In mathematics a '''completely regular semigroup''' is a semigroup in which every element is in some subgroup of the semigroup The Class (set theory)|class of completely regular semigroups forms an important subclass of the Class (set theory)|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
The name "completely regular semigroup" stems from Lyapin's book on semigroups
Completely regular semigroups are also called "Clifford semigroups"
In a completely regular semigroup
each Green H
-class is a group and the semigroup
is the union of these groups
Hence completely regular semigroups are also referred to as "unions of groups"