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Free Products of n-Tuple Semigroups

  • A.V. Zhuchok
  • J. Koppitz
Article

We construct a free product of arbitrary n-tuple semigroups, introduce the notion of n-bands of n-tuple semigroups and, in terms of this notion, describe the structure of the free product. We also construct a free commutative n-tuple semigroup of any rank and characterize one-generated free commutative n-tuple semigroups. Moreover, we describe the least commutative congruence on a free n-tuple semigroup and prove that the semigroups of the constructed free commutative n-tuple semigroup are isomorphic and that its automorphism group is isomorphic to a symmetric group.

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References

  1. 1.
    N. A. Koreshkov, “n-Tuple algebras of associative type,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 12, 34–42 (2008).MathSciNetzbMATHGoogle Scholar
  2. 2.
    N. A. Koreshkov, “On the nilpotency of n-tuple Lie algebras and associative n-tuple algebras,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 2, 33–38 (2010).MathSciNetzbMATHGoogle Scholar
  3. 3.
    N. A. Koreshkov, “Associative n-tuple algebras,” Mat. Zametki, 96, No. 1, 36–50 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A.V. Zhuchok, “Free products of doppelsemigroups,” Algebra Univers., 77, No. 3, 361–374 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A.V. Zhuchok, “Free left n-dinilpotent doppelsemigroups,” Comm. Algebra, 45, No. 11, 4960–4970 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A.V. Zhuchok and M. Demko, “Free n-dinilpotent doppelsemigroups,” Algebra Discrete Math., 22, No. 2, 304–316 (2016).MathSciNetzbMATHGoogle Scholar
  7. 7.
    A.V. Zhuchok, “Structure of free strong doppelsemigroups,” Comm. Algebra, 46, No. 8, 3262–3279 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Gould, K. A. Linton, and A. W. Nelson, “Interassociates of monogenic semigroups,” Semigroup Forum, 68, 186–201 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    B. N. Givens, K. A. Linton, A. Rosin, and L. Dishman, “Interassociates of the free commutative semigroup on n generators,” Semigroup Forum, 74, 370–378 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    B. N. Givens, A. Rosin, and K. Linton, “Interassociates of the bicyclic semigroup,” Semigroup Forum, 94, 104–122 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A.V. Zhuchok, “Commutative dimonoids,” Algebra Discrete Math., 3, 116–127 (2009).MathSciNetzbMATHGoogle Scholar
  12. 12.
    A.V. Zhuchok, “Dimonoids and bar-units,” Sib. Math. J., 56, No. 5, 827–840 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A.V. Zhuchok, “Trioids,” Asian-Eur. J. Math., 8, No. 4, 1550089-1–1550089-23 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    A.V. Zhuchok, “Free n-tuple semigroups,” Math. Notes, 103, No. 5, 737–744 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    B. M. Schein, “Restrictive semigroups and bisemigroups,” in: Tech. Rept. Univ. Arkansas (1989), pp. 1–23.Google Scholar
  16. 16.
    B. M. Schein, “Restrictive bisemigroups,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 1, No. 44, 168–179 (1965).Google Scholar
  17. 17.
    J.-L. Loday, “Dialgebras,” in: Lecture Notes in Mathematics, 1763, Springer, Berlin (2001), pp. 7–66.Google Scholar
  18. 18.
    J.-L. Loday and M. O. Ronco, “Trialgebras and families of polytopes,” Contemp. Math., 346, 369–398 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A. H. Clifford, “Bands of semigroups,” Proc. Amer. Math. Soc., 5, 499–504 (1954).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. S. Putcha, “Semilattice decompositions of semigroups,” Semigroup Forum, 6, 12–34 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. Petrich and P. V. Silva, “Structure of relatively free bands,” Comm. Algebra, 30, No. 9, 4165–4187 (2002).MathSciNetCrossRefzbMATHGoogle Scholar

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

Authors and Affiliations

  • A.V. Zhuchok
    • 1
  • J. Koppitz
    • 2
  1. 1.T. Shevchenko Lugansk National UniversityLuganskUkraine
  2. 2.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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