Skip to main content
SpringerLink
Log in

On the Lattice of Varieties of Completely Regular Semigroups

  • Chapter
Semigroups and Their Applications

Abstract

Completely regular semigroups are semigroups which are (disjoint) unions of groups. In this article we review the progress made in the study of the lattice L(CR) of varieties of completely regular semigroups. Various results describing the lattice of sub-varieties L(U) of some proper subvariety U of CR are given; for example, with U equal to the variety of bands, central completely simple semigroups and several other interesting varieties. In the final section, certain global decompositions of L(CR) using the concepts of kernel, left trace and right trace are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A.P. Birjukov, ‘Varieties of idempotent semigroups’, Algebra i Logika, 9 (1970), 255–273.

    MathSciNet  Google Scholar 

  2. A.H. Clifford, ‘Semigroups admitting relative inverses’, Annals of Math. 42 (1941), 1037–1049.

    Article  MathSciNet  MATH  Google Scholar 

  3. A.H. Clifford, ‘The structure of orthodox unions of groups’, Semigroup Forum, 3 (1972), 283–337.

    Article  MathSciNet  MATH  Google Scholar 

  4. A.H. Clifford, ‘A structure theorem for orthogroups’, J. Pure Appl. Math. 8 (1976), 23–50.

    MathSciNet  MATH  Google Scholar 

  5. A.H. Clifford, ‘The free completely regular semigroup on a set, J. Algebra 59 (1979), 434–451.

    Article  MathSciNet  MATH  Google Scholar 

  6. A.H. Clifford and M. Petrich, ‘Some classes of completely regular semigroups’, J. Algebra 46 (1977), 462–480.

    Article  MathSciNet  MATH  Google Scholar 

  7. P.H.H. Fantham, ‘On the classification of a certain type of semigroup’, Proc. London Math. Soc. (3) 10 (1960), 409–427.

    MathSciNet  MATH  Google Scholar 

  8. R. Feigenbaum, ‘Regular semigroup congruences’. Semigroup Forum 17 (1979), 373–377.

    Article  MathSciNet  MATH  Google Scholar 

  9. C.F. Fennemore, ‘All varieties of bands’, Math Nachr. 48 (1971) I: 237–252

    Article  MathSciNet  MATH  Google Scholar 

  10. C.F. Fennemore, ‘All varieties of bands’, Math Nachr. 48 (1971) II: 253–262

    MathSciNet  MATH  Google Scholar 

  11. J.A. Gerhard, ‘The lattice of equational classes of idempotent semigroups’, J. Algebra, 15 (1970), 195–224.

    Article  MathSciNet  MATH  Google Scholar 

  12. J.A. Gerhard, ‘Subdirectly irreducible idempotent semigroups’. Pacific Journal of Mathematics, 39 (1971) 669–676.

    MathSciNet  MATH  Google Scholar 

  13. J.A. Gerhard, ‘Free completely regular semigroups I. Representation’, J. Algebra 82 (1983), 135–142.

    Article  MathSciNet  Google Scholar 

  14. J.A. Gerhard, ‘Free completely regular semigroups II. Word Problem’, J. Algebra 82 (1983), 143–156.

    Article  MathSciNet  Google Scholar 

  15. J.A. Gerhard and M. Petrich, ‘All varieties of regular orthogroups’, Semigroup Forum 31 (1985), 311–351.

    Article  MathSciNet  MATH  Google Scholar 

  16. J.A. Gerhard and M. Petrich, ‘Varieties of bands revisited’, Preprint.

    Google Scholar 

  17. J.A. Gerhard and M. Petrich, ‘Free bands and free *-bands’, Glasgow Math. J.

    Google Scholar 

  18. J.A. Green and D. Rees, ‘On semigroups in which xr = x’, Proc. Cambridge Phil. Soc. 48 (1952), 35–40.

    Article  MathSciNet  MATH  Google Scholar 

  19. T.E. Hall and P.R. Jones, ‘On the lattice of varieties of bands of groups’, Pac. J. Math. 91 (1980), 327–337.

    MathSciNet  MATH  Google Scholar 

  20. P.R. Jones, ‘Completely simple semigroups: free products, free semigroups and varieties’, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 293–313.

    MathSciNet  MATH  Google Scholar 

  21. P.R. Jones, Mal’cev products of varieties of completely regular semigroups’, preprint.

    Google Scholar 

  22. J. Kaďurek and Libor Polák, On the word problem for free completely regular semigroups, preprint.

    Google Scholar 

  23. L.G. Kovács and M.F. Newman, ‘Hanna Neumann’s problem on varieties of groups’, Proceedings, Second International Conference on Group Theory, Canberra, 1973, Lecture Notes in Mathematics No. 372, 221–225.

    Google Scholar 

  24. J. Leech, ‘The structure of a band of groups’, Mem. Amer. Math. Soc. 1 (1975), No. 157, 67–95.

    MathSciNet  Google Scholar 

  25. G.I. Maševickiǐ, ‘On identities in varieties of completely simple semigroups over abelian groups’, Contemporary Algebra, Leningrad (1978), 81–89 (Russian).

