Advertisement

Survey of Global Semigroup Theory

  • John Rhodes
Chapter

Abstract

What is global semigroup theory? Is it ‘truly’ global? What is its relation to other areas? For other surveys of global semigroup theory, see [Mar], [Chico], [B-Mar].

Keywords

Word Problem Inverse Semigroup Regular Semigroup Wreath Product Finite Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    J. Almeida and A. Azevedo, “Implicit operations on certain classes of semigroups”, in Semigroups and Their Applications, editors S. M. Goberstein and P. M. Higgins (D. Reidel Publishing Co., Dordrecht/ Boston, 1986), pp. 1–12, andGoogle Scholar
  2. J. Almeida and A. Azevedo, “Some pseudo variety joins involving the pseudo variety of finite groups”, Semigroup Forum 37 (1988), 53–57.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [AB]
    R. Alperin and H. Bass, “Length functions of group action on A-trees”, in [GS], pp. 265-378.Google Scholar
  4. [Ale]
    S. V. Aleshin, Mat. Zanetki 11 (1972), 319; English translation in Math. Notes 11 (1972), and “A free group of finite automata”, Mathematika 38, no. 4 (1983), pp. 12-14 (Russian translation).zbMATHGoogle Scholar
  5. [Al]
    D. Allen, Jr., “A generalization of the Rees theorem to a class of regular semigroups”, Semigroup Forum 2 (1971), 321–331.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [ABR]
    D. Albert, R. Baldinger and J. Rhodes, “Undecidability of the identity problem for finite semigroups with applications”, in preparation.Google Scholar
  7. [Ash]
    C. J. Ash, “Finite semigroups with commuting idempotents”, Journal Australian Math. Soc. (Series A) 43 (1987).Google Scholar
  8. [Ash-Chico]
    C. J. Ash, “Finite idempotent-commuting semigroups”, in Semigroups and Their Applications, editors S. M. Goberstein and P. Higgins (D. Reidel Publishing Co., Dordrecht/Boston, 1986), pp. 13–25.Google Scholar
  9. [AN]
    S. I. Adian, The Burnside Problem and Identities in Groups (Springer-Verlag, Berlin/New York, 1979). (Ergebnisse der Math. U. ihren Grenzgebiete 95); editor M. Menicke, Springer Lecture Notes in Math. #806 (1980).CrossRefGoogle Scholar
  10. [B]
    T. C. Brown, “An interesting combinatorial method in the theory of locally finite semigroups”, Pacific J. of Math. 36 (1971), 285–289.zbMATHCrossRefGoogle Scholar
  11. [B-arb vs reg]
    J. C. Birget, “Arbitrary vs. regular semigroups”, Jour. Pure and Applied Algebra 34 (1984), 57–115.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [B-Mar]
    J. C. Birget, “Structure of finite semigroups and generalizations”, in Proceedings of the 1984 Marquette Conference on Semigroups, editors K. Byleen, P. Jones, F. Pastijn (Dept. of Mathematics, Marquette Univ., Milwaukee, WI 53233) (1984), pp. 1-16.Google Scholar
  13. [BG]
    S. W. Margolis and J. E. Pin, “Varieties of finite monoids and topology on the free monoid”, in Proceedings of the 1984 Marquette Conference on Semigroups, editors K. Byleen, P. Jones, F. Pastijn (Dept. of Mathematics, Marquette Univ., Milwaukee, WI 53233) (1984), pp. 113-130.Google Scholar
  14. [BMR]
    J. C. Birget, S. W. Margolis and J. Rhodes, “Decidability results for finite semigroups whose idempotents form a subsemigroup and generalizations”, to appear in Bulletin of the Australian Math. Soc.Google Scholar
  15. [BMR-Chico]
    J. C. Birget, S. W. Margolis and J. Rhodes, “Finite semigroups whose idempotents commute or form a subsemigroup”, in Semigroups and Their Applications, editors S. M. Goberstein and P. M. Higgins (D. Reidel Publishing Co., Dordrecht/Boston, 1986), pp. 25–36.Google Scholar
  16. [BR-GS]
