completely regular semigroup

{{Use dmy dates|date=November 2022}}

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.{{cite journal|doi=10.2307/1968781|author1=Clifford, A. H.|date=1941|title=Semigroups admitting relative inverses|journal=Annals of Mathematics|publisher=American Mathematical Society|volume=42|issue=4|pages=1037–1049|jstor=1968781|hdl=10338.dmlcz/100110|hdl-access=free}} The name "completely regular semigroup" stems from Lyapin's book on semigroups.{{cite book|last=E S Lyapin|title=Semigroups|publisher=American Mathematical Society|date=1963}}{{cite book|last=Mario Petrich|author2=Norman R Reilly|title=Completely regular semigroups |publisher= Wiley-IEEE|date=1999|page=1|isbn=0-471-19571-5}} In the Russian literature, completely regular semigroups are often called "Clifford semigroups".{{cite book|last=Mario Petrich|author2=Norman R Reilly|title=Completely regular semigroups |publisher= Wiley-IEEE|date=1999|page=63|isbn=0-471-19571-5}}

In the English literature, the name "Clifford semigroup" is used synonymously to "inverse Clifford semigroup", and refers to a completely regular inverse semigroup.{{cite book|last=Mario Petrich|author2=Norman R Reilly|title=Completely regular semigroups |publisher= Wiley-IEEE|date=1999|page=65|isbn=0-471-19571-5}}

In a completely regular semigroup, each Green H-class is a group and the semigroup is the union of these groups.{{cite book|last=John M Howie|title=Fundamentals of semigroup theory|publisher=Oxford University Press|date=1995|series=Oxford Science Publications|isbn=0-19-851194-9}} (Chap. 4) Hence completely regular semigroups are also referred to as "unions of groups". Epigroups generalize this notion and their class includes all completely regular semigroups.