Free operated monoids and rewriting systems



The construction of bases for quotients is an important problem. In this paper, applying the method of rewriting systems, we give a unified approach to construct sections—an alternative name for bases in semigroup theory—for quotients of free operated monoids. As applications, we capture sections of free \(*\)-monoids and free groups, respectively.


Operated monoids Term-rewriting systems Free \(*\)-monoids Free groups 



The authors are supported by the National Natural Science Foundation of China (No. 11771191), the Fundamental Research Funds for the Central Universities (No. lzujbky-2017-162), and the Natural Science Foundation of Gansu Province (Grant No. 17JR5RA175) and Shandong Province (No. ZR2016AM02). We thank the anonymous referee for valuable suggestions helping to improve the paper.


  1. 1.
    Aguiar, M.: On the associative analog of Lie bialgebras. J. Algebra 244, 492–532 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefMATHGoogle Scholar
  3. 3.
    Bai, C.: A unified algebraic approach to the classical Yang–Baxter equations. J. Phys. A Math. Theory 40, 11073–11082 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bai, C., Bellier, O., Guo, L., Ni, X.: Splitting of operations, Manin products and Rota–Baxter operators. Int. Math. Res. Not. 3, 485–524 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Baxter, G.: An analytic problem whose solution follows from a simple algebraic identity. Pacific J. Math. 10, 731–742 (1960)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bokut, L.A., Chen, Y.: Gröbner–Shirshov bases and their calculation. Bull. Math. Sci. 4, 325–395 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bokut, L.A., Chen, Y., Qiu, J.: Gröbner–Shirshov bases for associative algebras with multiple operators and free Rota–Baxter algebras. J. Pure Appl. Algebra 214, 89–110 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cariñena, J., Grabowski, J., Marmo, G.: Quantum bi-Hamiltonian systems. Int. J. Mod. Phys. A 15, 4797–4810 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann–Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249–273 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cohn, P.M.: Further Algebra and Applications. Springer, London (2003)CrossRefMATHGoogle Scholar
  11. 11.
    Ebrahimi-Fard, K., Guo, L., Kreimer, D.: Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT. J. Phys. A: Math. Gen. 37, 11037–11052 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gao, X., Guo, L.: Rota’s classification problem, rewriting systems and Gröbner–Shirshov bases. J. Algebra 470, 219–253 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gao, X., Guo, L., Sit, W., Zheng, S.: Rota–Baxter type operators, rewriting systems and Gröbner-Shirshov bases. J. Symb. Comput. arXiv:1412.8055v1
  14. 14.
    Gao, X., Zhang, T.: Averaging algebras, rewriting systems and Gröbner–Shirshov bases. J. Algeb. Appl. 16(2), 1850130 (2018)Google Scholar
  15. 15.
    Guo, L.: Operated semigroups, Motzkin paths and rooted trees. J. Algebr. Comb. 29, 35–62 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Guo, L.: An Introduction to Rota-Baxter Algebra, International Press (US) and Higher Education Press (China), (2012)Google Scholar
  17. 17.
    Guo, L., Sit, W., Zhang, R.: Differential type operators and Gröbner–Shirshov bases. J. Symb. Comput. 52, 97–123 (2013)CrossRefMATHGoogle Scholar
  18. 18.
    Guo, L., Zhang, B.: Renormalization of multiple zeta values. J. Algebra 319, 3770–3809 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hungerford, T.W.: Hungerford, Algebras. Springer, New York (1974)Google Scholar
  20. 20.
    Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)MATHGoogle Scholar
  21. 21.
    Kolchin, E.: Differential Algebraic Groups. Academic Press Inc, Orlando (1985)MATHGoogle Scholar
  22. 22.
    Kurosh, A.G.: Free sums of multiple operator algebras. Siberian. Math. J. 1, 62–70 (1960). (in Russian)MathSciNetMATHGoogle Scholar
  23. 23.
    Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1998)CrossRefMATHGoogle Scholar
  24. 24.
    Leroux, P.: On some remarkable operads constructed from Baxter operators (2003), arXiv:math.QA/0311214
  25. 25.
    Miller, J.B.: Averaging and Reynolds operators on Banach algebra I, representation by derivation and antiderivations. J. Math. Anal. Appl. 14, 527–548 (1966)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, New York (2002)CrossRefMATHGoogle Scholar
  27. 27.
    Ritt, J.F.: Differential Algebra. American Mathematical Society, New York (1950)CrossRefMATHGoogle Scholar
  28. 28.
    Zheng, S., Guo, L.: Relative locations of subwords in free operated semigroups and Motzkin words. Frontier Math. 10, 1243–1261 (2015)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhouPeople’s Republic of China

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