    J. C. Birget and J. Rhodes, “Group theory via global semigroup theory”, to appear in Journal of Algebra, 1989-90.Google Scholar
  17. [BR-Exp]
    J. C. Birget and J. Rhodes, “Almost finite expansions of arbitrary semigroups”, Journal of Pure and Applied Algebra 32 (1984), 239–287.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Cat]
    B. Tilson, “Categories as algebra: an essential ingredient in the theory of monoids”, Journal of Pure and Applied Algebra 48 (1987), 83–198.MathSciNetCrossRefGoogle Scholar
  19. [Chico]
    J. Rhodes, “New techniques in global semigroup theory” in Semigroups and Their Applications, editors S. M. Goberstein and P. M. Higgins, (D. Reidel Publishing Co., Dordrecht/Boston, 1986), pp. 169–182.Google Scholar
  20. [Ei, vol B]
    S. Eilenberg, Automata, Languages and Machines, vol. B (Academic Press, New York 1976).zbMATHGoogle Scholar
  21. [FLC 1]
    J. Rhodes, “The fundamental lemma of complexity for arbitrary finite semigroups”, Bull. Amer. Math. Soc. (1968), 1104-1109.Google Scholar
  22. [FLC 2]
    J. Rhodes, “Proof of the fundamental lemma of complexity (weak version) for arbitrary finite semigroups”, J. Combinatorial Theory, Series A 10 (1971), 22–73.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [FLC 3]
    J. Rhodes, “A proof of the fundamental lemma of complexity (strong version) for arbitrary finite semigroups”, J. Combinatorial Theory, Series 16 (1974), 209–214.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [FT]
    W. Feit, and J. G. Thompson, “Solvability of groups of odd order”, Pacific Jour, of Math. 13 (1963), 775–1029.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [Go]
    J. A. Goguen, “Realization is universal”, Math. System Theory 6, no. 4 (1973), 359–374.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [Gr]
    R. I. Grigorchuk, “Degrees of growth of finitely generated groups and the theory of invariant means”, Math. URSS Izv. 25, no. 2 (1985), 259–300; translation Amer. Math. Soc. 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [Gro]
    M. Gromov, “Hyperbolic groups” in Essays in Group Theory (S. Gersten, editor) (Springer-Verlag) and MSRI publication #8 (1987), pp. 75-263.Google Scholar
  28. [GRS]
    R. L. Graham, B. L. Rothschild and J. H. Spencer, Ramsey Theory (Wiley, New York, 1980).zbMATHGoogle Scholar
  29. [GS]
    S. M. Gersten and J. R. Stallings (editors), “Combinatorial group theory and topology”, Annals of Mathematics Studies 111 (Princeton University Press, 1987).Google Scholar
  30. [HLR]
    K. Henckell, S. Lazarus and J. Rhodes, “Prime decomposition theorem for arbitrary semigroups: General holonomy decomposition and synthesis theorem”, Jour, of Pure and Applied Algebra 55 (1988), 127–172.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [HMR]
    K. Henckell, S. Margolis and J. Rhodes, “A characterization of Type-II construct for finite monoids”, to be submitted to Journal of Algebra.Google Scholar
  32. [John]
    P. T. Johnstone, Topos Theory (Academic Press, New York, 1977).zbMATHGoogle Scholar
  33. [J]
    P. Jones, in this volume.Google Scholar
  34. [KR]
    J. Karnofsky and J. Rhodes, “Decidability of complexity one-half for finite semigroups”, Semigroup Forum 24 (1984), 55–66.MathSciNetCrossRefGoogle Scholar
  35. [KS]
    J. Kadourek and M. B. Szendrei, “A new approach in the theory of orthodox semigroups”, preprint 1988.Google Scholar
  36. [L]
    G. Lallement, Semigroups and Combinatorial Applications, (Wiley, New York, 1971).Google Scholar
  37. [Leung]
