Lifespan in a primitive Boolean linear dynamical system
Summary: Let \(\mathcal F\) be a set of \(k\) by \(k\) nonnegative matrices such that every ``long'' product of elements of \(\mathcal F\) is positive. \textit{J. E. Cohen} and \textit{P. H. Sellers} [Linear Algebra Appl. 47, 185--192 (1982; Zbl 0495.15013)] proved that, then, every such product of length \(2^k-2\) over \(\mathcal F\) must be positive. They suggested to investigate the minimum size of such \(\mathcal F\) for which there exists a non-positive product of length \(2^k-3\) over \(\mathcal F\) and they constructed one example of size \(2^k-2\). We construct one of size \(k\) and further discuss relevant basic problems in the framework of Boolean linear dynamical systems. We also formulate several primitivity properties for general discrete dynamical systems.
scientific article; zbMATH DE number 6534967
scientific article