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.