A note on irreducibility for topical maps

Abstract

Topical maps are a nonlinear generalization of nonnegative matrices acting on the interior of the standard cone $\mathbb{R}^n_{\ge 0}$. Several analogues of irreducibility have been defined for topical maps, and all are sufficient to guarantee the existence of entrywise positive eigenvectors. In this note, we organize several of these notions, showing which conditions are stronger and when different types of irreducibility are equivalent. We also consider how to computationally check the conditions. We show that certain irreducibility conditions can be expressed as Boolean satisfiability problems that can be checked using SAT solvers. This can be used to confirm the existence of entrywise positive eigenvectors when the dimension is large.