Namespaces
Variants
Actions

Difference between revisions of "Simple semi-group"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (Undo revision 48706 by Ulf Rehmann (talk))
Tag: Undo
m (tex encoded by computer)
Line 1: Line 1:
A [[Semi-group|semi-group]] not containing proper ideals or congruences of some fixed type. Various kinds of simple semi-groups arise, depending on the type considered: ideal-simple semi-groups, not containing proper two-sided ideals (the term simple semi-group is often used for such semi-groups only); left (right) simple semi-groups, not containing proper left (right) ideals; (left, right) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853002.png" />-simple semi-groups, semi-groups with a zero not containing proper non-zero two-sided (left, right) ideals and not being two-element semi-groups with zero multiplication; bi-simple semi-groups, consisting of one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853003.png" />-class (cf. [[Green equivalence relations|Green equivalence relations]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853005.png" />-bi-simple semi-groups, consisting of two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853006.png" />-classes one of which is the null class; and congruence-free semi-groups, not having congruences other than the universal relation and the equality relation.
+
<!--
 +
s0853002.png
 +
$#A+1 = 48 n = 0
 +
$#C+1 = 48 : ~/encyclopedia/old_files/data/S085/S.0805300 Simple semi\AAhgroup
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Every left or right simple semi-group is bi-simple; every bi-simple semi-group is ideal-simple, but there are ideal-simple semi-groups that are not bi-simple (and even ones for which all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853007.png" />-classes consist of one element). The most important type of ideal-simple semi-groups (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853008.png" />-simple semi-groups) are the completely-simple semi-groups (completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853009.png" />-simple semi-groups, cf. [[Completely-simple semi-group|Completely-simple semi-group]]). The most important examples of bi-simple but not completely-simple semi-groups are: the bicyclic semi-groups and the four-spiral semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530010.png" /> (cf. [[Bicyclic semi-group|Bicyclic semi-group]]; [[#References|[11]]]). The latter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530011.png" />, is given by generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530012.png" /> and defining relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530023.png" />. It is isomorphic to a [[Rees semi-group of matrix type|Rees semi-group of matrix type]] over a bicyclic semi-group with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530025.png" />, with sandwich-matrix
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530026.png" /></td> </tr></table>
+
A [[Semi-group|semi-group]] not containing proper ideals or congruences of some fixed type. Various kinds of simple semi-groups arise, depending on the type considered: ideal-simple semi-groups, not containing proper two-sided ideals (the term simple semi-group is often used for such semi-groups only); left (right) simple semi-groups, not containing proper left (right) ideals; (left, right)  $  0 $-
 +
simple semi-groups, semi-groups with a zero not containing proper non-zero two-sided (left, right) ideals and not being two-element semi-groups with zero multiplication; bi-simple semi-groups, consisting of one  $  {\mathcal D} $-
 +
class (cf. [[Green equivalence relations|Green equivalence relations]]); 0 $-
 +
bi-simple semi-groups, consisting of two  $  {\mathcal D} $-
 +
classes one of which is the null class; and congruence-free semi-groups, not having congruences other than the universal relation and the equality relation.
  
In a sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530027.png" /> is minimal among the bi-simple not completely-simple semi-groups generated by a finite number of idempotents, and quite often it arises as a sub-semi-group of those semi-groups.
+
Every left or right simple semi-group is bi-simple; every bi-simple semi-group is ideal-simple, but there are ideal-simple semi-groups that are not bi-simple (and even ones for which all the  $  {\mathcal D} $-
 +
classes consist of one element). The most important type of ideal-simple semi-groups ( $  0 $-
 +
simple semi-groups) are the completely-simple semi-groups (completely  $  0 $-
 +
simple semi-groups, cf. [[Completely-simple semi-group|Completely-simple semi-group]]). The most important examples of bi-simple but not completely-simple semi-groups are: the bicyclic semi-groups and the four-spiral semi-group  $  \mathop{\rm Sp} _ {4} $(
 +
cf. [[Bicyclic semi-group|Bicyclic semi-group]]; [[#References|[11]]]). The latter,  $  \mathop{\rm Sp} _ {4} $,
 +
is given by generators  $  a , b , c , d $
 +
and defining relations  $  a ^ {2} = a $,
 +
$  b  ^ {2} = b $,
 +
$  c  ^ {2} = c $,
 +
$  d  ^ {2} = d $,
 +
$  b a = a $,
 +
$  a b = b $,
 +
$  b c = b $,
 +
$  c b = c $,
 +
$  d c = c $,
 +
$  c d = d $,
 +
$  d a = d $.
 +
It is isomorphic to a [[Rees semi-group of matrix type|Rees semi-group of matrix type]] over a bicyclic semi-group with generators  $  u , v $,
 +
where  $  u v = 1 $,
 +
with sandwich-matrix
  
Right simple semi-groups are also called semi-groups with right division, or semi-groups with right invertibility. The reason for this terminology is the following equivalent property of such semi-groups: For any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530028.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530030.png" />. The right simple semi-groups containing idempotents are precisely the right groups (cf. [[Right group|Right group]]). An important example of a right simple semi-group without idempotents is given by the semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530031.png" /> of all transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530032.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530033.png" /> such that: 1) the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530034.png" /> is the equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530036.png" />; 2) the cardinality of the quotient set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530037.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530038.png" />; 3) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530039.png" /> intersects each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530040.png" />-class in at most one element; and 4) the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530041.png" />-classes disjoint from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530042.png" /> has infinite cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530043.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530044.png" />. The semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530045.png" /> is called a Teissier semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530046.png" />, and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530047.png" /> is the equality relation, it is called a Baer–Levi semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530048.png" /> (cf. [[#References|[6]]], [[#References|[7]]]). A Teissier semi-group is an example of a right simple semi-group without idempotents that does not necessarily satisfy the right cancellation law. Every right simple semi-group without idempotents can be imbedded in a suitable Teissier semi-group, while every such semi-group with the right cancellation law can be imbedded in a suitable Baer–Levi semi-group (in both cases one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530049.png" />).
+
$$
 +
\left \|
  
Various types of simple semi-groups often arise as  "blocks"  from which one can construct the semi-groups under consideration. For classical examples of simple semi-groups see [[Completely-simple semi-group|Completely-simple semi-group]]; [[Brandt semi-group|Brandt semi-group]]; [[Right group|Right group]]; for bi-simple inverse semi-groups (including structure theorems under certain restrictions on the semi-lattice of idempotents) see [[#References|[1]]], [[#References|[8]]], [[#References|[9]]]. There are ideal-simple inverse semi-groups with an arbitrary number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530050.png" />-classes. In the study of imbedding of semi-groups in simple semi-groups one usually either indicates conditions for the possibility of the corresponding imbedding, or establishes that any semi-group can be imbedded in a semi-group of the type considered. E.g., any semi-group can be imbedded in a bi-simple semi-group with an identity (cf. [[#References|[1]]]), in a bi-simple semi-group generated by idempotents (cf. [[#References|[10]]]), and in a semi-group that is simple relative to congruences (which may have some property given in advance: the presence or absence of a zero, completeness, having an empty Frattini sub-semi-group, etc., cf. [[#References|[3]]]–[[#References|[5]]]).
+
\begin{array}{cc}
 +
1  & v  \\
 +
1  & 1  \\
 +
\end{array}
 +
\right \| .
 +
$$
 +
 
 +
In a sense,  $  \mathop{\rm Sp} _ {4} $
 +
is minimal among the bi-simple not completely-simple semi-groups generated by a finite number of idempotents, and quite often it arises as a sub-semi-group of those semi-groups.
 +
 
 +
Right simple semi-groups are also called semi-groups with right division, or semi-groups with right invertibility. The reason for this terminology is the following equivalent property of such semi-groups: For any elements  $  a , b $
 +
there is an  $  x $
 +
such that  $  a x = b $.
 +
The right simple semi-groups containing idempotents are precisely the right groups (cf. [[Right group|Right group]]). An important example of a right simple semi-group without idempotents is given by the semi-groups  $  T ( M , \delta , p , q ) $
 +
of all transformations  $  \phi $
 +
of a set  $  M $
 +
such that: 1) the kernel of  $  \phi $
 +
is the equivalence relation  $  \delta $
 +
on  $  M $;
 +
2) the cardinality of the quotient set  $  M / \delta $
 +
is  $  p $;
 +
3) the set  $  M \phi $
 +
intersects each  $  \delta $-
 +
class in at most one element; and 4) the set of  $  \delta $-
 +
classes disjoint from  $  M \phi $
 +
has infinite cardinality  $  q $,
 +
and  $  q \leq  p $.
 +
The semi-group  $  T ( M , \delta , p , q ) $
 +
is called a Teissier semi-group of type  $  ( p , q ) $,
 +
and, if  $  \delta $
 +
is the equality relation, it is called a Baer–Levi semi-group of type  $  ( p , q ) $(
 +
cf. [[#References|[6]]], [[#References|[7]]]). A Teissier semi-group is an example of a right simple semi-group without idempotents that does not necessarily satisfy the right cancellation law. Every right simple semi-group without idempotents can be imbedded in a suitable Teissier semi-group, while every such semi-group with the right cancellation law can be imbedded in a suitable Baer–Levi semi-group (in both cases one can take  $  p = q $).
 +
 
 +
Various types of simple semi-groups often arise as  "blocks"  from which one can construct the semi-groups under consideration. For classical examples of simple semi-groups see [[Completely-simple semi-group|Completely-simple semi-group]]; [[Brandt semi-group|Brandt semi-group]]; [[Right group|Right group]]; for bi-simple inverse semi-groups (including structure theorems under certain restrictions on the semi-lattice of idempotents) see [[#References|[1]]], [[#References|[8]]], [[#References|[9]]]. There are ideal-simple inverse semi-groups with an arbitrary number of $  {\mathcal D} $-
 +
classes. In the study of imbedding of semi-groups in simple semi-groups one usually either indicates conditions for the possibility of the corresponding imbedding, or establishes that any semi-group can be imbedded in a semi-group of the type considered. E.g., any semi-group can be imbedded in a bi-simple semi-group with an identity (cf. [[#References|[1]]]), in a bi-simple semi-group generated by idempotents (cf. [[#References|[10]]]), and in a semi-group that is simple relative to congruences (which may have some property given in advance: the presence or absence of a zero, completeness, having an empty Frattini sub-semi-group, etc., cf. [[#References|[3]]]–[[#References|[5]]]).
  
 
====References====
 
====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">  E.S. Lyapin,  "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Bokut',  "Some embedding theorems for rings and semigroups"  ''Sibirsk. Mat. Zh.'' , '''4''' :  3  (1963)  pp. 500–518  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.G. Shutov,  "Embeddings of semigroups in simple and complete semigroups"  ''Mat. Sb.'' , '''62''' :  4  (1963)  pp. 496–511  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.N. Klimov,  "Embedding of semigroups in factorizable semigroups"  ''Sib. Math. J.'' , '''14''' :  5  (1973)  pp. 715–722  ''Sibirsk. Mat. Zh.'' , '''14''' :  5  (1973)  pp. 1025–1036</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Baer,  F. Levi,  "Vollständige irreduzibele Systeme von Gruppenaxiomen"  ''Sitzungsber. Heidelb. Akad. Wissenschaft. Math.-Nat. Kl.'' , '''2'''  (1932)  pp. 3–12</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M. Teissier,  "Sur les demi-groupes admettant l'existence du quotient d'un cote"  ''C.R. Acad. Sci. Paris'' , '''236''' :  11  (1953)  pp. 1120–1122</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  W.D. Munn,  "Some recent results on the structure of inverse semigroups"  K.W. Folley (ed.) , ''Semigroups'' , Acad. Press  (1969)  pp. 107–123</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J.M. Howie,  "An introduction to semigroup theory" , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  F. Pastijn,  "Embedding semigroups in semibands"  ''Semigroup Forum'' , '''14''' :  3  (1977)  pp. 247–263</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  K. Byleen,  J. Meakin,  F. Pastijn,  "The fundamental four-spiral semigroup"  ''J. of Algebra'' , '''54'''  (1978)  pp. 6–26</TD></TR></table>
 
