Free n-nilpotent trioids

Yul. V. Zhuchok

doi:10.15330/ms.43.1.3-11

We introduce the notion of a nilpotent trioid, construct a free n-nilpotent trioid and describe its structure. We also characterize the least n-nilpotent congruence on a free trioid and give examples of nilpotent trioids of nilpotency index 2.

1. J.-L. Loday, M.O. Ronco, Trialgebras and families of polytopes, Contemp. Math., 346 (2004), 369–398.

2. J.-L. Loday, Dialgebras, In: Dialgebras and related operads, Lect. Notes Math., Springer-Verlag, Berlin, 1763 (2001), 7–66.

3. A.V. Zhuchok, Dimonoids, Algebra and Logic, 50 (2011), №4, 323–340.

4. A.V. Zhuchok, Trioids, Asian-European Journal of Math., (2015), to appear.

5. Yul.V. Zhuchok, Free rectangular tribands, Buletinul Academiei de Stiinte a Republicii Moldova, Matematica, (2014), submitted.

6. A.I. Malcev, Nilpotent semigroups, Uchen. Zap. Ivanov. Gos. Ped. Inst., 4 (1953), 107–111. (in Russian)

7. B.H. Neumann, T. Taylor, Subsemigroups of nilpotent groups, Proc. Royal Soc. London, Ser. A, 274 (1963), 1–4.

8. E. Jespers, J. Okninski, Nilpotent semigroups and semigroup algebras, Journal of Algebra, 169 (1994), 984–1011.

9. R.S. Kruse, D.T. Price, On the classification of nilpotent rings, Mathematische Zeitschrift, 113 (1970), .3, 215–223.

10. A.V. Zhuchok, Free n-nilpotent dimonoids, Algebra and Discrete Math., 16 (2013), №2, 299–310.

11. A.V. Zhuchok, Free n-dinilpotent dimonoids, Problems of Physics, Mathematics and Technics, 17 (2013), .4, 43–46.

12. A.H. Clifford, Bands of semigroups, Proc. Amer. Math. Soc., 5 (1954), 499–504.

13. Yu.V. Zhuchok, On definibility of free trioids by endomorhism semigroups, Dopovidi NANU, 4 (2015), to appear. (in Russian)

14. A.V. Zhuchok, Free rectangular dibands and free dimonoids, Algebra and Discrete Math., 11 (2011), №2, 92–111.

15. A.V. Zhuchok, Tribands of subtrioids, Proc. Inst. Applied Math. and Mech., 21 (2010), 98–106.

16. A.V. Zhuchok, On combinatorial properties of operations on trioids, Naukovyy Chasopys NPU im. Dragomanova, Ser. 1, Phis.-Math. Nauku, 14 (2013), 77–83. (in Ukrainian)

17. A.V. Zhuchok, Some congruences on trioids, Journal of Mathematical Sciences, 187 (2012), №2, 138–145.

18. A.V. Zhuchok, Semiretractions of trioids, Ukr. Math. J., 66 (2014), №2, 218–231.

19. L.N. Shevrin, Semigroups, In the book: V. Artamonov, V. Salii, L. Skornyakov and others, General algebra, Sect. IV, 2 (1991), 11–191.

	Free $n$-nilpotent trioids
Author	Yul. V. Zhuchok yulia.mih@mail.ru Department of Algebra and System Analysis, Luhansk Taras Shevchenko National University
Abstract	We introduce the notion of a nilpotent trioid, construct a free n-nilpotent trioid and describe its structure. We also characterize the least n-nilpotent congruence on a free trioid and give examples of nilpotent trioids of nilpotency index 2.
Keywords	trioid; nilpotent trioid; free n-nilpotent trioid; dimonoid; semigroup; congruence
DOI	doi:10.15330/ms.43.1.3-11
Reference	1. J.-L. Loday, M.O. Ronco, Trialgebras and families of polytopes, Contemp. Math., 346 (2004), 369–398. 2. J.-L. Loday, Dialgebras, In: Dialgebras and related operads, Lect. Notes Math., Springer-Verlag, Berlin, 1763 (2001), 7–66. 3. A.V. Zhuchok, Dimonoids, Algebra and Logic, 50 (2011), №4, 323–340. 4. A.V. Zhuchok, Trioids, Asian-European Journal of Math., (2015), to appear. 5. Yul.V. Zhuchok, Free rectangular tribands, Buletinul Academiei de Stiinte a Republicii Moldova, Matematica, (2014), submitted. 6. A.I. Malcev, Nilpotent semigroups, Uchen. Zap. Ivanov. Gos. Ped. Inst., 4 (1953), 107–111. (in Russian) 7. B.H. Neumann, T. Taylor, Subsemigroups of nilpotent groups, Proc. Royal Soc. London, Ser. A, 274 (1963), 1–4. 8. E. Jespers, J. Okninski, Nilpotent semigroups and semigroup algebras, Journal of Algebra, 169 (1994), 984–1011. 9. R.S. Kruse, D.T. Price, On the classification of nilpotent rings, Mathematische Zeitschrift, 113 (1970), .3, 215–223. 10. A.V. Zhuchok, Free n-nilpotent dimonoids, Algebra and Discrete Math., 16 (2013), №2, 299–310. 11. A.V. Zhuchok, Free n-dinilpotent dimonoids, Problems of Physics, Mathematics and Technics, 17 (2013), .4, 43–46. 12. A.H. Clifford, Bands of semigroups, Proc. Amer. Math. Soc., 5 (1954), 499–504. 13. Yu.V. Zhuchok, On definibility of free trioids by endomorhism semigroups, Dopovidi NANU, 4 (2015), to appear. (in Russian) 14. A.V. Zhuchok, Free rectangular dibands and free dimonoids, Algebra and Discrete Math., 11 (2011), №2, 92–111. 15. A.V. Zhuchok, Tribands of subtrioids, Proc. Inst. Applied Math. and Mech., 21 (2010), 98–106. 16. A.V. Zhuchok, On combinatorial properties of operations on trioids, Naukovyy Chasopys NPU im. Dragomanova, Ser. 1, Phis.-Math. Nauku, 14 (2013), 77–83. (in Ukrainian) 17. A.V. Zhuchok, Some congruences on trioids, Journal of Mathematical Sciences, 187 (2012), №2, 138–145. 18. A.V. Zhuchok, Semiretractions of trioids, Ukr. Math. J., 66 (2014), №2, 218–231. 19. L.N. Shevrin, Semigroups, In the book: V. Artamonov, V. Salii, L. Skornyakov and others, General algebra, Sect. IV, 2 (1991), 11–191.
Pages	3-11
Volume	43
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Free $n$-nilpotent trioids