    Google Scholar 

  26. H. Neumann, ‘Varieties of groups’, Springer, New York 1967.

    Google Scholar 

  27. F. Pastijn, ‘The lattice of completely regular semigroup varieties’, preprint.

    Google Scholar 

  28. F. Pastijn and M. Petrich, ‘Congruences on regular semigroups’, Trans. Amer. Math. Soc., 295 (1986), 607–633.

    Article  MathSciNet  MATH  Google Scholar 

  29. F. Pastijn and M. Petrich, ‘The congruence lattice of a regular semigroup’, J. Pure Appl. Math.

    Google Scholar 

  30. F. Pastijn and P.G. Trotter, ‘Lattices of completely regular semigroup varieties’, Pacific J. Math. 119 (1985), 191–214.

    MathSciNet  MATH  Google Scholar 

  31. M. Petrich, ‘The structure of completely regular semigroups’, Trans. Amer. Math. Soc. 189 (1974), 211–236.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Petrich, ‘Varieties of orthodox bands of groups’, Pacific J. Math., 58 (1975), 209–217.

    MathSciNet  MATH  Google Scholar 

  33. M. Petrich, ‘Certain varieties and quasivarieties of completely regular semigroups’. Can. J. Math. XXIX (1977) 1171–1197.

    Article  MathSciNet  Google Scholar 

  34. M. Petrich, ‘On the varieties of completely regular semigroups’, Semigroup Forum 25 (1982), 153–169.

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Petrich, ‘A structure theorem for completely regular semigroups’, preprint.

    Google Scholar 

  36. M. Petrich and N.R. Reilly, ‘Varieties of groups and completely simple semigroups’. Bull. Austral. Math. Soc. 23 (1981), 339–359.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Petrich and N.R. Reilly, ‘Near varieties of idempotent generated completely simple semigroups’, Algebra Universalis 16 (1973), 83–104.

    Article  MathSciNet  Google Scholar 

  38. M. Petrich and N.R. Reilly, ‘All varieties of central completely simple semigroups’. Trans. Amer. Math. Soc. 280 (1983), 623–636.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Petrich and N.R. Reilly, ‘Certain homomorphisms of the lattice of varieties of completely simple semigroups’, J. Austral. Math. Soc. (Series A) 37 (1984), 287–306.

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Petrich and N.R. Reilly, ‘A semigroup of operators on the lattice of varieties of completely regular semigroups’, preprint.\

    Google Scholar 

  41. L. Polák, ‘On varieties of completely regular semigroups I’, Semigroup Forum 32 (1985), 97–123.

    Article  MathSciNet  MATH  Google Scholar 

  42. V.V. Rasin, ‘On the lattice of varieties of completely simple semigroups’. Semigroup Forum 17 (1979), 113–122.

    Article  MathSciNet  MATH  Google Scholar 

  43. V.V. Rasin, ‘Free completely simple semigroups’, Mat. Zapiski Ural. Univ. 11 (1979), 140–151 (Russian).

    Google Scholar 

  44. V.V. Rasin, ‘On the varieties of Cliffordian semigroups’, Semigroup Forum 23 (1981), 201–220.

    Article  MathSciNet  MATH  Google Scholar 

  45. V.V. Rasin, ‘Varieties of orthodox Clifford semigroups’, Izv. Vyssh. Uchebn. Zaved. Mat., 1982, No. 11, 82–85 (Russian)

    MathSciNet  Google Scholar 

  46. D. Rees, ‘On semi-groups’, Proc. Cambridge Phil. Soc. 36 (1940), 387–400.

    Article  MathSciNet  Google Scholar 

  47. N.R. Reilly, ‘Varieties of completely regular semigroups’, J. Austral. Math. Soc. (Series A) 38 (1985), 372–393.

    Article  MathSciNet  MATH  Google Scholar 

  48. P.G. Trotter, ‘Free completely regular semigroups’, Glasgow Math. J.

    Google Scholar 

  49. P.G. Trotter, ‘Relatively free bands of groups’, preprint.

    Google Scholar 

  50. R.J. Warne, ‘On the structure of semigroups which are unions of groups’. Trans. Amer. Math. Soc. 186 (1973), 385–401.

    Article  MathSciNet  Google Scholar 

  51. S. Wismath, ‘The lattice of varieties and pseudovarieties of band monoids’. Semigroup Forum, 33 (1986), 187–198.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Yamada, ‘Construction of inversive semigroups’, Mem. Fac. Lit. Sci. Shimane Univ., Nat. Sci 4 (1971), 1–9.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, British Columbia, Canada, V5A 1S6

    Norman R. Reilly

Authors
  1. Norman R. Reilly
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, California State University, Chico, USA

    Simon M. Goberstein  & Peter M. Higgins  & 

  2. Division of Computing and Mathematics, Deakin University, USA

    Peter M. Higgins

Rights and permissions

Reprints and permissions

Copyright information

© 1987 D. Reidel Publishing Company

About this chapter

Cite this chapter

Reilly, N.R. (1987). On the Lattice of Varieties of Completely Regular Semigroups. In: Goberstein, S.M., Higgins, P.M. (eds) Semigroups and Their Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3839-7_19

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-3839-7_19

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8209-9

  • Online ISBN: 978-94-009-3839-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation