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A [[Semi-group|semi-group]] in which every element is regular (see [[Regular element|Regular element]]).
 
A [[Semi-group|semi-group]] in which every element is regular (see [[Regular element|Regular element]]).
  
An arbitrary regular semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808501.png" /> contains idempotents (see [[Idempotent|Idempotent]]), and the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808502.png" /> is determined to a considerable extent by the  "structure"  and the  "distribution"  in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808503.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808504.png" /> 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808505.png" /> can be regarded as a partially ordered set in a natural way. There are known structure theorems describing a regular semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808506.png" /> with certain natural restrictions on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808507.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808508.png" /> are equivalent: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r0808509.png" /> is a primitive regular semi-group; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085010.png" /> is a regular semi-group equal to the union of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085011.png" />-minimal (right) ideals (see [[Minimal ideal|Minimal ideal]]); and c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085012.png" /> is an [[O-direct union|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085013.png" />-direct union]] of completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085014.png" />-simple semi-groups. The structure of regular semi-groups is also known in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085015.png" /> is a chain with the order type of the negative integers [[#References|[2]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085016.png" /> is obtained if one defines a partial operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085017.png" /> on it in the following way. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085018.png" /> are such that at least one of the products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085020.png" /> is equal to either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085023.png" />; one then sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085024.png" />. The resulting partial algebra can be axiomatized in terms of two quasi-order relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085026.png" />. These are closely related to the given partial operation (the realization of these relations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085027.png" /> is as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085028.png" /> means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085030.png" /> means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085031.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085032.png" /> is the natural partial order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085033.png" />). 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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085034.png" /> separates idempotents if and only if it is contained in the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085035.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085036.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085037.png" /> it is possible to construct in a canonical way a fundamental regular semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085039.png" /> is the bi-ordered set of all idempotents, and for any regular semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085040.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085041.png" /> there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085042.png" /> that separates idempotents and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085043.png" /> is a sub-semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085044.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085045.png" /> (for various constructions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085046.png" />, see [[#References|[3]]], [[#References|[5]]], [[#References|[8]]], [[#References|[10]]]). A regular semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085047.png" /> is fundamental if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085048.png" /> is injective.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085049.png" /> is a regular semi-group, then the sub-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085050.png" /> generated by its idempotents is also regular. The sub-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085051.png" /> exerts an essential influence on the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085052.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085053.png" />, any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085054.png" /> can be written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085057.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085058.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085060.png" /> are Green equivalence relations, [[#References|[5]]]). A sequence of idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085061.png" /> with the above property is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085063.png" />-chain. In a bi-simple idempotently-generated semi-group, any two idempotents are connected by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085064.png" />-chain, and if they are comparable in the sense of the natural partial order, then such a chain has length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085065.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085066.png" />, that is, the product of any two idempotents is again an idempotent, then the regular semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085067.png" /> 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]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085068.png" /> can be extended to the regular semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085069.png" /> in the following way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085070.png" /> if there are idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085072.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085073.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085074.png" /> is inverse, the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085075.png" /> becomes the natural partial order, and it is also called the natural partial order for an arbitrary regular semi-group. The relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085076.png" /> on the regular semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085077.png" /> is compatible with the multiplication if and only if, for any idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085078.png" />, the sub-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085079.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085080.png" /> is orthodox for any idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080850/r08085081.png" />). 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.
+
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]].
 
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]].

Latest revision as of 08:10, 6 June 2020


A semi-group in which every element is regular (see Regular element).

An arbitrary regular semi-group $ S $ contains idempotents (see 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). 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); 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); and c) $ S $ is an $ 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 [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 [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 [7]), that is, those whose only subgroups consist of one element.

A homomorphic image of a regular semi-group is regular. Every normal complex of a regular semi-group which is a sub-semi-group contains an idempotent. An arbitrary congruence (cf. 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). 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). 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 [3], [5], [8], [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 [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, [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 [4], [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 [6] (cf. 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) is a sub-semi-group. There are structure theorems for pseudo-inverse, pseudo-orthodox [11] and natural [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 or of the sum of the direct spectrum of groups (see Clifford semi-group), and are based on various representations of semi-groups and their decomposition into subdirect products (see [1], [13]). See also Semi-group.

References

[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2] W.D. Munn, "Regular -semigroups" Glasgow Math. J. , 9 : 1 (1968) pp. 46–66
[3] A. Clifford, "The fundamental representation of a regular semigroup" Semigroup Forum , 10 (1975) pp. 84–92
[4] A. Clifford, "A structure theorem for orthogroups" J. Pure Appl. Algebra , 8 (1976) pp. 23–50
[5] K.S.S. Nambooripad, "Structure of regular semigroups, I" Mem. Amer. Math. Soc. , 22 : 224 (1979)
[6] K.S.S. Nambooripad, "The natural partial order on a regular semigroup" Proc. Edinburgh Math. Soc. , 23 : 3 (1980) pp. 249–260
[7] K.S.S. Nambooripad, A.R. Rajan, "Structure of combinatorial regular semigroups" Quart. J. Math. , 29 : 116 (1978) pp. 489–504
[8] P.A. Grillet, "The structure of regular semigroups, I-IV" Semigroup Forum , 8 (1974) pp. 177–183; 254–265; 368–373
[9] T.E. Hall, "Orthodox semigroups" Pacific. J. Math. , 39 (1971) pp. 677–686
[10] T.E. Hall, "On regular semigroups" J. of Algebra , 24 (1973) pp. 1–24
[11] J. Meakin, K.S.S. Nambooripad, "Coextensions of pseudo-inverse semigroups by rectangular bands" J. Austral. Math. Soc. , 30 (1980/81) pp. 73–86
[12] R.J. Warne, "Natural regular semigroups" G. Pollák (ed.) , Algebraic Theory of Semigroups , North-Holland (1979) pp. 685–720
[13] G. Lallement, "Structure theorems for regular semigroups" Semigroup Forum , 4 (1972) pp. 95–123
How to Cite This Entry:
Regular semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_semi-group&oldid=48487
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article