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A [[Semi-group|semi-group]] in which every element is regular (see [[Regular element|Regular element]]).

An arbitrary regular semi-group  $  S $
contains idempotents (see [[Idempotent|Idempotent]]), and the structure of  $  S $
is determined to a considerable extent by the  "structure"  and the  "distribution"  in  $  S $
of the set  $  E ( S) $
of all its idempotents (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]). Regular semi-groups with a unique idempotent are just groups. In the first place,  $  E ( S) $
can be regarded as a partially ordered set in a natural way. There are known structure theorems describing a regular semi-group  $  S $
with certain natural restrictions on the set  $  E ( S) $. 
One such restriction (for semi-groups with zero) is that all non-zero idempotents are primitive (see [[Completely-simple semi-group|Completely-simple semi-group]]); a semi-group with this property is called primitive. The following conditions on a semi-group  $  S $
are equivalent: a)  $  S $
is a primitive regular semi-group; b)  $  S $
is a regular semi-group equal to the union of its  $  O $-
minimal (right) ideals (see [[Minimal ideal|Minimal ideal]]); and c)  $  S $
is an [[O-direct union| $  O $-
direct union]] of completely  $  0 $-
simple semi-groups. The structure of regular semi-groups is also known in the case when  $  E ( S) $
is a chain with the order type of the negative integers [[#References|[2]]].

A more informative view of  $  E ( S) $
is obtained if one defines a partial operation  $  \circ $
on it in the following way. If  $  e , f \in E ( S) $
are such that at least one of the products  $  e f $, 
$  f e $
is equal to either  $  e $
or  $  f $, 
then  $  e f \in E ( S) $; 
one then sets  $  e \circ f = e f $. 
The resulting partial algebra can be axiomatized in terms of two quasi-order relations  $  \omega  ^ {r} $
and  $  \omega  ^ {l} $. 
These are closely related to the given partial operation (the realization of these relations in  $  E ( S) $
is as follows:  $  e \omega  ^ {r} f $
means  $  f e = e $, 
$  e \omega  ^ {l} f $
means  $  e f = e $; 
then  $  \omega  ^ {r} \cap \omega  ^ {l} $
is the natural partial order on  $  E ( S) $). 
Such a partial algebra is called a bi-ordered set (see [[#References|[5]]]). An arbitrary regular semi-group can be constructed in a specific way from a bi-ordered set and groups. It is thus possible to classify regular semi-groups in terms of bi-ordered sets. Among the types of semi-groups that have been investigated in this way are combinatorial regular semi-groups (see [[#References|[7]]]), that is, those whose only subgroups consist of one element.

A homomorphic image of a regular semi-group is regular. Every [[Normal complex|normal complex]] of a regular semi-group which is a sub-semi-group contains an idempotent. An arbitrary congruence (cf. [[Congruence (in algebra)|Congruence (in algebra)]]) on a regular semi-group is uniquely determined by its classes that contain idempotents. A congruence on a regular semi-group  $  S $
separates idempotents if and only if it is contained in the relation  $  {\mathcal H} $(
see [[Green equivalence relations|Green equivalence relations]]). The set of such congruences forms a modular sublattice with a zero and a unit element in the lattice of all congruences on  $  S $(
cf. also [[Modular lattice|Modular lattice]]). A regular semi-group is called fundamental if this sublattice contains only the equality relation. Every combinatorial regular semi-group is fundamental. Fundamental regular semi-groups are important, not only as one of the more visible types of regular semi-groups, but also because of their  "universality"  property in the class of all semi-groups. More precisely, for any bi-ordered set  $  E $
it is possible to construct in a canonical way a fundamental regular semi-group  $  T _ {E} $
such that  $  E $
is the bi-ordered set of all idempotents, and for any regular semi-group  $  S $
with  $  E ( S) = E $
there is a homomorphism  $  \phi :  S \rightarrow T _ {E} $
that separates idempotents and is such that  $  \phi ( S) $
is a sub-semi-group of  $  T _ {E} $
containing  $  E $(
for various constructions of  $  T _ {E} $, 
see [[#References|[3]]], [[#References|[5]]], [[#References|[8]]], [[#References|[10]]]). A regular semi-group  $  S $
is fundamental if and only if  $  \phi $
is injective.