    Hing Leung, “An algebraic method for solving decision problems in finite automata theory”, Ph. D. thesis, Math. Dept., Penn State University, 1987.Google Scholar
  38. [LB I]
    J. Rhodes and B. Tilson, “Lower bounds for complexity of finite semigroups”, Jour, of Pure and Applied Algebra 1 (1971), 79–95.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [LB II]
    J. Rhodes and B. Tilson, “Improved lower bounds for complexity of finite semigroups”, Jour, of Pure and Applied Algebra 2 (1972), 13–71.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [LS]
    R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory (Springer-Verlag, 1977).Google Scholar
  41. [Mac]
    S. MacLane, Categories for the Working Mathematician, (Springer-Verlag, New York, 1971).Google Scholar
  42. [Mar]
    J. Rhodes, “Global structure theorems for arbitrary semigroups”, in Proceedings of the Marquette Conference on Semigroups, editors K. Byleen, P. Jones, F. Pastijn, (Dept. Math., Marquette University, Milwaukee, WI 53233) (1984), pp. 197-228.Google Scholar
  43. [Mc]
    D. B. McAlister, “Regular semigroups, fundamental semigroups, and groups”, Jour. Australian Math. Soc. (Series A) 39 (1980), 475–503.MathSciNetCrossRefGoogle Scholar
  44. [MM]
    S. W. Margolis and J. C. Meakin, “E-unitary inverse monoids and the Caley graph of a group presentation”, to appear in Journal of Pure and Applied Algebra, 1989.Google Scholar
  45. [MM2]
    S. W. Margolis and J. C. Meakin, “Inverse presentations, context free languages and Rabin’s theorem”, preprint 1988.Google Scholar
  46. [MMS]
    S. W. Margolis, J. C. Meakin and J. B. Stephen, “Some decision problems for inverse monoid presentations”, in Semigroups and Their Applications, editors S. M. Goberstein and P. M. Higgins (D. Reidel Publishing Co., Dordrecht/Boston, 1986), pp. 99–110.Google Scholar
  47. [MP]
    S. Margolis and J. E. Pin, “Inverse semigroups and extensions of groups by semilattices”, Jour, of Algebra 110, no. 2 (1987), 277–297.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [Mu]
    V. L. Murskii, “Examples of varieties of semigroups”, Mathematical Notes 31 (1968), 423–427 (Russian translation).CrossRefGoogle Scholar
  49. [Munn]
    W. D. Munn, “Free inverse semigroups”, Proc. London Math. Soc. 30 (1974), 385–404.MathSciNetCrossRefGoogle Scholar
  50. [Nam]
    V. S. S. Nambooripad, “Structure of regular semigroups, F, Mem. Amer. Math. Soc. 224 (1979).Google Scholar
  51. [01]
    A. V. Ol’shanskii, “On the Novikov-Adyan theorem”, AMS Russian Translations (1983), 203–236.Google Scholar
  52. [01 2]
    A. V. OPshanskii, “On a geometric method in the combinatorial group theory”, Proceedings of IMC, August 16-24, 1983, Warsaw, vol. 1 published 1984, pp. 415-423.Google Scholar
  53. [Pet]
    M. Petrich, Inverse Semigroups, (Wiley, New York, 1984).zbMATHGoogle Scholar
  54. [Part I]
    J. Rhodes, “Infinite iteration of matrix semigroups, I: Structure theorem for torsion semigroup”, Journal of Algebra 98 (1986), 442–451.MathSciNetCrossRefGoogle Scholar
  55. [Part II]
    J. Rhodes, “Infinite iteration of matrix semigroups, II: Structure theorem for arbitrary semigroups up to aperiodic morphism”, Journal of Algebra 100 (1986), pp. 25–137.MathSciNetzbMATHCrossRefGoogle Scholar
  56. [Pin 1]
    J. E. Pin, “Finite group topology and p-adic topology for free monoids”, 12th ICALP, Lecture Notes in Computer Science 199 (Springer-Verlag, Berlin, 1985), pp. 285–299.Google Scholar
  57. [Pin 2]
    J. E. Pin, “A topological approach to a conjecture of Rhodes”, to appear in Bulletin of the Australian Math. Soc.Google Scholar
  58. [Pin 3]
