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Operators related to idempotent generated and monoid completely regular semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Mario Petrich  and
Norman R. Reilly
Mario Petrich
Affiliation:
Simon Fraser UniversityBurnaby, B.C., Canada
Norman R. Reilly
Affiliation:
Simon Fraser UniversityBurnaby, B.C., Canada
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Abstract

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The class CR of completely regular semigroups (unions of groups or algebras with the associative binary operation of multiplication and the unary operation of inversion subject to the laws x = xx-1, (x−1)-1 = x and xx-1 = x-1x) is a variety. Among the important subclasses of CR are the classes M of monoids and I of idempotent generated members. For each C ∈ {I, M}, there are associated mappings ν → ν ∩ C and ν → (Ν ∩ C), the variety generated by ν ∩ C. The lattice theoretic properties of these mappings and the interactions between these mappings are studied.

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups 20M10: General structure theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 1 , August 1990 , pp. 1 - 23
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Clifford, A. H., ‘The free completely regular semigroup on a set’, J. Algebra 49 (1979), 434–451.CrossRefGoogle Scholar
[2]Gerhard, J. A. and Petrich, M., ‘All varieties of regular orthogroups’, Semigroup Forum 31 (1985), 311–351.CrossRefGoogle Scholar
[3]Hall, T. E., ‘On regular semigroups’, J. Algebra 24 (1973), 1–24.Google Scholar
[4]Jones, P. R., ‘Completely simple semigroups: free products, free semigroups and varieties’, Proc. Royal Soc. Edinburgh Sect. A 88 (1981), 293–313.CrossRefGoogle Scholar
[5]Jones, P. R., ‘On the lattice of varieties of completely regular semigroups’, J. Austral. Math. Soc. Ser. A 35 (1983), 227–235.Google Scholar
[6]Pastijn, F. and Petrich, M., ‘Congruences on regular semigroups’, Trans. Amer. Math. Soc. 295 (1986), 607–633.Google Scholar
[7]Petrich, M., ‘Varieties of orthodox bands of groups’, Pacific J. Math. 58 (1975), 209–217.CrossRefGoogle Scholar
[8]Petrich, M. and Reilly, N. R., ‘Near varieties of idempotent generated completely simple semigroups’, Algebra Universalis 16 (1983), 83–104.CrossRefGoogle Scholar
[9]Petrich, M. and Reilly, N. R., ‘Semigroups generated by certain operators on varieties of completely regular semigroups’, Pacific J. Math. 132 (1988), 151–175.CrossRefGoogle Scholar
[10]Petrich, M. and Reilly, N. R., ‘Operators related to E-disjunctive and fundamental completely regular semigroups’ (to appear, J. of Algebra).Google Scholar
[11]Polák, L., ‘On varieties of completely regular semigroups III’, Semigroup Forum 37 (1988), 1–30.Google Scholar
[12]Rasin, V. V., ‘Free completely simple semigroups’, Ural. Gos. Univ. Mat. Zap. 11 (1979), 140–151 (Russian).Google Scholar
[13]Reilly, N. R., ‘Varieties of completely regular semigroups’, J. Austral Math. Soc. Ser. A 38 (1985), 372–393.Google Scholar
[14]Wismath, S., ‘The lattice of varieties and pseudovarieties of band monoids’, Semigroup Forum 33 (1986), 187–198.CrossRefGoogle Scholar