On decomposability of periodic semigroups of non-negative matrices (Q1826845)

The \(m\times m\) matrices whose entries are nonnegative elements of an ordered field \(\mathbb{F}\) form a semigroup \(\mathbb{M}^+_m\) under multiplication. A subsemigroup \(S\) is called decomposable if there exists a permutation matrix \(P\) such that \(P^{-1}SP\) has block triangular form \(\left[\begin{smallmatrix} * & *\\ 0 & *\end{smallmatrix}\right]\). Let \(S\) be an indecomposable subsemigroup of \(\mathbb{M}^+_m\) and let \(n\) be a positive integer. Let \(r\) be the smallest rank of any nonzero element of \(S\). The author's main result (Corollary 4.4) is that the identity \(X^{n+1}= X\) holds for all \(X\in S\) if and only if every prime divisor of \(r\) divided \(n\). Much of the hard work of the proof is done in the preceding Theorems 3.1 and 3.2, in which the form of an individual matrix \(X\) in \(\mathbb{M}^+_n\) satisfying \(X^{n+1}= X\) (with \(n\) minimal \(>0\)) is investigated.