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COMPLETELY REGULAR MONOIDS WITH TWO GENERATORS

Part of: Semigroups

Published online by Cambridge University Press:  14 June 2011

MARIO PETRICH
MARIO PETRICH*
Affiliation:
21 420 Bol, Brač, Croatia
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Abstract

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We classify semigroups in the title according to whether they have a finite or an infinite number ofℒ-classes or ℛ-classes. For each case, we provide a concrete construction using Rees matrix semigroups and their translational hulls. An appropriate relatively free semigroup is used to complete the classification. All this is achieved by first treating the special case in which one of the generators is idempotent. We conclude by a discussion of a possible classification of 2-generator completely regular semigroups.

Keywords

completely regularmonoidRees constructiontranslational hullideal extensiongenerated byclassification

MSC classification

Secondary: 20M17: Regular semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 2 , April 2011 , pp. 271 - 287
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Clifford, A. H., ‘The free completely regular semigroup on a set’, J. Algebra 59 (1979), 434–451.CrossRefGoogle Scholar
[2]Petrich, M., Introduction to Semigroups (Merrill, Columbus, 1973).Google Scholar
[3]Petrich, M., ‘A condition on the natural order for regular semigroups’, Comm. Algebra 30 (2002), 517–542.CrossRefGoogle Scholar
[4]Petrich, M., ‘Characterizing cryptogroups with a finite number of ℋ-classes in each 𝒟-class by their subsemigroups’, Ann. Mat. Pura Appl. 187 (2008), 119–136.CrossRefGoogle Scholar
[5]Petrich, M. and Reilly, N. R., Completely Regular Semigroups (Wiley, New York, 1999).Google Scholar