Abstract
A semigroup S is completely regular if and only if it is a disjoint union of groups. This concept, so simple in its formulation, has intrigued investigators for over forty years. In the early days, the underlying proposition was that the theory of such objects must necessarily be readily derivable from that for groups. The early work of Rees and Clifford gave some support to this notion. However, the work of recent years, especially that on varieties, has shown that the study of completely regular semigroups requires its own ingenious arsenal of tools.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.P. Birjukov, Varieties of idempotent semigroups, Algebra i Logika, 9 (1970), 255–273 (Russian).
A.H. Clifford, Semigroups admitting relative inverses, Annals of Math., 42 (1941), 1037–1049.
R. Feigenbaum, Regular semigroup congruences, Semigroup Forum 17 (1979), 373–377.
C.F. Fennemore, All varieties of bands, Math. Nachr., 48 (1971), I: 237–252, II: 253-262.
G. Grätzer, General lattice theory, Academic Press, New York, 1978.
J. A. Gerhard, The lattice of equational classes of idempotent semigroups, J. Algebra, 15 (1970), 195–224.
T.E. Hall and P.R. Jones, On the lattice of varieties of bands of groups, Pacific J. Math. 91 (1980), 327–337.
J.M. Howie, An introduction to semigroup theory, Academic Press, London, 1976.
P.R. Jones, Completely simple semigroups: free products, free semigroups and varieties, Proc. Roy. Soc. Edinburgh Sct. A 88 (1981), 293–313.
P.R. Jones, On the lattice of varieties of completely regular semigroups, J. Austral. Math. Soc, (Series A) 35 (1983), 227–235.
J. Kadourek, On the word problem for free bands of groups and for free objects in some other varieties of completely regular semigroups, Semigroup Forum 38 (1989), 1–55.
F. Pastijn, The lattice of completely regular semigroup varieties, preprint.
F. Pastijn and M. Petrich, Congruences on regular semigroups, Trans. Amer. Math. Soc, 295 (1986), 607–633.
M. Petrich, Varieties of orthodox bands of groups, Pacific J. Math., 58 (1975), 209–217.
M. Petrich, A structure theorem for completely regular semigroups, Proc. Amer. Math. Soc., 99 (1987), 617–622.
M. Petrich, Congruences on completely regular semigroups, preprint.
M. Petrich and N.R. Reilly, All varieties of central completely simple semigroups, Trans. Amer. Math. Soc, 280 (1983), 623–636.
L. Polák, On varieties of completely regular semigroups I, Semigroup Forum, 32 (1985), 97–123.
L. Polák, On varieties of completely regular semigroups II, Semigroup Forum, 36 (1987), 253–284.
L. Polák, On varieties of completely regular semigroups III, Semigroup Forum, 37 (1988), 1–30.
V.V. Rasin, On the lattice of varieties of completely simple semigroups, Semigroup Forum, 17 (1979), 113–122.
V.V. Rasin, On the varieties of Cliffordian semigroups, Semigroup Forum, 23 (1981), 201–220.
V.V. Rasin, Varieties of orthodox Clifford semigroups, Izv. Vyssh. Uchebn. Zaved. Mat., 11 (1982), 82–85.
D. Rees, On semi-groups, Proc. Cambridge Phil. Soc, 36 (1940), 387–400.
N.R. Reilly, Varieties of completely regular semigroups, J. Austral. Math. Soc (Series A), 38 (1985), 372–393.
P.G. Trotter, Subdirect decompositions of the lattice of varieties of completely regular semigroups (to appear).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer Science+Business Media New York
About this chapter
Cite this chapter
Reilly, N.R. (1990). Completely Regular Semigroups. In: Almeida, J., Bordalo, G., Dwinger, P. (eds) Lattices, Semigroups, and Universal Algebra. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-2608-1_24
Download citation
DOI: https://doi.org/10.1007/978-1-4899-2608-1_24
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-2610-4
Online ISBN: 978-1-4899-2608-1
eBook Packages: Springer Book Archive