Advertisement

Free commutative trioids

  • Anatolii V. ZhuchokEmail author
Research Article
  • 27 Downloads

Abstract

Loday and Ronco introduced the notion of a trioid and constructed the free trioid of rank 1. We construct the free commutative trioid of rank 1 and prove that the free commutative trioid of rank \(n>1\) is a subdirect product of the free commutative semigroup of rank n and the free commutative trioid of rank 1. We also characterize the least commutative congruence, the least commutative dimonoid congruences, and the least commutative semigroup congruence on the free trioid.

Keywords

Trioid Commutative trioid Free commutative trioid Free trioid 

Notes

Acknowledgements

The author is grateful to the anonymous referee for helpful suggestions.

References

  1. 1.
    Bagherzadeha, F., Bremnera, M., Madariagab, S.: Jordan trialgebras and post-Jordan algebras. J. Algebra 486, 360–395 (2017)MathSciNetGoogle Scholar
  2. 2.
    Bokut, L.A., Chen, Y., Liu, C.: Gröbner–Shirshov bases for dialgebras. Int. J. Algebra Comput. 20(3), 391–415 (2010)zbMATHGoogle Scholar
  3. 3.
    Casas, J.M.: Trialgebras and Leibniz 3-algebras. Bol. Soc. Mat. Mex. 12(2), 165–178 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ebrahimi-Fard, K.J.: Loday-type algebras and the Rota–Baxter relation. Lett. Math. Phys. 61(2), 139–147 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gubarev, V., Kolesnikov, P.: Embedding of dendriform algebras into Rota–Baxter algebras. Cent. Eur. J. Math. 11(2), 226–245 (2013).  https://doi.org/10.2478/s11533-012-0138-z MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kolesnikov, P.S., Voronin, V.Y.: On special identities for dialgebras. Linear Multilinear Algebra 61(3), 377–391 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kolesnikov, P.S.: Varieties of dialgebras and conformal algebras. Sib. Math. J. 49(2), 257–272 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Loday, J.-L.: Dialgebras. In: Dialgebras and Related Operads: Lecture Notes in Mathematics, vol. 1763, pp. 7–66. Springer, Berlin (2001)Google Scholar
  9. 9.
    Loday, J.-L., Ronco, M.O.: Trialgebras and families of polytopes. Contemp. Math. 346, 369–398 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Novelli, J.-C., Thibon, J.-Y.: Polynomial realizations of some trialgebras. In: 18th Conference on Formal Power Series and Algebraic Combinatorics. San Diego, USA, pp. 243–254 (2006)Google Scholar
  11. 11.
    Pozhidaev, A.P.: Dialgebras and related triple systems. Sib. Math. J. 49(4), 696–708 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Vallette, B.: Homology of generalized partition posets. J. Pure Appl. Algebra 208(2), 699–725 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Zhuchok, A.V.: Commutative dimonoids. Algebra Discrete Math. 2, 116–127 (2009)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Zhuchok, A.V.: Dimonoids and bar-units. Sib. Math. J. 56(5), 827–840 (2015).  https://doi.org/10.1134/S0037446615050055 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Zhuchok, A.V.: Elements of dimonoid theory. In: Mathematics and its applications. Proceedings of Institute of Mathematics of NAS of Ukraine, Monograph, Kiev, vol. 98 (2014) (in Ukrainian)Google Scholar
  16. 16.
    Zhuchok, A.V.: Free commutative dimonoids. Algebra Discrete Math. 9(1), 109–119 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zhuchok, A.V.: Free dimonoids. Ukr. Math. J. 63(2), 196–208 (2011).  https://doi.org/10.1007/s11253-011-0498-8 MathSciNetzbMATHGoogle Scholar
  18. 18.
    Zhuchok, A.V.: Free products of dimonoids. Quasigroups Relat. Syst. 21(2), 273–278 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Zhuchok, A.V.: Free products of doppelsemigroups. Algebra Univers. 77(3), 361–374 (2017).  https://doi.org/10.1007/s00012-017-0431-6 MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zhuchok, A.V.: Structure of relatively free dimonoids. Commun. Algebra 45(4), 1639–1656 (2017).  https://doi.org/10.1080/00927872.2016.1222404 MathSciNetzbMATHGoogle Scholar
  21. 21.
    Zhuchok, A.V.: Tribands of subtrioids. Proc. Inst. Appl. Math. Mech. 21, 98–106 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Zhuchok, A.V.: Trioids. Asian-Eur. J. Math. 8(4), 1550089 (2015).  https://doi.org/10.1142/S1793557115500898 MathSciNetzbMATHGoogle Scholar
  23. 23.
    Zhuchok, A.V., Gorbatkov, A.B.: On the structure of dimonoids. Semigroup Forum 94(2), 194–203 (2017).  https://doi.org/10.1007/s00233-016-9795-8 MathSciNetzbMATHGoogle Scholar
  24. 24.
    Zhuchok, A.V., Zhuchok, Yul.V.: Free left \(n\)-dinilpotent dimonoids. Semigroup Forum 93(1), 161–179 (2016).  https://doi.org/10.1007/s00233-015-9743-z
  25. 25.
    Zhuchok, Yul.V.: Free \(n\)-nilpotent trioids. Mat. Stud. 43(1), 3–11 (2015)Google Scholar
  26. 26.
    Zhuchok, Yul.V.: Free rectangular tribands. Bul. Acad. Stiinte Repub. Mold. Mat. 78(2), 61–73 (2015)Google Scholar
  27. 27.
    Zhuchok, Y.V.: On the determinability of free trioids by semigroups of endomorphisms. Rep. NAS Ukr. 4, 7–11 (2015) (in Russian)Google Scholar
  28. 28.
    Zhuchok, Y.V.: The endomorphism monoid of a free trioid of rank 1. Algebra Univers. 76(3), 355–366 (2016).  https://doi.org/10.1007/s00012-016-0392-1 MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Algebra and System AnalysisLuhansk Taras Shevchenko National UniversityStarobilskUkraine

Personalised recommendations