<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">  E.S. Lyapin,  "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Bokut',  "Some embedding theorems for rings and semigroups"  ''Sibirsk. Mat. Zh.'' , '''4''' :  3  (1963)  pp. 500–518  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.G. Shutov,  "Embeddings of semigroups in simple and complete semigroups"  ''Mat. Sb.'' , '''62''' :  4  (1963)  pp. 496–511  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.N. Klimov,  "Embedding of semigroups in factorizable semigroups"  ''Sib. Math. J.'' , '''14''' :  5  (1973)  pp. 715–722  ''Sibirsk. Mat. Zh.'' , '''14''' :  5  (1973)  pp. 1025–1036</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Baer,  F. Levi,  "Vollständige irreduzibele Systeme von Gruppenaxiomen"  ''Sitzungsber. Heidelb. Akad. Wissenschaft. Math.-Nat. Kl.'' , '''2'''  (1932)  pp. 3–12</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M. Teissier,  "Sur les demi-groupes admettant l'existence du quotient d'un cote"  ''C.R. Acad. Sci. Paris'' , '''236''' :  11  (1953)  pp. 1120–1122</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  W.D. Munn,  "Some recent results on the structure of inverse semigroups"  K.W. Folley (ed.) , ''Semigroups'' , Acad. Press  (1969)  pp. 107–123</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J.M. Howie,  "An introduction to semigroup theory" , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  F. Pastijn,  "Embedding semigroups in semibands"  ''Semigroup Forum'' , '''14''' :  3  (1977)  pp. 247–263</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  K. Byleen,  J. Meakin,  F. Pastijn,  "The fundamental four-spiral semigroup"  ''J. of Algebra'' , '''54'''  (1978)  pp. 6–26</TD></TR></table>

Revision as of 14:55, 7 June 2020


A semi-group not containing proper ideals or congruences of some fixed type. Various kinds of simple semi-groups arise, depending on the type considered: ideal-simple semi-groups, not containing proper two-sided ideals (the term simple semi-group is often used for such semi-groups only); left (right) simple semi-groups, not containing proper left (right) ideals; (left, right) $ 0 $- simple semi-groups, semi-groups with a zero not containing proper non-zero two-sided (left, right) ideals and not being two-element semi-groups with zero multiplication; bi-simple semi-groups, consisting of one $ {\mathcal D} $- class (cf. Green equivalence relations); $ 0 $- bi-simple semi-groups, consisting of two $ {\mathcal D} $- classes one of which is the null class; and congruence-free semi-groups, not having congruences other than the universal relation and the equality relation.

Every left or right simple semi-group is bi-simple; every bi-simple semi-group is ideal-simple, but there are ideal-simple semi-groups that are not bi-simple (and even ones for which all the $ {\mathcal D} $- classes consist of one element). The most important type of ideal-simple semi-groups ( $ 0 $- simple semi-groups) are the completely-simple semi-groups (completely $ 0 $- simple semi-groups, cf. Completely-simple semi-group). The most important examples of bi-simple but not completely-simple semi-groups are: the bicyclic semi-groups and the four-spiral semi-group $ \mathop{\rm Sp} _ {4} $( cf. Bicyclic semi-group; [11]). The latter, $ \mathop{\rm Sp} _ {4} $, is given by generators $ a , b , c , d $ and defining relations $ a ^ {2} = a $, $ b ^ {2} = b $, $ c ^ {2} = c $, $ d ^ {2} = d $, $ b a = a $, $ a b = b $, $ b c = b $, $ c b = c $, $ d c = c $, $ c d = d $, $ d a = d $. It is isomorphic to a Rees semi-group of matrix type over a bicyclic semi-group with generators $ u , v $, where $ u v = 1 $, with sandwich-matrix

$$ \left \| \begin{array}{cc} 1 & v \\ 1 & 1 \\ \end{array} \right \| . $$

In a sense, $ \mathop{\rm Sp} _ {4} $ is minimal among the bi-simple not completely-simple semi-groups generated by a finite number of idempotents, and quite often it arises as a sub-semi-group of those semi-groups.