    J. E. Pin, “Topologies for the free monoid”, to appear in Journal of Algebra, preprint (1988).Google Scholar
  59. [Pin 4]
    J. E. Pin, “On a conjecture of Rhodes”, to appear in Semigroup Forum.Google Scholar
  60. [Po]
    E. Post, “Recursive un decidability of a problem of Thue”, Jour, of Symbolic Logic 12 (1947), 1–11.MathSciNetzbMATHCrossRefGoogle Scholar
  61. [Re]
    J. Reiterman, “The Birkhoff theorem for finite algebras”, Algebra Universalis 14 (1982), 1–10.MathSciNetzbMATHCrossRefGoogle Scholar
  62. [Red]
    K. Henckell and J. Rhodes, “Reduction theorem for the Type-II conjecture for finite monoids”, to appear (preprint May 1988).Google Scholar
  63. [Redux]
    B. Tilson, “Type II re dux”, in Semigroups and Their Applications, editors S. M. Goberstein and P. M. Higgins (D. Reidel Publishing Co., Dordrecht/Boston, 1986), pp. 201–206.Google Scholar
  64. [R-hats]
    J. Rhodes, “A short proof that \(\hat S_{_{\text{A}}^ + } \) is finite if S is finite”, Journal of Pure and Applied Algebra 55 (1988), 197–198.MathSciNetzbMATHCrossRefGoogle Scholar
  65. [R-trees]
    J. Rhodes, “Monoids acting on trees”, in progress.Google Scholar
  66. [Scheib]
    H. E. Scheiblich, “Free inverse semigroups”, Semigroup Forum 4 (1972), 385–404.MathSciNetCrossRefGoogle Scholar
  67. [Schein]
    B. Schein, “Free inverse semigroups are not finitely presentable”, AMS Translations 113 (1979), 89–122, original in Russian, 1965. Also Acta Math. Sci. Hungar. 26 (1975), 41-52.zbMATHGoogle Scholar
  68. [Si]
    I. Simon, “Recognizable sets with multiplicities in the tropical semiring”, preprint July 1988.Google Scholar
  69. [Step]
    J. B. Stephen, “Presentations of inverse monoids”, to appear in Journal of Pure and Applied Algebra.Google Scholar
  70. [St]
    H. Straubing, “The Burnside problem for semigroups of matrices”, in Combinatorics on Words (Academic Press, New York, 1983), 279–295.Google Scholar
  71. [St Im]
    J. R. Stallings, “Finite graphs and free groups”, Contemporary Math. 44 (1985), 79–84; and “Topology of finite graphs”, Invent. Math. 71 (1983), 551-565.MathSciNetCrossRefGoogle Scholar
  72. [Syn-RA]
    J. Rhodes and D. Allen, “Synthesis of the classical and modern theory of finite semigroups”, Advances in Math. 11 (1973), 238–266.MathSciNetCrossRefGoogle Scholar
  73. [Syn-B]
    J. C. Birget, “The synthesis theorem for finite regular semigroups and its generalization”, Jour, of Pure and Applied Algebra 55 (1988), 1–80.MathSciNetzbMATHCrossRefGoogle Scholar
  74. [T]
    D. Thérien, “On the equation x t = x t +q in categories”, Semigroup Forum, vol. 37, no. 3 (1988), 265–272.MathSciNetzbMATHCrossRefGoogle Scholar
  75. [Th]
    D. Thérien and A. Weiss, “Graph congruences and wreath products”, Jour, of Pure and Applied Algebra 36 (1985), 205–215.zbMATHCrossRefGoogle Scholar
  76. [Thur]
    J. W. Cannon, D. B. A. Epstein, D. F. Holt, M. S. Paterson and W. P. Thurston, “Word processing and group theory”, preprint Jan. 1988.Google Scholar
  77. [Trees]
    J. P. Serre, Trees (Springer-Verlag, New York, 1980).Google Scholar
  78. [T, chl2]
    B. Tilson, chapter 12 in [E].Google Scholar
  79. [Unamb-B]
    J. C. Birget, “Iteration of expansions-unambiguous semigroups”, Jour. Pure and Appl. Algebra 34 (1984), 1–56.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [II-g]_K. Henckell and J. Rhodes, “Type-II conjecture is true for g-trivial monoids”, preprint April 1988, to be submitted to Journal of Algebra.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • John Rhodes
    • 1
  1. 1.Dep. of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations