values of \( k \), and for those that strongly preserve the set of nearly reducible primitive matrices.