Completely regular semigroup

In mathematics, a completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. The class of completely regular semigroups forms an important subclass of the class of regular semigroups, the class of inverse semigroups being another such subclass. A H Clifford was the first to publish a major paper on completely regular semigroups though he used the terminology "semigroups admitting relative inverses" to refer to such semigroups.^[1] The name "completely regular semigroup" stems from Lyapin's book on semigroups.^[2][3] Completely regular semigroups are also called Clifford semigroups.^[4] In a completely regular semigroup, each Green H-class is a group and the semigroup is the union of these groups. ^[5] Hence completely regular semigroups are also referred to as "unions of groups".

Examples

"While there is an abundance of natural examples of inverse semigroups, for completely regular semigroups the examples (beyond completely simple semigroups) are mostly artificially constructed: the minimum ideal of a finite semigroup is completely simple, and the various relatively free completely regular semigroups are the other more or less natural examples." ^[6]

See also

• Special classes of semigroups

References

1. ^ A H Clifford (1941). "Semigroups admitting relative inverses". Annals of Mathematics (American Mathematical Society) 42 (4): 1037–1049. doi:10.2307/1968781. JSTOR 1968781. 
2. ^ E S Lyapin (1963). Semigroups. American Mathematical Society. 
3. ^ Mario Petrich; Norman R Reilly (1999). Completely regular semigroups. Wiley-IEEE. pp. 1. ISBN 0471195715. 
4. ^ Mario Petrich; Norman R Reilly (1999). Completely regular semigroups. Wiley-IEEE. pp. 63. ISBN 0471195715. 
5. ^ John M Howie (1995). Fundamentals of semigroup theory. Oxford Science Publications. Oxford University Press. ISBN 0198511949.  (Chap. 4)
6. ^ Zbl 0967.20034 (accessed on 05 May 2009)