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Finite basis problem for semigroups of order six

  • Edmond W. H. Lee (a1) and Wen Ting Zhang (a2) (a3)

Abstract

Two semigroups are distinct if they are neither isomorphic nor anti-isomorphic. Although there exist  $15\,973$  pairwise distinct semigroups of order six, only four are known to be non-finitely based. In the present article, the finite basis property of the other $15\,969$  distinct semigroups of order six is verified. Since all semigroups of order five or less are finitely based, the four known non-finitely based semigroups of order six are the only examples of minimal order.

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References

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1. Almeida, J., Finite semigroups and universal algebra (World Scientific, Singapore, 1994).
2. Bahturin, Yu. A. and Ol’šanskiĭ, A. Yu., ‘Identical relations in finite Lie rings’, Math. USSR-Sb. 25 (1975) 507523 (Engl. transl. of Mat. Sb. 96 (1975) no. 138, 543–559).
3. Birkhoff, G., ‘On the structure of abstract algebras’, Proc. Cambridge Philos. Soc. 31 (1935) 433454.
4. Bol’bot, A. D., ‘Finite basing of identities of four-element semigroups’, Sib. Math. J. 20 (1979) no. 2, 323.
5. Burris, S. and Sankappanavar, H. P., A course in universal algebra (Springer, New York, 1981).
6. Distler, A. and Kelsey, T. W., ‘The monoids of orders eight, nine & ten’, Ann. Math. Artif. Intell. 56 (2009) 325.
7. Distler, A. and Mitchell, J. D., ‘Smallsemi—a GAP package, version 0.6.2’, 2010, http://www.gap-system.org/Packages/smallsemi.html.
8. Edmunds, C. C., ‘Varieties generated by semigroups of order four’, Semigroup Forum 21 (1980) 6781.
9. Edmunds, C. C., Lee, E. W. H. and Lee, K. W. K., ‘Small semigroups generating varieties with continuum many subvarieties’, Order 27 (2010) 83100.
10. Golubov, È. A. and Sapir, M. V., ‘Varieties of finitely approximable semigroups’, Soviet Math. (Iz. VUZ) 26 (1982) no. 11, 2536 (Engl. transl. of Izv. Vyssh. Uchebn. Zaved. Mat. (1982) no. 11, 21–29).
11. Ježek, J., ‘Nonfinitely based three-element idempotent groupoids’, Algebra Universalis 20 (1985) 292301.
12. Karnofsky, J., ‘Finite equational bases for semigroups’, Notices Amer. Math. Soc. 17 (1970) 813814.
13. Kruse, R. L., ‘Identities satisfied by a finite ring’, J. Algebra 26 (1973) 298318.
14. Lee, E. W. H., ‘Identity bases for some non-exact varieties’, Semigroup Forum 68 (2004) 445457.
15. Lee, E. W. H., ‘On identity bases of exclusion varieties for monoids’, Comm. Algebra 35 (2007) 22752280.
16. Lee, E. W. H., ‘Combinatorial Rees–Sushkevich varieties are finitely based’, Internat. J. Algebra Comput. 18 (2008) 957978.
17. Lee, E. W. H., ‘Finite basis problem for 2-testable monoids’, Cent. Eur. J. Math. 9 (2011) 122.
18. Lee, E. W. H., ‘Finite basis problem for semigroups of order five or less: generalization and revisitation’, Studia Logica 101 (2013) 95115.
19. Lee, E. W. H. and Li, J. R., ‘Minimal non-finitely based monoids’, Dissertationes Math. (Rozprawy Mat.) 475 (2011) 365.
20. Lee, E. W. H., Li, J. R. and Zhang, W. T., ‘Minimal non-finitely based semigroups’, Semigroup Forum 85 (2012) 577580.
21. Lee, E. W. H. and Volkov, M. V., ‘On the structure of the lattice of combinatorial Rees–Sushkevich varieties’, Semigroups and formal languages, Proceedings of the International Conference, Lisboa, 2005 (eds André, J. M., Fernandes, V. H., Branco, M. J. J., Gomes, G. M. S., Fountain, J. and Meakin, J. C.; World Scientific, Singapore, 2007) 164187.
22. Lee, E. W. H. and Volkov, M. V., ‘Limit varieties generated by completely 0-simple semigroups’, Internat. J. Algebra. Comput. 21 (2011) 257294.
23. Luo, Y. F. and Zhang, W. T., ‘On the variety generated by all semigroups of order three’, J. Algebra 334 (2011) 130.
24. L’vov, I. V., ‘Varieties of associative rings. I’, Algbra. Logic 12 (1973) 150167 (Engl. transl. of Algebra i Logika 12 (1973) 269–297).
25. Lyndon, R. C., ‘Identities in two-valued calculi’, Trans. Amer. Math. Soc. 71 (1951) 457465.
