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On the lattice of varieties of completely regular semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

P. R. Jones
P. R. Jones
Affiliation:
Department of Mathematics, Statistics and Computer Science Marquette UniversityMilwaukee, Wisconsin 53233, U.S.A.
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Abstract

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Several morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 2 , October 1983 , pp. 227 - 235
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Birjukov, A. P., ‘Varieties of idempotent semigroups’, Algebra i Logika 9 (1970), 255–273.Google Scholar
[2]Fennemore, C. F., ‘All varieties of bands,’ Math. Nachr. 48 (1971), I: 237–252, II: 253–262.CrossRefGoogle Scholar
[3]Gerhard, J. A., ‘The lattice of equational classes of idempotent semigroups,’ J. Algebra 15 (1970), 195–224.CrossRefGoogle Scholar
[4]Gützer, G., General lattice theory (Birkhauser Verlag, Basel, 1978).Google Scholar
[5]Hall, T. E., ‘On regular semigroups,’ J. Algebra 24 (1973), 1–24.CrossRefGoogle Scholar
[6]Hall, T. E. and Jones, P. R., ‘On the lattice of varieties of bands of groups,’, Pacific J. Math. 91 (1980) 327–337.CrossRefGoogle Scholar
[7]Howie, J. M., An introduction to semigroup theory (Academic Press, London, 1976).Google Scholar
[8]Jones, P. R., ‘Completely simple semigroups: free products, free semigroups and varieties’, Proc. Royal Soc. Edinburgh A 88 (1981), 293–313.CrossRefGoogle Scholar
[9]Masevickii, C. I., ‘On identities in varieties of completely simple semigroups over abelian groups’, Contemporary algebra, Leningrad (1978), pp. 81–89 (Russian).Google Scholar
[10]Neumann, H., Varieties of groups (Springer-Verlag, New York, 1967).CrossRefGoogle Scholar
[11]Petrich, M., ‘Certain varieties and quasivarieties of completely regular semigroups,’ Canad. J. Math. 29 (1977), 1171–1197.Google Scholar
[12]Petrich, M., ‘On the varieties of completely regular semigroups,’ Semigroup Forum 25 (1982), 153–170.CrossRefGoogle Scholar
[13]Petrich, M. and Reilly, N. R., ‘Varieties of groups and of completely simple semigroups,’ Bull. Austral. Math. Soc. 23 (1981), 339–359.CrossRefGoogle Scholar
[14]Petrich, M. and Reilly, N. R., ‘Near varieties of idempotent generated completely simple semigroups,’ Algebra Universalis, to appear.Google Scholar
[15]Petrich, M. and Reilly, N. R., “All variables of central simple semigroups”, Trans. Amer. Math. Soc., tp appear.Google Scholar
[16]Petrich, M. and Reilly, N. R., “Certain homomorphisms of the lattice of varieties of completely simple semigroups,’ J. Austral. Math. Soc., to appear.Google Scholar
[17]Rasin, V. V., ‘On the lattice of varieties of completely simple semigroups’, Semigroup Forum 17 (1979), 113–122.CrossRefGoogle Scholar