Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T13:33:14.806Z Has data issue: false hasContentIssue false

Operators related to idempotent generated and monoid completely regular semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Mario Petrich
Affiliation:
Simon Fraser UniversityBurnaby, B.C., Canada
Norman R. Reilly
Affiliation:
Simon Fraser UniversityBurnaby, B.C., Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Clifford, A. H., ‘The free completely regular semigroup on a set’, J. Algebra 49 (1979), 434451.CrossRefGoogle Scholar
[2]Gerhard, J. A. and Petrich, M., ‘All varieties of regular orthogroups’, Semigroup Forum 31 (1985), 311351.CrossRefGoogle Scholar
[3]Hall, T. E., ‘On regular semigroups’, J. Algebra 24 (1973), 124.Google Scholar
[4]Jones, P. R., ‘Completely simple semigroups: free products, free semigroups and varieties’, Proc. Royal Soc. Edinburgh Sect. A 88 (1981), 293313.CrossRefGoogle Scholar
[5]Jones, P. R., ‘On the lattice of varieties of completely regular semigroups’, J. Austral. Math. Soc. Ser. A 35 (1983), 227235.Google Scholar
[6]Pastijn, F. and Petrich, M., ‘Congruences on regular semigroups’, Trans. Amer. Math. Soc. 295 (1986), 607633.Google Scholar
[7]Petrich, M., ‘Varieties of orthodox bands of groups’, Pacific J. Math. 58 (1975), 209217.CrossRefGoogle Scholar
[8]Petrich, M. and Reilly, N. R., ‘Near varieties of idempotent generated completely simple semigroups’, Algebra Universalis 16 (1983), 83104.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), 151175.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), 130.Google Scholar
[12]Rasin, V. V., ‘Free completely simple semigroups’, Ural. Gos. Univ. Mat. Zap. 11 (1979), 140151 (Russian).Google Scholar
[13]Reilly, N. R., ‘Varieties of completely regular semigroups’, J. Austral Math. Soc. Ser. A 38 (1985), 372393.Google Scholar
[14]Wismath, S., ‘The lattice of varieties and pseudovarieties of band monoids’, Semigroup Forum 33 (1986), 187198.CrossRefGoogle Scholar