Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Relatively inherently nonfinitely q-based semigroups


Authors: Marcel Jackson and Mikhail Volkov
Journal: Trans. Amer. Math. Soc. 361 (2009), 2181-2206
MSC (2000): Primary 08C15, 20M20
DOI: https://doi.org/10.1090/S0002-9947-08-04798-3
Published electronically: November 25, 2008
MathSciNet review: 2465833
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that every semigroup $ {\bf S}$ whose quasivariety contains a 3-nilpotent semigroup or a semigroup of index more than 2 has no finite basis for its quasi-identities provided that one of the following properties holds:

  • $ {\bf S}$ is finite;
  • $ {\bf S}$ has a faithful representation by injective partial maps on a set;
  • $ {\bf S}$ has a faithful representation by order preserving maps on a chain.
As a corollary it is shown that, in an asymptotic sense, almost all finite semigroups and finite monoids admit no finite basis for their quasi-identities.


References [Enhancements On Off] (What's this?)

  • 1. S. Burris and H.P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics 78, Springer-Verlag, Berlin-Heidelberg-New York, 1980. MR 648287 (83k:08001)
  • 2. D.M. Clark, B.A. Davey, M.G. Jackson and J.G. Pitkethly, The axiomatisability of topological prevarieties, Adv. Math. 218 (2008), 1604-1653. MR 2419934
  • 3. A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups. Vols. I, II. Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I. 1961, 1967. MR 0132791 (24:A2627); MR 0218472 (36:1558)
  • 4. D.F. Cowan, N.R. Reilly, P.G. Trotter and M.V. Volkov, The finite basis problem for quasivarieties and pseudovarieties generated by regular semigroups I. Quasivarieties generated by regular semigroups, J. Algebra 267 (2003), 635-653. MR 2003345 (2004m:20108)
  • 5. J. Dénes and A.D. Keedwell, Latin Squares and Their Applications, Academic Press, New York-London, 1974. MR 0351850 (50:4338)
  • 6. V.A. Gorbunov, Algebraic Theory of Quasivarieties, Consultants Bureau, New York, 1998. MR 1654844 (2001a:08004)
  • 7. M. Jackson, Small Semigroup Related Structures with Infinite Properties, Ph.D. thesis, University of Tasmania, 1999.
  • 8. M. Jackson, Flat algebras and the translation of universal Horn logic to equational logic, J. Symbolic Logic. 73 (2008), 90-128. MR 2387934
  • 9. M. Jackson, Residual bounds for compact totally disconnected algebras, Houston J. Math. 34 (2008), 33-67. MR 2383696
  • 10. M. Jackson and T. Stokes, Algebras of partial maps, to appear in the Proceedings of the Sydney Conference on Semigroup Theory and Related Mathematics.
  • 11. D.J. Kleitman, B.R. Rothschild and J.H. Spencer, The number of semigroups of order $ n$, Proc. Amer. Math. Soc. 55 (1976), 227-232. MR 0414380 (54:2483)
  • 12. V. Koubek and V. Rödl, Note on the number of monoids of order $ n$, Comment. Math. Univ. Carolin. 26 (1985), 309-314. MR 803927 (87a:05016)
  • 13. J. Lawrence and R. Willard, On finitely based groups and nonfinitely based quasivarieties, J. Algebra 203 (1998), 1-11. MR 1620693 (99g:20047)
  • 14. A.I. Mal'cev, On the immersion of the associative systems in groups. I, Mat. Sb. 6 (1939), 331-336 [Russian].
  • 15. A.I. Mal'cev, On the immersion of the associative systems in groups. II, Mat. Sb. 8 (1940), 251-264 [Russian].
  • 16. A.I. Mal'cev, On the general theory of algebraic systems, Mat. Sb. 35 (1954), 3-20 [Russian]. MR 0065533 (16:440e)
  • 17. S.W. Margolis and M.V. Sapir, Quasi-identities of finite semigroups and symbolic dynamics, Israel J. Math. 92 (1995), 317-331. MR 1357761 (96i:20075)
  • 18. V.L. Murskiı, The existence of a finite basis of identities, and other properties of ``almost all'' finite algebras, Problemy Kibernet. 30 (1975), 43-56 [Russian]. MR 0401606 (53:5433)