26. Lyndon, R. C., ‘Identities in finite algebras’, Proc. Amer. Math. Soc. 5 (1954) 89.
27. Mashevitskiĭ, G. I., ‘An example of a finite semigroup without an irreducible basis of identities in the class of completely 0-simple semigroups’, Russian Math. Surveys 38 (1983) no. 2, 192193 (Engl. transl. of Uspekhi Mat. Nauk 38 (1983) no. 2, 211–212).
28. McKenzie, R., ‘Equational bases for lattice theories’, Math. Scand. 27 (1970) 2438.
29. Murskiĭ, V. L., ‘The existence in the three-valued logic of a closed class with a finite basis, having no finite complete system of identities’, Soviet Math. Dokl. 6 (1965) 10201024 (Engl. transl. of Dokl. Akad. Nauk SSSR 163 (1965) 815–818).
30. Oates, S. and Powell, M. B., ‘Identical relations in finite groups’, J. Algebra 1 (1964) 1139.
31. Perkins, P., ‘Bases for equational theories of semigroups’, J. Algebra 11 (1969) 298314.
32. Petrich, M. and Reilly, N. R., Completely regular semigroups (Wiley & Sons, New York, 1999).
33. Plemmons, R. J., ‘There are 15973 semigroups of order 6’, Math. Alg. 2 (1967) 217.
34. Pollák, G., ‘On two classes of hereditarily finitely based semigroup identities’, Semigroup Forum 25 (1982) 933.
35. Pollák, G., ‘Some sufficient conditions for hereditarily finitely based varieties of semigroups’, Acta Sci. Math. (Szeged) 50 (1986) no. 3–4, 299330.
36. Pollák, G. and Volkov, M. V., ‘On almost simple semigroup identities’, Colloq. Math. Soc. János Bolyai 39 (North-Holland, Amsterdam, 1985) 287323.
37. Rasin, V. V., ‘Varieties of orthodox Clifford semigroups’, Soviet Math. (Iz. VUZ) 26 (1982) no. 11, 107110 (Engl. transl. of Izv. Vyssh. Uchebn. Zaved. Mat. (1982) no. 11, 82–85).
38. Sapir, M. V., ‘Problems of Burnside type and the finite basis property in varieties of semigroups’, Math. USSR-Izv. 30 (1988) 295314 (Engl. transl. of Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 319–340).
39. Shevrin, L. N. and Volkov, M. V., ‘Identities of semigroups’, Soviet Math. (Iz. VUZ) 29 (1985) no. 11, 164 (Engl. transl. of Izv. Vyssh. Uchebn. Zaved. Mat. (1985) no. 11, 3–47).
40. Tarski, A., ‘Equational logic and equational theories of algebras’, Contributions to mathematical logic, Proceedings of the Logic Colloquium, Hannover, 1966 (eds Schmidt, H. A., Schütte, K. and Thiele, H. J.; North-Holland, Amsterdam, 1968) 275288.
41. Tishchenko, A. V., ‘The finiteness of a base of identities for five-element monoids’, Semigroup Forum 20 (1980) 171186.
42. Trahtman, A. N., ‘A basis of identities of the five-element Brandt semigroup’, Ural. Gos. Univ. Mat. Zap. 12 (1981) no. 3, 147149 (in Russian).
43. Trahtman, A. N., ‘The finite basis question for semigroups of order less than six’, Semigroup Forum 27 (1983) 387389.
44. Trahtman, A. N., ‘Some finite infinitely basable semigroups’, Ural. Gos. Univ. Mat. Zap. 14 (1987) no. 2, 128131 (in Russian).
45. Trahtman, A. N., ‘Finiteness of identity bases of five-element semigroups’, Semigroups and their homomorphisms (ed. Lyapin, E. S.; Ross. Gos. Ped. Univ., Leningrad, 1991) 7697 (in Russian).
46. Višin, V. V., ‘Identity transformations in a four-valued logic’, Soviet Math. Dokl. 4 (1963) 724726 (Engl. transl. of Dokl. Akad. Nauk SSSR 150 (1963) 719–721).
47. Volkov, M. V., ‘The finite basis question for varieties of semigroups’, Math. Notes 45 (1989) no. 3, 187194 (Engl. transl. of Mat. Zametki 45 (1989) no. 3, 12–23).
48. Volkov, M. V., ‘“Forbidden divisor” characterizations of epigroups with certain properties of group elements’, RIMS Kôkyûroku Bessatsu 1166 (2000) 226234.
49. Volkov, M. V., ‘The finite basis problem for finite semigroups’, Sci. Math. Jpn. 53 (2001) 171199.
50. Zhang, W. T. and Luo, Y. F., ‘A new example of a minimal non-finitely based semigroup’, Bull. Aust. Math. Soc. 84 (2011) 484491.
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Finite basis problem for semigroups of order six

  • Edmond W. H. Lee (a1) and Wen Ting Zhang (a2) (a3)

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