mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a completely regular semigroup is a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...

in which every element is in some

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

of the semigroup. The

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

of completely regular semigroups forms an important subclass of the

regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...

s, the class of

inverse semigroup In group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that and , i.e. a regular semigr ...

s 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. The name "completely regular semigroup" stems from Lyapin's book on semigroups. In the Russian literature, completely regular semigroups are often called "Clifford semigroups". In the English literature, the name " Clifford semigroup" is used synonymously to "inverse Clifford semigroup", and refers to a completely regular

. In a completely regular semigroup, each

Green Green is the color between cyan and yellow on the visible spectrum. It is evoked by light which has a dominant wavelength of roughly 495570 nm. In subtractive color systems, used in painting and color printing, it is created by a com ...

''H''-class is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

and the semigroup is the union of these groups. Hence completely regular semigroups are also referred to as "unions of groups".

Epigroup In abstract algebra, an epigroup is a semigroup in which every element has a power that belongs to a subgroup. Formally, for all ''x'' in a semigroup ''S'', there exists a positive integer ''n'' and a subgroup ''G'' of ''S'' such that ''x'n'' be ...

s generalize this notion and their class includes all completely regular semigroups.

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." (Retrieved 5 May 2009)

References

{{Reflist Algebraic structures Semigroup theory

Examples

See also

References