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Semiring identities of the Brandt monoid

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Abstract

The 6-element Brandt monoid \(B_2^1\) admits a unique addition under which it becomes an additively idempotent semiring. We show that this addition is a term operation of \(B_2^1\) as an inverse semigroup. As a consequence, we exhibit an easy proof that the semiring identities of \(B_2^1\) are not finitely based.

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References

  1. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, Berlin (1981)

    Book  Google Scholar 

  2. Dolinka, I.: A nonfinitely based finite semiring. Int. J. Algebra Comput. 17, 1537–1551 (2007)

    Article  MathSciNet  Google Scholar 

  3. Jackson, M., Ren, M., Zhao, X.: Nonfinitely based ai-semirings with finitely based semigroup reducts (in preparation)

  4. Jackson, M., Zhang, W.T.: From \(A\) to \(B\) to \(Z\). Semigroup Forum (2021). https://doi.org/10.1007/s00233-021-10180-3

    Article  MATH  Google Scholar 

  5. Kad’ourek, J.: On varieties of combinatorial inverse semigroups. I. Semigroup Forum 43, 305–330 (1991)

    Article  MathSciNet  Google Scholar 

  6. Kleiman, E.I.: On bases of identities of Brandt semigroups. Semigroup Forum 13, 209–218 (1977)

    Article  Google Scholar 

  7. Kleiman, E.I.: Bases of identities of varieties of inverse semigroups. Sib. Math. J. 20, 530–543 (1979) [Translated from Sibirskii Matematicheskii Zhurnal 20, 760–777 (1979)]

  8. Kleiman, E.I.: A pseudovariety generated by a finite semigroup. Ural. Gos. Univ. Mat. Zap. 13(1), 40–42 (1982) (Russian)

  9. Kuřil, M., Polák, L.: On varieties of semilattice-ordered semigroups. Semigroup Forum 71, 27–48 (2005)

    Article  MathSciNet  Google Scholar 

  10. Lawson, M.V.: Inverse Semigroups. The Theory of Partial Symmetries. World Scientific, Singapore (1999)

    MATH  Google Scholar 

  11. Lee, E.W.H., Zhang, W.T.: Finite basis problem for semigroups of order six. Lond. Math. Soc. J. Comput. Math. 18, 1–129 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Leech, J.: Inverse monoids with a natural semilattice ordering. Proc. London Math. Soc. s3-70(1), 146–182 (1995)

  13. McAlister, D.B.: Semilattice ordered inverse semigroups. In: André, J.M., et al. (eds.) Semigroups and Formal Languages, pp. 205–218. World Scientific, New Jersey (2007)

    Chapter  Google Scholar 

  14. Perkins, P.: Decision problems for equational theories of semigroups and general algebras. PhD thesis, University of California, Berkeley (1966)

  15. Perkins, P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969)

    Article  MathSciNet  Google Scholar 

  16. Petrich, M.: Inverse Semigroups. Wiley, New York (1984)

    MATH  Google Scholar 

  17. Schein, B.M.: Completions, translational hulls and ideal extensions of inverse semigroups. Czechoslov. Math. J. 23, 575–610 (1973)

    Article  MathSciNet  Google Scholar 

  18. Solomon, L.: Representations of the rook monoid. J. Algebra 256, 309–342 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is thankful to the anonymous referee for his/her constructive remarks.

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Authors and Affiliations

  1. Institute of Natural Sciences and Mathematics, Ural Federal University, 620000, Ekaterinburg, Russia

    Mikhail V. Volkov

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  1. Mikhail V. Volkov
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Correspondence to Mikhail V. Volkov.

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Presented by M. Jackson.

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Supported by the Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2020-1537/1)

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Volkov, M.V. Semiring identities of the Brandt monoid. Algebra Univers. 82, 42 (2021). https://doi.org/10.1007/s00012-021-00731-8

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