On finite presentations of inverse semigroups with zero having polynomial growth

  • L. M. Shneerson
  • D. EasdownEmail author
Research Article


We study growth of inverse semigroups defined by finite presentations. Let S be a finitely presented Rees quotient of a free inverse semigroup given by an irredundant presentation with n generators and m relators. We show that if S has polynomial growth, then \(m\ge n^2-1\) and this estimate is sharp. For any positive integer n, we also find, up to isomorphism, syntactic descriptions of all presentations that achieve this sharp lower bound. As part of the process, we describe all irredundant presentations of finite Rees quotients of free inverse semigroups having rank n, with the smallest number, namely \(n^2\), of relators.


Free inverse semigroup Rees quotient Inverse semigroup presentation Growth 



The authors wish to thank the anonymous referee for his or her careful reading of the manuscript, comments and suggestions leading to improvements in the final version. The authors gratefully acknowledge support from the School of Arts and Sciences, Hunter College, and the School of Mathematics and Statistics, University of Sydney, resulting in this ongoing collaborative work. L. M. Shneerson gratefully acknowledges partial support from the PSC-CUNY Grant Program.


  1. 1.
    Adjan, S.I.: Identities of special semigroups. Dokl. Acad. Nauk. 143, 499–502 (1962). (in Russian) MathSciNetGoogle Scholar
  2. 2.
    Anick, D.: Generic algebras and CW complexes. In: Browder, W. (ed.) Algebraic Topology and Algebraic K-Theory, pp. 247–321. Princeton University Press, Princeton (1987)Google Scholar
  3. 3.
    Baumslag, B., Pride, S.: Groups with two more generators than relators. J. Lond. Math. Soc. 17, 425–426 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bass, H.: The degree of polynomial growth of finitely generated nilpotent groups. Proc. Lond. Math. Soc. 25, 603–614 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bieri, R.: Deficiency and the geometric invariants of a group (with an appendix by Pascal Schweiger). J. Pure Appl. Algebra 208, 951–959 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Mathematical Surveys No. 7, vol. 1. American Mathematical Society, Providence (1961)Google Scholar
  7. 7.
    de Bruijn, N.G.: A combinatorial problem. K. Ned. Akad. Wet. 49, 758–764 (1946)zbMATHGoogle Scholar
  8. 8.
    de la Harpe, P.: Topics in Geometric Group Theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2000)Google Scholar
  9. 9.
    Easdown, D., Shneerson, L.M.: Principal Rees quotients of free inverse semigroups. Glasg. Math. J. 45, 263–267 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Easdown, D., Shneerson, L.M.: Growth of Rees quotients of free inverse semigroups defined by small numbers of relators. Int. J. Algebra Comput. 23, 521–545 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gilman, R.H.: Presentations of groups and monoids. J. Algebra 57, 544–554 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Golod, E.S., Shafarevich, I.R.: On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. 28, 261–272 (1964). (in Russian) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Grigorchuk, R.I.: On Milnor’s problem on the growth of groups. Dokl. Akad. Nauk. 271, 31–33 (1983). (in Russian) Google Scholar
  14. 14.
    Grigorchuk, R.I.: Milnor’s problem on the growth of groups and its consequences. In: Frontiers in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday. Princeton Mathematical Series, vol. 51, pp. 705–773. Princeton University Press, Princeton (2014)Google Scholar
  15. 15.
    Gromov, M.: Groups of polynomial growth and expanding maps. IHES Publ. Math. 53, 53–73 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Higgins, P.M.: Techniques of Semigroup Theory. Oxford University Press, Oxford (1992)zbMATHGoogle Scholar
  17. 17.
    Howie, J.M.: An Introduction to Semigroup Theory. Academic Press, New York (1976)zbMATHGoogle Scholar
  18. 18.
    Iyudu, N., Shkarin, S.: Finite dimensional semigroup quadratic algebras with minimal number of relations. Monatsh. Math. 168(2), 239–252 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand–Kirillov Dimension. Graduate Studies in Mathematics, vol. 22. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  20. 20.
    Lau, J.: Growth of a class of inverse semigroups. Ph.D. thesis, University of Sydney, Sydney (1997)Google Scholar
  21. 21.
