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Matrix semigroups with commutable rank

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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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Authors and Affiliations

  1. Department of Mathematics, Colby College, 04901, Waterville, ME, USA

    Leo Livshits

  2. Department of Math and CS, University of Prince Edward Island, C1A 4P3, Charlottetown, PEI, Canada

    Gordon MacDonald

  3. Department of Math and CS, Colby College, 04901, Waterville, ME, USA

    Ben Mathes

  4. Department of Mathematics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Jan Okniński

  5. Department of Math, Stats and CS, Dalhousie University, B3H 3J3, Halifax, NS, Canada

    Heydar Radjavi

Authors
  1. Leo Livshits
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  2. Gordon MacDonald
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  3. Ben Mathes
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  4. Jan Okniński
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  5. Heydar Radjavi
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Corresponding authors

Correspondence to Leo Livshits, Gordon MacDonald, Ben Mathes, Jan Okniński or Heydar Radjavi.

Additional information

Communicated by Norman R. Reilly

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Livshits, L., MacDonald, G., Mathes, B. et al. Matrix semigroups with commutable rank. Semigroup Forum 67, 288–316 (2003). https://doi.org/10.1007/s00233-002-0013-5

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  • DOI: https://doi.org/10.1007/s00233-002-0013-5

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