Right simple semi-groups are also called semi-groups with right division, or semi-groups with right invertibility. The reason for this terminology is the following equivalent property of such semi-groups: For any elements $ a , b $ there is an $ x $ such that $ a x = b $. The right simple semi-groups containing idempotents are precisely the right groups (cf. Right group). An important example of a right simple semi-group without idempotents is given by the semi-groups $ T ( M , \delta , p , q ) $ of all transformations $ \phi $ of a set $ M $ such that: 1) the kernel of $ \phi $ is the equivalence relation $ \delta $ on $ M $; 2) the cardinality of the quotient set $ M / \delta $ is $ p $; 3) the set $ M \phi $ intersects each $ \delta $- class in at most one element; and 4) the set of $ \delta $- classes disjoint from $ M \phi $ has infinite cardinality $ q $, and $ q \leq p $. The semi-group $ T ( M , \delta , p , q ) $ is called a Teissier semi-group of type $ ( p , q ) $, and, if $ \delta $ is the equality relation, it is called a Baer–Levi semi-group of type $ ( p , q ) $( cf. [6], [7]). A Teissier semi-group is an example of a right simple semi-group without idempotents that does not necessarily satisfy the right cancellation law. Every right simple semi-group without idempotents can be imbedded in a suitable Teissier semi-group, while every such semi-group with the right cancellation law can be imbedded in a suitable Baer–Levi semi-group (in both cases one can take $ p = q $).

Various types of simple semi-groups often arise as "blocks" from which one can construct the semi-groups under consideration. For classical examples of simple semi-groups see Completely-simple semi-group; Brandt semi-group; Right group; for bi-simple inverse semi-groups (including structure theorems under certain restrictions on the semi-lattice of idempotents) see [1], [8], [9]. There are ideal-simple inverse semi-groups with an arbitrary number of $ {\mathcal D} $- classes. In the study of imbedding of semi-groups in simple semi-groups one usually either indicates conditions for the possibility of the corresponding imbedding, or establishes that any semi-group can be imbedded in a semi-group of the type considered. E.g., any semi-group can be imbedded in a bi-simple semi-group with an identity (cf. [1]), in a bi-simple semi-group generated by idempotents (cf. [10]), and in a semi-group that is simple relative to congruences (which may have some property given in advance: the presence or absence of a zero, completeness, having an empty Frattini sub-semi-group, etc., cf. [3][5]).

References

[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[3] L.A. Bokut', "Some embedding theorems for rings and semigroups" Sibirsk. Mat. Zh. , 4 : 3 (1963) pp. 500–518 (In Russian)
[4] E.G. Shutov, "Embeddings of semigroups in simple and complete semigroups" Mat. Sb. , 62 : 4 (1963) pp. 496–511 (In Russian)
[5] V.N. Klimov, "Embedding of semigroups in factorizable semigroups" Sib. Math. J. , 14 : 5 (1973) pp. 715–722 Sibirsk. Mat. Zh. , 14 : 5 (1973) pp. 1025–1036
[6] R. Baer, F. Levi, "Vollständige irreduzibele Systeme von Gruppenaxiomen" Sitzungsber. Heidelb. Akad. Wissenschaft. Math.-Nat. Kl. , 2 (1932) pp. 3–12
[7] M. Teissier, "Sur les demi-groupes admettant l'existence du quotient d'un cote" C.R. Acad. Sci. Paris , 236 : 11 (1953) pp. 1120–1122
[8] W.D. Munn, "Some recent results on the structure of inverse semigroups" K.W. Folley (ed.) , Semigroups , Acad. Press (1969) pp. 107–123
[9] J.M. Howie, "An introduction to semigroup theory" , Acad. Press (1976)
[10] F. Pastijn, "Embedding semigroups in semibands" Semigroup Forum , 14 : 3 (1977) pp. 247–263
[11] K. Byleen, J. Meakin, F. Pastijn, "The fundamental four-spiral semigroup" J. of Algebra , 54 (1978) pp. 6–26
How to Cite This Entry:
Simple semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple_semi-group&oldid=49585
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article