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''completely-regular semi-group''

A semi-group in which every element is a group element, that is, lies in some subgroup. An element of a semi-group is a group element if and only if it is completely regular (see [[Regular element|Regular element]]). A semi-group  $  S $
is a Clifford semi-group if and only if either of the following conditions holds: 1) For every  $  a \in S $, 
$  a \in a  ^ {2} S \cap Sa  ^ {2} $; 
or 2) each one-sided ideal  $  I $
of  $  S $
is isolated (or semi-prime), that is, if  $  x \notin I $
then  $  x  ^ {n} \notin I $
for all natural numbers  $  n $.

Together with inverse semi-groups (cf. [[Inversion semi-group|Inversion semi-group]]), Clifford semi-groups represent one of the most important types of regular semi-groups. Their study was begun in the fundamental paper [[#References|[1]]] of A.H. Clifford. Every Clifford semi-group has a (unique) decomposition into groups, the classes of which are exactly the  $  {\mathcal H} $-
classes (cf. [[Green equivalence relations|Green equivalence relations]]). Such a decomposition is not necessarily a band (see [[Band of semi-groups|Band of semi-groups]]); conditions for this to be so are known (see [[#References|[3]]]). The Green relations  $  {\mathcal J} $
and  $  {\mathcal D} $
on a Clifford semi-group coincide. Every [[Completely-simple semi-group|completely-simple semi-group]] is a Clifford semi-group; a Clifford semi-group is completely simple if and only if it is simple (see [[Simple semi-group|Simple semi-group]]). Every Clifford semi-group  $  S $
can be decomposed into a semi-lattice of completely-simple semi-groups; this decomposition is unique, its components are just the  $  {\mathcal D} $-
classes, and the corresponding quotient semi-lattice is isomorphic to the semi-lattice of principal ideals of  $  S $. 
Conversely, every semi-group decomposable into a semi-lattice of completely-simple semi-groups is a Clifford semi-group.

The following conditions are equivalent for a Clifford semi-group: 1)  $  S $
is inverse; 2) every idempotent of  $  S $
lies in the centre, that is, it commutes with every element of  $  S $; 
3) every one-sided ideal of  $  S $
is two-sided; 4) the Green relations  $  {\mathcal H} $
and  $  {\mathcal D} $
on  $  S $
coincide; 5)  $  S $
is a semi-lattice of groups; and 6)  $  S $
is a subdirect product of groups and groups with zero.

The decomposition of an arbitrary Clifford semi-group into a semi-lattice of completely-simple semi-groups defines its  "global structure" . The multiplication law for elements within the components of this decomposition is given by Rees' theorem (see [[Completely-simple semi-group|Completely-simple semi-group]]). Further investigations into Clifford semi-groups are to a large extent directed towards a clarification of their  "fine structure" , that is, to determining multiplication laws of elements from different components. When the components are groups, that is, for inverse Clifford semi-groups, there is a constructive description in terms of a so-called sum of a direct spectrum of groups. Let  $  \{ G _  \alpha  \} _ {\alpha \in A }  $
be a family of pairwise disjoint groups, let  $  A $
be a semi-lattice (see [[Idempotents, semi-group of|Idempotents, semi-group of]]) such that for every pair of elements  $  \alpha , \beta \in A $
such that  $  \alpha \geq  \beta $, 
there is a homomorphism  $  \phi _ {\alpha , \beta }  :  G _  \alpha  \rightarrow G _  \beta  $
such that  $  \phi _ {\alpha , \alpha }  $
is the identity automorphism for every  $  \alpha $
and  $  \phi _ {\alpha , \beta }  \circ \phi _ {\beta , \gamma }  = \phi _ {\alpha , \gamma }  $
whenever  $  \alpha \geq  \beta \geq  \gamma $. 
A product  $  \cdot $
is defined on the union  $  S = \cup _ {\alpha \in A }  G _  \alpha  $
by setting  $  a \cdot b = a \phi _ {\alpha , \alpha \beta }  b \phi _ {\beta , \alpha \beta }  $
for arbitrary  $  a \in G _  \alpha  $
and  $  b \in G _  \beta  $.

Then  $  S $
becomes an inverse Clifford semi-group. Conversely, every inverse Clifford semi-group can be obtained in this way.

The problem of the  "fine structure"  of Clifford semi-groups is in general extremely complicated, and to date (1987) there is no satisfactory solution to it. Certain very complex constructions, describing Clifford semi-groups in terms of completely-simple semi-groups, their translation hulls, semi-lattices, and mappings with special properties, are to be found in [[#References|[5]]]. Great progress has been achieved in the case of orthodox Clifford semi-groups (see [[Regular semi-group|Regular semi-group]]); such semi-groups are called orthogroups. There are a few clear, if somewhat cumbersome, constructions for them (see [[#References|[2]]]). All the constructions mentioned generalize, in some way, the description of inverse Clifford semi-groups obtained in [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   "Semigroups admitting relative inverses"  ''Ann. of Math.'' , '''42''' :  4  (1941)  pp. 1037–1049</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.H. Clifford,   "A structure theorem for orthogroups"  ''J. Pure Appl. Algebra'' , '''8''' :  1  (1976)  pp. 23–50</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Petrich,   "The structure of completely regular semigroups"  ''Trans. Amer. Math. Soc.'' , '''189'''  (1974)  pp. 211–236</TD></TR></table>

====Comments====
The function symbol is written after the arguments in the article above. This is common practice in the theory of semi-groups.

An extensive bibliography on recent work concerning Clifford semi-groups can be found in [[#References|[a1]]] and in the paper of J. Meakin and K.S.S. Nambooripad in [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Petrich,   "Inverse semigroups" , Wiley  (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Pollák (ed.)  St. Schwartz (ed.)  O. Steinfeld (ed.) , ''Semigroups'' , ''Coll. Math. Soc. J. Bolyai'' , '''39''' , North-Holland  (1985)</TD></TR></table>