Semigroups, Algebras and Operator Theory pp 27-38 | Cite as

# A Nonfinitely Based Semigroup of Triangular Matrices

## 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.

### Keywords

Semigroup reduct Involution semigroup semigroup variety## Notes

### 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.

### 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)MATHMathSciNetCrossRefGoogle 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)MATHMathSciNetCrossRefGoogle 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]MATHMathSciNetGoogle 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)MATHCrossRefGoogle Scholar
- 8.Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. 1. American Mathematical Society, Providence (1961)MATHGoogle 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)MATHMathSciNetCrossRefGoogle 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)]MathSciNetGoogle Scholar - 12.Mal’cev, A.I.: Nilpotent semigroups. Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz.-Mat. Nauki
**4**, 107–111 (1953). [Russian]MathSciNetGoogle Scholar - 13.McKenzie, R.N.: Equational bases for lattice theories. Math. Scand.
**27**, 24–38 (1970)MATHMathSciNetGoogle Scholar - 14.Oates, S., Powell, M.B.: Identical relations in finite groups. J. Algebra
**1**, 11–39 (1964)MATHMathSciNetCrossRefGoogle 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)MATHMathSciNetCrossRefGoogle 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)]MathSciNetGoogle 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