Abstract
We focus on matrix semigroups (and algebras) on which rank is commutable [rank(AB) = rank(BA)]. It is shown that in a number of cases (for example, in dimensions less than 6), but not always, commutativity of rank entails permutability of rank [rank(A 1 A 2... A n ) = rank(A σ(1) A σ(2)... A σ(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.
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