Advertisement

The monoid of \(2 \times 2\) triangular boolean matrices under skew transposition is non-finitely based

  • Meng Gao
  • Wen Ting ZhangEmail author
  • Yan Feng Luo
Research Article
  • 19 Downloads

Abstract

Let \(\mathscr {T\!B}_n\) be the involution semigroup of all upper triangular boolean \(n\times n\) matrices under the ordinary matrix multiplication and the skew transposition. It is shown by Auinger et al. that the involution semigroup \(\mathscr {T\!B}_n\) is non-finitely based if \(n > 2\), but the case when \(n=2\) still remains open. In this paper, we give a sufficient condition under which an involution semigroup is non-finitely based. As an application, we show that the involution semigroup \(\mathscr {T\!B}_2\) is non-finitely based. Hence \(\mathscr {T\!B}_n\) is non-finitely based for all \(n \ge 2\).

Keywords

Semigroup Boolean matrix Identity basis Finite basis problem Variety 

Notes

Acknowledgements

The authors are very grateful to the anonymous referees for valuable comments and pertinent suggestions, especially for pointing out the gap described in Remark 16, and also to Professor Mikhail Volkov for his valuable remarks and suggestions.

References

  1. 1.
    Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)zbMATHGoogle Scholar
  2. 2.
    Auinger, K., Dolinka, I., Volkov, M.V.: Matrix identities involving multiplication and transposition. J. Eur. Math. Soc. 14(3), 937–969 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Auinger, K., Dolinka, I., Volkov, M.V.: Equational theories of semigroups with involution. J. Algebra 369, 203–225 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Auinger, K., Dolinka, I., Pervukhina, T.V., Volkov, M.V.: Unary enhancements of inherently non-finitely based semigroups. Semigroup Forum 89(1), 41–51 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)CrossRefGoogle Scholar
  6. 6.
    Crvenković, S., Dolinka, I., Ésik, Z.: The variety of Kleene algebras with conversion is not finitely based. Theor. Comput. Sci. 230(1–2), 235–245 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jackson, M., Volkov, M.V.: The algebra of adjacency patterns: Rees matrix semigroups with reversion. In: Blass, A., Dershowitz, N., Reisig, W. (eds.) Fields of Logic and Computation. Lecture Notes in Computer Science, vol. 6300, pp. 414–443. Springer (2010)Google Scholar
  8. 8.
    Lee, E.W.H.: Finitely based finite involution semigroups with non-finitely based reducts. Quaest. Math. 39(2), 217–243 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lee, E.W.H.: Finite involution semigroups with infinite irredundant bases of identities. Forum Math. 28(3), 587–607 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lee, E.W.H.: Equational theories of unstable involution semigroups. Electron. Res. Announc. Math. Sci. 24, 10–20 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lee, E.W.H.: Non-finitely based finite involution semigroups with finitely based semigroup reducts. Korean J. Math. 27, 53–62 (2019)MathSciNetGoogle Scholar
  12. 12.
    Li, J.R., Luo, Y.F.: On the finite basis problem for the monoids of triangular boolean matrices. Algebra Univers. 65, 353–362 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pin, J.-E., Straubing, H.: Monoids of upper triangular matrices. In: Pollák, G., et al. (eds.) Semigroups Structure and Universal Algebraic Problems, Colloquium Mathematics Society, vol. 39, pp. 259–272. János Bolyai, North-Holland (1985)Google Scholar
  14. 14.
    Perkins, P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Volkov, M.V.: The finite basis problem for finite semigroups. Sci. Math. Jpn. 53, 171–199 (2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Volkov, M.V., Goldberg, I.A.: The finite basis problem for monoids of triangular boolean matrices. In: Algebraic Systems, Formal Languages, and Conventional and Unconventional Computation Theory [Research Institute for Mathematical Sciences, Kyoto University, 205–214 (2004)]Google Scholar
  17. 17.
    Zhang, W.T., Ji, Y.D., Luo, Y.F.: The finite basis problem for infinite involution semigroups of triangular \(2 \times 2\) matrices. Semigroup Forum 94, 426–441 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, W.T., Luo, Y.F.: The finite basis problem for involution semigroups of triangular \( 2 \times 2\) matrices. Bull. Aust. Math. Soc.  https://doi.org/10.1017/S0004972719001035

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Key Laboratory of Applied Mathematics and Complex SystemsLanzhouPeople’s Republic of China

Personalised recommendations