    Lau, J.: Rational growth of a class of inverse semigroups. J. Algebra 204, 406–425 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lau, J.: Degree of growth of some inverse semigroups. J. Algebra 204, 426–439 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  25. 25.
    Mann, A.: How Groups Grow. London Mathematical Society Lecture Note Series, vol. 395. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  26. 26.
    Milnor, J.W.: A note on curvature and fundamental group. J. Diff. Geom. 2, 1–7 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Munn, W.D.: Free inverse semigroups. Proc. Lond. Math. Soc. 29(3), 385–404 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Neumann, B.H.: Some remarks on semigroup presentations. Can. J. Math. 19, 1018–1026 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Okninski, J.: Semigroup Algebras. Marcel Dekker, New York (1980)zbMATHGoogle Scholar
  30. 30.
    Piochi, B.: Quasi-abelian and quasi-solvable regular semigroup. Le Mat. 51, 167–182 (1996)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Rauzy, G.: Suite à termes dans un alphabet fini. Séminaire de Théorie des Nombres de Bordeaux Exposé 25, 1–16 (1982–1983)Google Scholar
  32. 32.
    Reilly, N.R.: Free generators in free inverse semigroups. Bull. Austral. Math. Soc. 7, 407–424 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Romanovskii, N.S.: Free subgroups in finitely presented groups. Algebra Log. 16, 88–97 (1977). (in Russian) CrossRefGoogle Scholar
  34. 34.
    Sapir, M.: Combinatorial Algebra: Syntax and Semantics. With Contributions by Victor S. Guba and Mikhail V. Volkov. Springer Monographs in Mathematics (2014)Google Scholar
  35. 35.
    Schein, B.M.: Representations of generalised groups. Izv. Vyss. Ucebn. Zav. Matem. 3(28), 164–176 (1962). (in Russian) Google Scholar
  36. 36.
    Shirayanagi, K.: A classification of finite-dimensional monomial algebras. In: Effective Methods in Algebraic Geometry (Castiglioncello, 1990). Progress in Mathematics, vol. 94, pp. 469–481. Birkhäuser, Boston (1991)Google Scholar
  37. 37.
    Shirshov, A.I.: On rings with identity relations. Mat. Sb. 43, 277–283 (1957). (in Russian) MathSciNetzbMATHGoogle Scholar
  38. 38.
    Shneerson, L.M.: Identities in finitely presented semigroups. Log. Algebra Comput. Math. 2(1), 163–201 (1973). (in Russian) MathSciNetGoogle Scholar
  39. 39.
    Shneerson, L.M.: Free subsemigroups of finitely presented semigroups. Sib. Mat. Zh. 15, 450–454 (1974). (in Russian) MathSciNetCrossRefGoogle Scholar
  40. 40.
    Shneerson, L.M.: Finitely presented semigroups with nontrivial identities. Sib. Mat. Zh. 23, 124–133 (1982). (in Russian) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Shneerson, L.M.: Identities and a bounded height condition for semigroups. Int. J. Algebra Comput. 13, 565–583 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Shneerson, L.M.: On growth, identities and free subsemigroups for inverse semigroups of deficiency one. Int. J. Algebra Comput. 25, 233–258 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Shneerson, L.M., Easdown, D.: Growth and existence of identities in a class of finitely presented inverse semigroups with zero. Int. J. Algebra Comput. 6, 105–121 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Shneerson, L.M., Easdown, D.: Growth of finitely presented Rees quotients of free inverse semigroups. Int. J. Algebra Comput. 21, 315–328 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Shvarts, A.: A volume invariant of coverings. Dokl. Akad. Nauk. SSSR 105, 32–34 (1955). (in Russian) MathSciNetzbMATHGoogle Scholar
  46. 46.
    Stöhr, R.: Groups with one more generator than relators. Math. Z. 182, 45–47 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Ufnarovsky, V.A.: Growth criterion for graphs and algebras given by words. Mat. Zam. 31(3), 465–472 (1982). (in Russian) Google Scholar
  48. 48.
    Ufnarovsky, V.A.: Combinatorial and Asymptotic Methods in Algebra. Algebra VI, Encyclopedia of Mathematical Sciences, Springer, Berlin (1995)Google Scholar
  49. 49.
    Vinberg, E.V.: On the theorem concerning the infinite-dimensionality of an associative algebra. Izv. Akad. Nauk. SSSR Ser. Mat. 29(1), 209–214 (1965). (in Russian) MathSciNetzbMATHGoogle Scholar
  50. 50.
    Wilson, J.S.: Soluble groups of deficiency \(1\). Bull. Lond. Math. Soc. 28, 476–480 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Wolf, J.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Differ. Geom. 2, 421–446 (1968)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Hunter CollegeCity University of New YorkNew YorkUSA
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

Personalised recommendations