  • 19. S. Oates and M. B. Powell, Identical relations in finite groups, J. Algebra 1 (1964), 11-39. MR 0161904 (28:5108)
  • 20. A.Yu. Ol'shanskiı, Conditional identities in finite groups, Sibirsk. Mat. Zh. 15 (1974), 1409-1413 [Russian; English translation in Siberian Math. J. 15 (1975), 1000-1003]. MR 0367068 (51:3310)
  • 21. M. Petrich, Inverse Semigroups, Wiley Interscience, New York, 1984. MR 752899 (85k:20001)
  • 22. V.B. Repnitskiı and A.S. Vernitskiı, Semigroups of order-preserving mappings, Comm. Algebra 28 (2000), 3635-3641. MR 1767577 (2001b:20109)
  • 23. M.V. Sapir, On the quasivarieties generated by finite semigroups, Semigroup Forum 20 (1980), 73-88. MR 572536 (81g:08016)
  • 24. M.V. Sapir, An implication characterisation of prevarieties of semigroups and rings, Ural. Gos. Univ. Mat. Zap. 13, no. 1 (1982), 121-132 [Russian]. MR 694237 (84i:20063)
  • 25. M.V. Sapir, Semigroup Quasivarieties, Ph.D. thesis, Moscow State Pedagogical Institute, 1983 [Russian].
  • 26. B.M. Schein, A system of axioms for semigroups embeddable in generalized groups, Doklady Akad. Nauk SSSR 134 (1960), 1030-1034 [Russian; English translation in Soviet Math. Doklady 1 (1960), 1180-1183]. MR 0141719 (25:5116)
  • 27. B.M. Schein, Embedding semigroups in generalized groups, Mat. Sb. 55 (1961), 379-400 [Russian; English translation in Translations of the Amer. Math. Soc. (2) 139 (1962), 164-176]. MR 0139673 (25:3104)
  • 28. B.M. Schein, An Abstract Theory of Semigroups of One-to-One Transformations, Ph.D. thesis, Saratov State University, 1962 [Russian].
  • 29. B.M. Schein, Relation algebras and function semigroups, Semigroup Forum 1 (1970), 1-62. MR 0285638 (44:2856)
  • 30. B.M. Schein, Subsemigroups of inverse semigroups, Matematiche (Catania) 51 (1996), 205-227. MR 1485712 (98m:20078)
  • 31. B.M. Schein, On finite semigroups embeddable in inverse semigroups, Semigroup Forum 62 (2001), 329-330. MR 1831515
  • 32. L.N. Shevrin and M.V. Volkov, Identities of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. (1985), no. 11, 3-47 [Russian; English translation in Soviet Math. (Iz. VUZ) 29 (1985), no. 11, 1-64]. MR 829099 (87f:20094)
  • 33. J.D.H. Smith, Mal'cev Varieties, Lecture Notes in Mathematics 554, Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 0432511 (55:5499)
  • 34. A.S. Vernitskiı, The finite basis problem for the semigroups of order-preserving mappings, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 641-647. MR 1693605 (2000e:20092)
  • 35. M.V. Volkov, The finite basis problem for finite semigroups, Sci. Math. Jpn. 53 (2001), 171-199. MR 1821612 (2002a:20069)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 08C15, 20M20

Retrieve articles in all journals with MSC (2000): 08C15, 20M20


Additional Information

Marcel Jackson
Affiliation: Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
Email: M.G.Jackson@latrobe.edu.au

Mikhail Volkov
Affiliation: Department of Mathematics, Ural State University, Ekaterinburg 620083, Russia
Email: Mikhail.Volkov@usu.ru

DOI: https://doi.org/10.1090/S0002-9947-08-04798-3
Keywords: Quasi-identity, quasivariety, universal class, semigroup, injective map, order preserving map, finite q-basis property, inherently nonfinitely q-based semigroup relative to a class, 3-nilpotent semigroup, homotopy embedding
Received by editor(s): June 4, 2007
Published electronically: November 25, 2008
Additional Notes: The first author was supported by ARC Discovery Project Grant DP0342459
The second author acknowledges support from the Russian Foundation for Basic Research, grants 05-01-00540 and 06-01-00613. The paper was initiated during the second author’s Distinguished Fellowship at the Institute for Advanced Study of La Trobe University.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society