Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T23:43:49.722Z Has data issue: false hasContentIssue false
  • English
  • Français

Yet Another Solution to the Burnside Problem for Matrix Semigroups

Published online by Cambridge University Press:  20 November 2018

Benjamin Steinberg
Benjamin Steinberg*
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6 e-mail: bsteinbg@math.carleton.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

Keywords

20M3020M32Burnside problemkernel category
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 1 , 01 March 2012 , pp. 188 - 192
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Almeida, J., Margolis, S. W., Steinberg, B., and Volkov, M. V., Representation theory of finite semigroups, semigroup radicals and formal language theory. Trans. Amer. Math. Soc. 361(2009), no. 3, 1429–1461. doi:10.1090/S0002-9947-08-04712-0Google Scholar
[2] Berstel, J. and Reutenauer, C., Rational Series and Their Languages. EATCS Monographs on Theoretical Computer Science 12, Springer-Verlag, Berlin, 1988.Google Scholar
[3] Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras. John Wiley & Sons, New York, 1988. Reprint of the 1962 original.Google Scholar
[4] de Luca, A. and Varricchio, S., Finiteness and regularity in semigroups and formal languages. EATCS Monographs in Theoretical Computer Science, Springer-Verlag, Berlin, 1999.Google Scholar
[5] Freedman, A., Gupta, R. N., and Guralnick, R. M., Shirshov's theorem and representations of semigroups. Olga Taussky-Todd: in memoriam. Pacific J. Math. (1997) Special Issue, 159–176.Google Scholar
[6] Le Saëc, B., Pin, J.-E., and Weil, P., Semigroups with idempotent stabilizers and applications to automata theory. Internat. J. Algebra Comput. 1(1991), no. 3, 291–314. doi:10.1142/S0218196791000195Google Scholar
[7] McNaughton, R. and Zalcstein, Y., The Burnside problem for semigroups. J. Algebra 34(1975), 292–299. doi:10.1016/0021-8693(75)90184-2Google Scholar
[8] Okniński, J., Semigroups of Matrices. Series in Algebra 6, World Scientific Publishing, River Edge, NJ, 1998.Google Scholar
[9] Rhodes, J. and Steinberg, B., The q-Theory of Finite Semigroups. Springer Monographs in Mathematics, Springer, New York, 2009.Google Scholar
[10] Rhodes, J. and Tilson, B., The kernel of monoid morphisms. J. Pure Appl. Algebra 62(1989), no. 3, 227–268. doi:10.1016/0022-4049(89)90137-0Google Scholar
[11] Schützenberger, M. P., Finite counting automata. Information and Control 5(1962), 91–107. doi:10.1016/S0019-9958(62)90244-9Google Scholar
[12] Straubing, H., The Burnside problem for semigroups of matrices. In: Combinatorics on Words. Academic Press, Toronto, 1983, pp. 279–295.Google Scholar