If  $  S $
is a regular semi-group, then the sub-semi-group  $  \langle  E ( S) \rangle $
generated by its idempotents is also regular. The sub-semi-group  $  \langle  E ( S) \rangle $
exerts an essential influence on the structure of  $  S $. 
A regular semi-group is idempotently generated if and only if the same is true for each of its principal factors [[#References|[10]]]. In an idempotently-generated regular semi-group  $  S $, 
any element  $  x $
can be written in the form  $  x = e _ {1} \dots e _ {n} $, 
where  $  e _ {i} \in E ( S) $
and  $  e _ {i} ( {\mathcal L} \cup {\mathcal R} ) e _ {i+} 1 $
for  $  i = 1 \dots n - 1 $(
here  $  {\mathcal L} $
and  $  {\mathcal R} $
are Green equivalence relations, [[#References|[5]]]). A sequence of idempotents  $  e _ {1} \dots e _ {n} $
with the above property is called an  $  E $-
chain. In a bi-simple idempotently-generated semi-group, any two idempotents are connected by an  $  E $-
chain, and if they are comparable in the sense of the natural partial order, then such a chain has length  $  \geq  4 $.

If  $  \langle  E ( S) \rangle = E( S) $, 
that is, the product of any two idempotents is again an idempotent, then the regular semi-group  $  S $
is called orthodox. The class of orthodox semi-groups contains, in particular, all inverse semi-groups. A semi-group is orthodox if and only if its principal factors are. There are structure theorems for orthodox semi-groups (see [[#References|[4]]], [[#References|[9]]]).

The natural partial order on  $  E ( S) $
can be extended to the regular semi-group  $  S $
in the following way:  $  x \leq  y $
if there are idempotents  $  e $
and  $  f $
such that  $  x = e y = y f $. 
If  $  S $
is inverse, the relation  $  \leq  $
becomes the natural partial order, and it is also called the natural partial order for an arbitrary regular semi-group. The relation  $  \leq  $
on the regular semi-group  $  S $
is compatible with the multiplication if and only if, for any idempotent  $  e $, 
the sub-semi-group  $  e S e $
is inverse [[#References|[6]]] (cf. [[Inversion semi-group|Inversion semi-group]]). Regular semi-groups with this property are called pseudo-inverse. A wider class is formed by pseudo-orthodox semi-groups (those in which the sub-semi-group  $  e S e $
is orthodox for any idempotent  $  e $). 
These classes of semi-groups are also called  "locally inverse regular semi-grouplocally inverse"  and  "locally orthodox regular semi-grouplocally orthodox" , respectively. A regular semi-group is called natural if the set of all its group elements (see [[Regular element|Regular element]]) is a sub-semi-group. There are structure theorems for pseudo-inverse, pseudo-orthodox [[#References|[11]]] and natural [[#References|[12]]] regular semi-groups.

Numerous structure theorems for various types of regular semi-groups represent (sometimes very remote) generalizations and modifications of the structure of a [[Rees semi-group of matrix type|Rees semi-group of matrix type]] or of the sum of the direct spectrum of groups (see [[Clifford semi-group|Clifford semi-group]]), and are based on various representations of semi-groups and their decomposition into subdirect products (see [[#References|[1]]], [[#References|[13]]]). See also [[Semi-group|Semi-group]].

====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  W.D. Munn,   "Regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085082.png" />-semigroups"  ''Glasgow Math. J.'' , '''9''' :  1  (1968)  pp. 46–66</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Clifford,   "The fundamental representation of a regular semigroup"  ''Semigroup Forum'' , '''10'''  (1975)  pp. 84–92</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Clifford,   "A structure theorem for orthogroups"  ''J. Pure Appl. Algebra'' , '''8'''  (1976)  pp. 23–50</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K.S.S. Nambooripad,   "Structure of regular semigroups, I"  ''Mem. Amer. Math. Soc.'' , '''22''' :  224  (1979)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K.S.S. Nambooripad,   "The natural partial order on a regular semigroup"  ''Proc. Edinburgh Math. Soc.'' , '''23''' :  3  (1980)  pp. 249–260</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  K.S.S. Nambooripad,   A.R. Rajan,   "Structure of combinatorial regular semigroups"  ''Quart. J. Math.'' , '''29''' :  116  (1978)  pp. 489–504</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  P.A. Grillet,   "The structure of regular semigroups, I-IV"  ''Semigroup Forum'' , '''8'''  (1974)  pp. 177–183; 254–265; 368–373</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  T.E. Hall,   "Orthodox semigroups"  ''Pacific. J. Math.'' , '''39'''  (1971)  pp. 677–686</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  T.E. Hall,   "On regular semigroups"  ''J. of Algebra'' , '''24'''  (1973)  pp. 1–24</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  J. Meakin,   K.S.S. Nambooripad,   "Coextensions of pseudo-inverse semigroups by rectangular bands"  ''J. Austral. Math. Soc.'' , '''30'''  (1980/81)  pp. 73–86</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  R.J. Warne,   "Natural regular semigroups"  G. Pollák (ed.) , ''Algebraic Theory of Semigroups'' , North-Holland  (1979)  pp. 685–720</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  G. Lallement,   "Structure theorems for regular semigroups"  ''Semigroup Forum'' , '''4'''  (1972)  pp. 95–123</TD></TR></table>