Matrix semigroups with commutable rank

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Title Matrix semigroups with commutable rank
Author(s) L. Livshits, Gordon W. MacDonald, B. Mathes, J. Okninski, H. Radjavi
Journal Semigroup Forum
Date 2003
Volume 67
Issue 2
Start page 288
End page 316
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(sigma(1))A(sigma(2))... A(sigma(n)))]. It is shown that a commutable-rank semigroup has a natural decomposition as a semi-lattice 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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