Main Definitions and Basic Facts

  • Mark V. Sapir
Part of the Springer Monographs in Mathematics book series (SMM)


This chapter introduces the main characters that will appear in this book: sets, words, graphs, automata, rewriting systems, various kinds of (universal) algebras, varieties, free algebras (including free semigroups and groups) and subshifts. We also introduce the main properties of algebras that we are interested in: the Burnside property, the finite basis property, properties of the growth function and the growth series, etc.


  1. 3.
    S.I. Adian, The Burnside Problem and Identities in Groups (Springer, Berlin/ New York, 1979)CrossRefGoogle Scholar
  2. 18.
    K.A. Baker, G.F. McNulty, H. Werner, The finitely based varieties of graph algebras. Acta Sci. Math. (Szeged) 51(1–2), 3–15 (1987)Google Scholar
  3. 19.
    K.A. Baker, G.F. McNulty, H. Werner, Shift-automorphism methods for inherently non-finitely based varieties of algebras. Czechoslov. Math. J. 39(1), 53–69 (1989)MathSciNetGoogle Scholar
  4. 20.
    Yu.A. Bakhturin, A.Yu. Ol’shanski, Identity relations in finite Lie algebras. Mat. Sb. 96(4), 543–559 (1975)Google Scholar
  5. 27.
    J.A. Beachy, Introductory Lectures on Rings and Modules. London Mathematical Society Student Texts, vol. 47 (Cambridge University Press, Cambridge, 1999), pp. viii+238Google Scholar
  6. 44.
    G.M. Bergman, The diamond lemma for ring theory. Adv. Math. 29(2), 178–218 (1978)CrossRefMathSciNetGoogle Scholar
  7. 49.
    R.V. Book, F. Otto, String-Rewriting Systems. Texts and Monographs in Computer Science (Springer, New York, 1993)Google Scholar
  8. 60.
    W. Burnside, On an unsettled question in the theory of discontinuous groups. Q. J. Pure Appl. Math. 33, 230–238 (1902)zbMATHGoogle Scholar
  9. 66.
    A. Church, J.B. Rosser, Some properties of conversion. Trans. Am. Math. Soc. 39(3), 472–482 (1936)CrossRefMathSciNetGoogle Scholar
  10. 68.
    A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups. I (American Mathematical Society, Providence, 1961)Google Scholar
  11. 70.
    P.M. Cohn, Embedding in semigroup with one-sided division. J. Lond. Math. Soc. 31, 169–181 (1956)CrossRefzbMATHGoogle Scholar
  12. 71.
    P.M. Cohn, Embeddings in sesquilateral division semigroups. J. Lond. Math. Soc. 31, 181–191 (1956)CrossRefzbMATHGoogle Scholar
  13. 84.
    N. Dershowitz, Orderings for term-rewriting systems. Theor. Comput. Sci. 17, 279–301 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 87.
    V. Diekert, M. Kufleitner, K. Reinhardt, T. Walter, Regular languages are Church–Rosser congruential, in Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 7392 (Springer, Berlin/New York, 2012), pp. 177–188Google Scholar
  15. 93.
    S. Eilenberg, Automata, Languages, and Machines, Vol. A. Pure and Applied Mathematics, vol. 58 (Academic [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, 1974)Google Scholar
  16. 102.
    N. Fine, H. Wilf, Uniqueness theorems for periodic functions. Proc. Am. Math. Soc. 16, 109–114 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 115.
    W.H. Gottschalk, G.A. Hedlund, Topological Dynamics. AMS Colloquium Publications, vol. 36 (American Mathematical Society, Providence, 1955)Google Scholar
  18. 129.
    V.S. Guba, On some properties of periodic words. Math. Notes 72(3-4), 301–307 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 142.
    M. Hall Jr., The Theory of Groups (Macmillan, New York, 1959)zbMATHGoogle Scholar
  20. 146.
    T. Harju, D. Nowotka, The equation \(x^{i} = y^{j}z^{k}\) in a free semigroup. Semigroup Forum 68(3), 488–490 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 147.
    T. Harju, D. Nowotka, On the equation \(x^{k} = z_{1}^{k_{1}}z_{2}^{k_{2}}\ldots z_{n}^{k_{n}}\) in a free semigroup. Theor. Comput. Sci. 330(1), 117–121 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 149.
    G.A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3, 320–375 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 162.
    N. Jacobson, Lie Algebras (Dover, New York, 1962)zbMATHGoogle Scholar
  24. 163.
    B. Jónsson, A. Tarski, On two properties of free algebras. Math. Scand. 9, 95–101 (1961)zbMATHMathSciNetGoogle Scholar
  25. 182.
    D. Knuth, P. Bendix, Simple word problems in universal algebras, in Computational Problems in Abstract Algebra, ed. by J. Leech (Pergamon, New York, 1970), pp. 263–297Google Scholar
