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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. Alfred 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] In the Russian literature, completely regular semigroups are often called "Clifford semigroups".[4] In the English literature, the name "Clifford semigroup" is used synonymously to "inverse Clifford semigroup", and refers to a completely regular inverse semigroup.[5] In a completely regular semigroup, each Green H-class is a group and the semigroup is the union of these groups.[6] Hence completely regular semigroups are also referred to as "unions of groups". Epigroups generalize this notion and their class includes all completely regular semigroups.

Examples

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"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."[7]

See also

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References

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  1. ^ Clifford, A. H. (1941). "Semigroups admitting relative inverses". Annals of Mathematics. 42 (4). American Mathematical Society: 1037–1049. doi:10.2307/1968781. hdl:10338.dmlcz/100110. JSTOR 1968781.
  2. ^ E S Lyapin (1963). Semigroups. American Mathematical Society.
  3. ^ Mario Petrich; Norman R Reilly (1999). Completely regular semigroups. Wiley-IEEE. p. 1. ISBN 0-471-19571-5.
  4. ^ Mario Petrich; Norman R Reilly (1999). Completely regular semigroups. Wiley-IEEE. p. 63. ISBN 0-471-19571-5.
  5. ^ Mario Petrich; Norman R Reilly (1999). Completely regular semigroups. Wiley-IEEE. p. 65. ISBN 0-471-19571-5.
  6. ^ John M Howie (1995). Fundamentals of semigroup theory. Oxford Science Publications. Oxford University Press. ISBN 0-19-851194-9. (Chap. 4)
  7. ^ Zbl 0967.20034 (Retrieved 5 May 2009)

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Special classes of semigroups

mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying

Inverse element

an I-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are completely regular semigroups; these are I-semigroups in which

Rees matrix semigroup

building new semigroups out of old ones. In his 1940 paper Rees proved the following theorem characterising completely simple semigroups: A semigroup is completely

Clifford semigroup

Clifford semigroup (sometimes also called "inverse Clifford semigroup") is a completely regular inverse semigroup. It is an inverse semigroup with x x

Automatic semigroup

In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating

Epigroup

quasi-periodic semigroup, group-bound semigroup, completely π-regular semigroup, strongly π-regular semigroup (sπr), or just π-regular semigroup (although

Four-spiral semigroup

examples of bi-simple but not completely-simple semigroups; it is also an important example of a fundamental regular semigroup; it is an indispensable building

Semiautomaton

alphabet Σ, or as the induced transformation semigroup of Q. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category