Free burnside semigroups
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 6, pp. 579-595.

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.

Classification: 20M05, 20F50
@article{ITA_2001__35_6_579_0,
     author = {Do Lago, Alair Pereira and Simon, Imre},
     title = {Free burnside semigroups},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {579--595},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {6},
     year = {2001},
     mrnumber = {1922297},
     zbl = {1061.20049},
     language = {en},
     url = {http://archive.numdam.org/item/ITA_2001__35_6_579_0/}
}
TY  - JOUR
AU  - Do Lago, Alair Pereira
AU  - Simon, Imre
TI  - Free burnside semigroups
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2001
SP  - 579
EP  - 595
VL  - 35
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/item/ITA_2001__35_6_579_0/
LA  - en
ID  - ITA_2001__35_6_579_0
ER  - 
%0 Journal Article
%A Do Lago, Alair Pereira
%A Simon, Imre
%T Free burnside semigroups
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2001
%P 579-595
%V 35
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/item/ITA_2001__35_6_579_0/
%G en
%F ITA_2001__35_6_579_0
Do Lago, Alair Pereira; Simon, Imre. Free burnside semigroups. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 6, pp. 579-595. http://archive.numdam.org/item/ITA_2001__35_6_579_0/

[1] 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. | Zbl

[2] S.I. Adyan, The Burnside problem and identities in groups. Izdat. “Nauka”, Moscow (1975). | Zbl

[3] 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. | MR

[4] J. Brzozowski, K. Čulík and A. Gabrielian, Classification of non-counting events. J. Comput. System Sci. 5 (1971) 41-53. | MR | Zbl

[5] J.A. Brzozowski and I. Simon, Characterizations of locally testable events. Discrete Math. 4 (1973) 243-271. | MR | Zbl

[6] W. Burnside, On an unsettled question in the theory of discontinuous groups. Quart. J. Math. 33 (1902) 230-238. | JFM

[7] 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. | Zbl

[8] A. De Luca and S. Varricchio, On non-counting regular classes. Theoret. Comput. Sci. 100 (1992) 67-104. | MR | Zbl

[9] A.P. Do Lago, Local groups in free groupoids satisfying certain monoid identities (to appear). | Zbl

[10] 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 São Paulo (1991).

[11] 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.

[12] A.P. Do Lago, On the Burnside semigroups x n =x n+m . Int. J. Algebra Comput. 6 (1996) 179-227. | MR | Zbl

[13] A.P. Do Lago, Grupos Maximais em Semigrupos de Burnside Livres, Ph.D. Thesis. Universidade de São Paulo (1998). Electronic version at http://www.ime.usp.br/~alair/Burnside/

[14] 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. | Zbl

[15] 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. | Zbl

[16] J.A. Green and D. Rees, On semigroups in which x r =x. Proc. Cambridge. Philos. Soc. 48 (1952) 35-40. | MR | Zbl

[17] V.S. Guba, The word problem for the relatively free semigroup satisfying t m =t m+n with m3. Int. J. Algebra Comput. 2 (1993) 335-348. | MR | Zbl

[18] V.S. Guba, The word problem for the relatively free semigroup satisfying t m =t m+n with m4 or m=3,n=1. Int. J. Algebra Comput. 2 (1993) 125-140. | MR | Zbl

[19] M. Hall, Solution of the Burnside problem for exponent six. Illinois J. Math. 2 (1958) 764-786. | MR | Zbl

[20] 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.

[21] S.V. Ivanov, The free Burnside groups of sufficiently large exponents. Int. J. Algebra Comput. 4 (1994) ii+308. | MR | Zbl

[22] J. Kaďourek and L. Polák, On free semigroups satisfying x r x. Simon Stevin 64 (1990) 3-19. | MR | Zbl

[23] 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. | Zbl

[24] G. Lallement, Semigroups and Combinatorial Applications. John Wiley & Sons, New York (1979). | MR | Zbl

[25] F.W. Levi and B.L. Van Der Waerden, Über eine besondere Klasse von Gruppen. Abh. Math. Sem. Hamburg 9 (1933) 154-158. | JFM

[26] I.G. Lysënok, Infinity of Burnside groups of period 2 k for k13. Uspekhi Mat. Nauk 47 (1992) 201-202. | MR | Zbl

[27] S. Maclane, Categories for the working mathematician. Springer-Verlag, New York, Grad. Texts in Math. 5 (1971). | MR | Zbl

[28] J. Mccammond, The solution to the word problem for the relatively free semigroups satisfying t a =t a+b with a6. Int. J. Algebra Comput. 1 (1991) 1-32. | MR | Zbl

[29] D. Mclean, Idempotent semigroups. Amer. Math. Monthly 61 (1954) 110-113. | MR | Zbl

[30] P.S. Novikov and S.I. Adjan, Infinite periodic groups. I. Izv. Akad. Nauk SSSR Ser. Mat. 32 212-244. | MR | Zbl

[31] P.S. Novikov and S.I. Adjan, Infinite periodic groups. II. Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 251-524. | MR | Zbl

[32] A.Y. Ol'Shanskiĭ, Geometry of defining relations in groups. Kluwer Academic Publishers Group, Dordrecht (1991). Translated from the 1989 Russian original by Yu.A. Bakhturin. | Zbl

[33] I. Sanov, Solution of Burnside’s problem for exponent 4. Leningrad. Gos. Univ. Uchen. Zap. Ser. Mat. 10 (1940) 166-170 (Russian). | Zbl

[34] I. Simon, Notes on non-counting languages of order 2. Manuscript (1970).

[35] H. Straubing, Finite automata, formal logic, and circuit complexity. Birkhäuser Boston Inc., Boston, MA (1994). | MR | Zbl

[36] A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I Mat. Nat. Kl. 1 (1912) 1-67. | JFM

[37] B. Tilson, Categories as algebra: An essential ingredient in the theory of monoids. J. Pure Appl. Algebra 48 (1987) 83-198. | MR | Zbl