Matrix semigroups with commutable rank. (Q1423724)

The authors study complex matrix semigroups (and algebras) on which rank is commutable (i.e., \(\text{rank}(AB)=\text{rank}(BA)\)). It is shown that in a number of cases (e.g., in dimension \(\leq 6\)), but not always, commutativity of rank entails permutability of rank (i.e., \(\text{rank}(A_1A_2\cdots A_n)=\text{rank}(A_{\sigma(1)}A_{\sigma(2)}\cdots A_{\sigma(n)})\)). It is shown that a commutable-rank semigroup has a natural decomposition as a semilattice of semigroups that have a simpler structure. While it is still unknown whether commutativity of rank entails permutability of rank for algebras, the question is reduced to the case of algebras of nilpotents.