Hostname: page-component-7b7567c8c-mnklg Total loading time: 0 Render date: 2026-11-12T08:01:37.929Z Has data issue: false hasContentIssue false
  • English
  • Français

Free Burnside Semigroups

Published online by Cambridge University Press:  15 July 2002

Alair Pereira do Lago and
Imre Simon
Alair Pereira do Lago
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil; (alair@ime.usp.br)
Imre Simon
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil; (is@ime.usp.br)
Get access

Abstract

This paper surveys the area of Free Burnside Semigroups. The theory of these semigroups, as is the case for groups, is far from being completely known. For semigroups, the most impressive results were obtained in the last 10 years. In this paper we give priority to the mathematical treatment of the problem and do not stress too much neither motivation nor the historical aspects. No proofs are presented in this paper, but we tried to give as many examples as was possible.

Keywords

Information

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 35 , Issue 6: A tribute to Aldo de Luca , November 2001, pp. 579 - 595
Copyright
© EDP Sciences, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

S.I. Adian, The Burnside problem and identities in groups. Springer-Verlag, Berlin-New York, Ergebnisse der Mathematik und ihrer Grenzgebiete 95 [Results in Mathematics and Related Areas] (1979). Translated from the Russian by John Lennox and James Wiegold.
S.I. Adyan, The Burnside problem and identities in groups. Izdat. ``Nauka'', Moscow (1975).
J. Brzozowski, Open problems about regular languages, edited by R.V. Book. Academic Press, New York, Formal Language Theory, Perspectives and Open Problems (1980) 23-47.
Brzozowski, J., Culík, K. and Gabrielian, A., Classification of non-counting events. J. Comput. System Sci. 5 (1971) 41-53. CrossRef
Brzozowski, J.A. and Simon, I., Characterizations of locally testable events. Discrete Math. 4 (1973) 243-271. CrossRef
Burnside, W., On an unsettled question in the theory of discontinuous groups. Quart. J. Math. 33 (1902) 230-238.
A. de Luca and S. Varricchio, On non-counting regular classes, edited by M.S. Paterson, Automata, Languages and Programming. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 443 (1990) 74-87.
de Luca, A. and Varricchio, S., On non-counting regular classes. Theoret. Comput. Sci. 100 (1992) 67-104. CrossRef
A.P. do Lago, Local groups in free groupoids satisfying certain monoid identities (to appear).
A.P. do Lago, Sobre os semigrupos de Burnside x n = x n+m , Master's Thesis. Instituto de Matemática e Estatística da Universidade de S ao Paulo (1991).
A.P. do Lago, On the Burnside semigroups x n = x n+m , in LATIN'92, edited by I. Simon. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 583 (1992) 329-343.
do Lago, A.P., On the Burnside semigroups x n =x n+m . Int. J. Algebra Comput. 6 (1996) 179-227. CrossRef
A.P. do Lago, Grupos Maximais em Semigrupos de Burnside Livres, Ph.D. Thesis. Universidade de S ao Paulo (1998). Electronic version at<http://www.ime.usp.br/ alair/Burnside>
A.P. do Lago, Maximal groups in free Burnside semigroups, in LATIN'98, edited by C.L. Lucchesi and A.V. Moura. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 1380 (1998) 70-81.
S. Eilenberg, Automata, languages, and machines, Vol. B. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1976). With two chapters (``Depth decomposition theorem'' and ``Complexity of semigroups and morphisms'') by B. Tilson, Pures Appl. Math. 59 .
Green, J.A. and Rees, D., On semigroups in which x r = x. Proc. Cambridge. Philos. Soc. 48 (1952) 35-40. CrossRef
Guba, V.S., The word problem for the relatively free semigroup satisfying t m = t m+n with m ≥ 3. Int. J. Algebra Comput. 2 (1993) 335-348. CrossRef
Guba, V.S., The word problem for the relatively free semigroup satisfying t m = t m+n with m ≥ 4 or m=3, n=1. Int. J. Algebra Comput. 2 (1993) 125-140. CrossRef
Hall, M., Solution of the Burnside problem for exponent six. Illinois J. Math. 2 (1958) 764-786.
G. Huet and D.C. Oppen, Equations and rewrite rules: A survey, edited by R.V. Book. Academic Press, New York, Formal Language Theory, Perspectives and Open Problems (1980) 349-405.
S.V. Ivanov, The free Burnside groups of sufficiently large exponents. Int. J. Algebra Comput. 4 (1994) ii+308.
Kadourek, J. and Polák, L., On free semigroups satisfying x r ≃ x. Simon Stevin 64 (1990) 3-19.
J.W. Klop, Term rewriting systems: From Church-Rosser to Knuth-Bendix and beyond, edited by M.S. Paterson, Automata, Languages and Programming. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 443 (1990) 350-369.
G. Lallement, Semigroups and Combinatorial Applications. John Wiley & Sons, New York (1979).
Levi, F.W. and van der Waerden, B.L., Über eine besondere Klasse von Gruppen. Abh. Math. Sem. Hamburg 9 (1933) 154-158. CrossRef
Lysënok, I.G., Infinity of Burnside groups of period 2 k for k ≥ 13. Uspekhi Mat. Nauk 47 (1992) 201-202.
S. MacLane, Categories for the working mathematician. Springer-Verlag, New York, Grad. Texts in Math. 5 (1971).
McCammond, J., The solution to the word problem for the relatively free semigroups satisfying t a = t a+b with a ≥ 6. Int. J. Algebra Comput. 1 (1991) 1-32. CrossRef
McLean, D., Idempotent semigroups. Amer. Math. Monthly 61 (1954) 110-113. CrossRef
P.S. Novikov and S.I. Adjan, Infinite periodic groups. I. Izv. Akad. Nauk SSSR Ser. Mat. 32 212-244.
Novikov, P.S. and Adjan, S.I., Infinite periodic groups. II. Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 251-524.
A.Y. Ol'shanski {\u{\i}}\kern.15em , Geometry of defining relations in groups. Kluwer Academic Publishers Group, Dordrecht (1991). Translated from the 1989 Russian original by Yu.A. Bakhturin.
Sanov, I., Solution of Burnside's problem for exponent 4. Leningrad. Gos. Univ. Uchen. Zap. Ser. Mat. 10 (1940) 166-170 (Russian).
I. Simon, Notes on non-counting languages of order 2. Manuscript (1970).
H. Straubing, Finite automata, formal logic, and circuit complexity. Birkhäuser Boston Inc., Boston, MA (1994).
Thue, A., Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I Mat. Nat. Kl. 1 (1912) 1-67.
Tilson, B., Categories as algebra: An essential ingredient in the theory of monoids. J. Pure Appl. Algebra 48 (1987) 83-198. CrossRef