Skip to main content
SpringerLink
Log in

A Nonfinitely Based Semigroup of Triangular Matrices

  • Conference paper
  • First Online:
Book cover Semigroups, Algebras and Operator Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 142))

Abstract

A new sufficient condition under which a semigroup admits no finite identity basis has been recently suggested in a joint paper by Karl Auinger, Yuzhu Chen, Xun Hu, Yanfeng Luo, and the author. Here we apply this condition to show the absence of a finite identity basis for the semigroup \(\mathrm {UT}_3(\mathbb {R})\) of all upper triangular real \(3\times 3\)-matrices with 0 s and/or 1 s on the main diagonal. The result holds also for the case when \(\mathrm {UT}_3(\mathbb {R})\) is considered as an involution semigroup under the reflection with respect to the secondary diagonal.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Auinger, K., Chen, Y., Hu, X., Luo, Y., Volkov, M.V.: The finite basis problem for Kauffman monoids. Algebra Universalis, accepted. [A preprint is available under http://arxiv.org/abs/1405.0783.]

  2. Auinger, K., Dolinka, I., Volkov, M.V.: Matrix identities involving multiplication and transposition. J. Eur. Math. Soc. 14, 937–969 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Austin, A.K.: A closed set of laws which is not generated by a finite set of laws. Q. J. Math. Oxf. Ser. (2) 17, 11–13 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bahturin, Yu.A., Ol’shanskii, A.Yu.: Identical relations in finite Lie rings. Mat. Sb., N. Ser. 96(138), 543–559 (1975). [Russian (English trans.) Math. USSR-Sbornik 25, 507–523 1975)]

    Google Scholar 

  5. Biryukov, A.P.: On infinite collections of identities in semigroups. Algebra i Logika 4(2), 31–32 (1965). [Russian]

    MATH  MathSciNet  Google Scholar 

  6. Blondel, V.D., Cassaigne, J., Karhumäki, J.: Freeness of multiplicative matrix semigroups, problem 10.3. In: Blonde, V.D., Megretski, A. (eds.) Unsolved Problems in Mathematical Systems and Control Theory, pp. 309–314. Princeton University Press, Princeton (2004)

    Google Scholar 

  7. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)

    Book  MATH  Google Scholar 

  8. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. 1. American Mathematical Society, Providence (1961)

    MATH  Google Scholar 

  9. Kim, K.H., Roush, F.: The semigroup of adjacency patterns of words. Algebraic Theory of Semigroups. Colloqium Mathematical Society János Bolyai, vol. 20, pp. 281–297. North-Holland, Amsterdam (1979)

    Google Scholar 

  10. Kruse, R.L.: Identities satisfied by a finite ring. J. Algebra 26, 298–318 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  11. L’vov, I.V.: Varieties of associative rings. I. Algebra i Logika 12, 269–297 (1973). [Russian (English trans.) Algebra and Logic 12, 150–167 (1973)]

    MathSciNet  Google Scholar 

  12. Mal’cev, A.I.: Nilpotent semigroups. Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz.-Mat. Nauki 4, 107–111 (1953). [Russian]

    MathSciNet  Google Scholar 

  13. McKenzie, R.N.: Equational bases for lattice theories. Math. Scand. 27, 24–38 (1970)

    MATH  MathSciNet  Google Scholar 

  14. Oates, S., Powell, M.B.: Identical relations in finite groups. J. Algebra 1, 11–39 (1964)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Sapir, M.V.: Problems of Burnside type and the finite basis property in varieties of semigroups. Izv. Akad. Nauk SSSR Ser. Mat. 51, 319–340 (1987). [Russian (English trans.) Math. USSR-Izv. 30, 295–314 (1987)]

    MathSciNet  Google Scholar 

  18. Volkov, M.V.: The finite basis problem for finite semigroups, Sci. Math. Jpn. 53,171–199 (2001). [A periodically updated version is avalaible under http://csseminar.imkn.urfu.ru/MATHJAP_revisited.pdf.]

  19. Zimin, A.I.: Semigroups that are nilpotent in the sense of Mal’cev. Izv. Vyssh. Uchebn. Zaved. Mat., no.6, 23–29 (1980). [Russian]

    Google Scholar 

Download references

Acknowledgments

The author gratefully acknowledges support from the Presidential Programme “Leading Scientific Schools of the Russian Federation”, project no. 5161.2014.1, and from the Russian Foundation for Basic Research, project no. 14-01-00524. In particular, the latter project was used to support the author’s participation in the International Conference on Semigroups, Algebras and Operator Theory (ICSAOT-2014) held at the Cochin University of Science and Technology as well as his participation in the International Conference on Algebra and Discrete Mathematics (ICADM-2014) held at Government College, Kattappana. Also the author expresses cordial thanks to the organizers of these two conferences, especially to Dr. P. G. Romeo, Convenor of ICSAOT-2014, and Sri. G. N. Prakash, Convenor of ICADM-2014, for their warm hospitality.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Ural Federal University, Lenina 51, 620000, Ekaterinburg, Russia

    M. V. Volkov

Authors
  1. M. V. Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Volkov .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology (CUSAT), Kochi, Kerala, India

    P G Romeo

  2. Department of Mathematics, University of Nebraska, Lincoln, Nebraska, USA

    John. C Meakin

  3. Department of Mathematics, University of Kerala, Thiruvananthapuram, Kerala, India

    A R Rajan

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer India

About this paper

Cite this paper

Volkov, M.V. (2015). A Nonfinitely Based Semigroup of Triangular Matrices. In: Romeo, P., Meakin, J., Rajan, A. (eds) Semigroups, Algebras and Operator Theory. Springer Proceedings in Mathematics & Statistics, vol 142. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2488-4_2

Download citation

Publish with us

Policies and ethics

Search

Navigation