  26. 188.
    R.L. Kruse, Identities satisfied by a finite ring. J. Algebra 26, 298–318 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 189.
    C. Kuratowski, Une méthode d’élimination des nombres transfinis des raisonnements mathématiques. Fundam. Math. 3, 76–108 (1922)zbMATHGoogle Scholar
  28. 191.
    G. Lallement, Semigroups and Combinatorial Applications (Wiley, New York/Chichester/Brisbane, 1979)zbMATHGoogle Scholar
  29. 194.
    M.V. Lawson, Finite Automata (Chapman & Hall/CRC, Boca Raton, 2004)zbMATHGoogle Scholar
  30. 195.
    P. Le Chenadec, Canonical Forms in Finitely Presented Algebras. Research Notes in Theoretical Computer Science (Pitman/Wiley, London/New York, 1986)Google Scholar
  31. 200.
    F.W. Levi, On semigroups. Bull. Calcutta Math. Soc. 36, 141–146 (1944)zbMATHGoogle Scholar
  32. 203.
    D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge/New York, 1995)CrossRefzbMATHGoogle Scholar
  33. 207.
    M. Lothaire. Algebraic Combinatorics on Words. Encyclopedia of Mathematics and Its Applications, vol. 90 (Cambridge University Press, Cambridge, 2002)Google Scholar
  34. 208.
    I.V. L’vov, Varieties of associative rings. I. Algebra i Logika 12(3), 269–297 (1973)Google Scholar
  35. 209.
    R.C. Lyndon, Identities in two-valued calculi. Trans. Am. Math. Soc. 71, 457–465 (1951)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 211.
    R.C. Lyndon, M.P. Schützenberger, The equation \(a^{M} = b^{N}c^{P}\) in a free group. Mich. Math. J. 9, 289–298 (1962)CrossRefzbMATHGoogle Scholar
  37. 215.
    W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring. Math. Ann. 111, 259–280 (1935)CrossRefMathSciNetGoogle Scholar
  38. 218.
    A.I. Mal’cev, Algebraic Systems (Springer, Berlin/New York, 1973)Google Scholar
  39. 225.
    R. McKenzie, Tarski’s finite basis problem is undecidable. Int. J. Algebra Comput. 6, 49–104 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 226.
    R. McNaughton, S. Papert, Counter-Free Automata. With an appendix by William Henneman. M.I.T. Research Monograph, vol. 65 (M.I.T., Cambridge/Mass.-London, 1971)Google Scholar
  41. 228.
    G.F. McNulty, C.R. Shallon, Inherently non-finitely based finite algebras, in Universal Algebra and Lattice Theory Puebla, 1982. Volume 1004 of Lecture Notes in Mathematics (Springer, Berlin/New York, 1983), pp. 206–231Google Scholar
  42. 238.
    M. Morse, G.A. Hedlund, Symbolic dynamics. Am. J. Math. 60, 815–866 (1938)MathSciNetGoogle Scholar
  43. 239.
    M. Morse, G.A. Hedlund, Symbolic dynamics II. Sturmian trajectories. Am. J. Math. 62, 1–42 (1940)CrossRefMathSciNetGoogle Scholar
  44. 242.
    V.L. Murskiǐ, The existence in the three-valued logic of a closed class with a finite basis having no finite complete system of identities. Dokl. Akad. Nauk SSSR 163, 815–818 (1965)MathSciNetGoogle Scholar
  45. 243.
    V.L. Murskiǐ, The number of k-element algebras with a binary operation which do not have a finite basis of identities. Probl. Kibernet. 35(5–27), 208 (1979)Google Scholar
  46. 248.
    H. Neumann, Varieties of Groups (Springer, New York, 1967)CrossRefzbMATHGoogle Scholar
  47. 250.
    M.H.A. Newman, On theories with a combinatorial definition of “equivalence”. Ann. Math. 43(2), 223–243 (1942)CrossRefzbMATHGoogle Scholar
  48. 252.
    S. Oates, M.B. Powell, Identical relations in finite groups. J. Algebra 1, 11–39 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 267.
    P. Perkins, Basic questions for general algebras. Algebra Universalis 19(1), 16–23 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  50. 275.
    J. Sakarovitch, Elements of Automata Theory (trans. from the 2003 French original by Reuben Thomas) (Cambridge University Press, Cambridge, 2009)Google Scholar
  51. 286.
    O. Schreier, Die Untergruppen der freien Gruppen. Abh. Math. Sem. Univ. Hambg. 5, 161–188 (1926)CrossRefMathSciNetGoogle Scholar
  52. 290.
    C. Shallon, Nonfinitely based binary algebra derived from lattices, Ph.D. thesis, University of California, Los Angeles, 1979Google Scholar
  53. 310.
    A.S. Švarč, A volume invariant of coverings. Dokl. Akad. Nauk SSSR 105, 32–34 (1955)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Mark V. Sapir
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations