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A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370101.png" /> containing the given semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370102.png" /> as a sub-semi-group. One is usually concerned with extensions that are in some way related to the given semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370103.png" />. The most well-developed theory is that of ideal extensions (those semi-groups containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370104.png" /> as an ideal). To each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370105.png" /> of an ideal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370106.png" /> of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370107.png" /> are assigned its left and right translations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e0370109.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701012.png" />); let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701013.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701014.png" /> is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701015.png" /> into the translation hull <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701017.png" />, and is an isomorphism in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701018.png" /> is weakly reductive (see [[Translations of semi-groups|Translations of semi-groups]]). The semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701019.png" /> is called the type of the ideal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701020.png" />. Among the ideal extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701022.png" />, one can distinguish strong extensions, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701023.png" />, and pure extensions, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701024.png" />. Every ideal extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701025.png" /> is a pure extension of one of its strong extensions.
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An ideal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701027.png" /> is called dense (or essential) if every homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701028.png" /> that is injective on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701029.png" /> is an isomorphism. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701030.png" /> has a maximal dense ideal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701031.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701032.png" /> is weakly reductive. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701033.png" /> is unique up to an isomorphism and is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701034.png" />. Also, in this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701035.png" /> is called a densely-imbedded ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701036.png" />. The sub-semi-groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701037.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701038.png" />, and only these, are isomorphic to dense ideal extensions of a weakly reductive semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701039.png" />.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701040.png" /> is an ideal extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701041.png" /> and if the quotient semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701042.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701044.png" /> is called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701045.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701046.png" />. The following cases have been studied extensively: ideal extensions of completely-simple semi-groups, of a group by a completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701047.png" />-simple semi-group, of a commutative semi-group with cancellation by a group with added zero, etc. In general, the problem of describing all ideal extensions of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701049.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701050.png" /> is far from being solved.
+
A semi-group  $  S $
 +
containing the given semi-group  $  A $
 +
as a sub-semi-group. One is usually concerned with extensions that are in some way related to the given semi-group  $  A $.  
 +
The most well-developed theory is that of ideal extensions (those semi-groups containing  $  A $
 +
as an ideal). To each element  $  s $
 +
of an ideal extension $  S $
 +
of a semi-group $  A $
 +
are assigned its left and right translations  $  \lambda _ {s} $,
 +
$  \rho _ {s} $:  
 +
$  \lambda _ {s} x = sx $,  
 +
$  x \rho _ {s} = xs $(
 +
$  x \in A $);
 +
let  $  \tau = \tau _ {s} = ( \lambda _ {s} , \rho _ {s} ) $.  
 +
The mapping  $  \tau $
 +
is a homomorphism of  $  S $
 +
into the translation hull  $  T ( A) $
 +
of  $  A $,
 +
and is an isomorphism in the case when  $  A $
 +
is weakly reductive (see [[Translations of semi-groups|Translations of semi-groups]]). The semi-group $  \tau S $
 +
is called the type of the ideal extension  $  S $.  
 +
Among the ideal extensions $  S $
 +
of $  A $,
 +
one can distinguish strong extensions, for which  $  \tau S = \tau A $,
 +
and pure extensions, for which  $  \tau  ^ {-} 1 \tau A = A $.  
 +
Every ideal extension of  $  A $
 +
is a pure extension of one of its strong extensions.
  
Among other types of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701051.png" /> one can mention semi-groups that have a congruence with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701052.png" /> as one of its classes, and in particular the so-called Schreier extensions of a semi-group with identity [[#References|[1]]], which are analogues of Schreier extensions of groups. In studying the various forms of extensions of a semi-group (in particular, for inverse semi-groups), one uses cohomology of semi-groups.
+
An ideal extension  $  S $
 +
of $  A $
 +
is called dense (or essential) if every homomorphism of $  S $
 +
that is injective on  $  A $
 +
is an isomorphism. $  A $
 +
has a maximal dense ideal extension  $  D $
 +
if and only if  $  A $
 +
is weakly reductive. In this case,  $  D $
 +
is unique up to an isomorphism and is isomorphic to  $  T ( A) $.  
 +
Also, in this case,  $  A $
 +
is called a densely-imbedded ideal in  $  D $.  
 +
The sub-semi-groups of  $  T ( A) $
 +
containing  $  \tau A $,  
 +
and only these, are isomorphic to dense ideal extensions of a weakly reductive semi-group  $  A $.
  
Another broad area in the theory of extensions of semi-groups is concerned with various problems on the existence of extensions of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701053.png" /> that belong to a given class. Thus, any semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701054.png" /> can be imbedded in a complete semi-group, in a simple semi-group (relative to congruences), or in a bi-simple semi-group with zero and an identity (see [[Simple semi-group|Simple semi-group]]), and any finite or countable semi-group can be imbedded in a semi-group with two generators. Conditions are known under which a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037010/e03701055.png" /> can be imbedded in a semi-group without proper left ideals, in an inverse semi-group (cf. [[Inversion semi-group|Inversion semi-group]]), in a group (see [[Imbedding of semi-groups|Imbedding of semi-groups]]), etc.
+
If  $  S $
 +
is an ideal extension of  $  A $
 +
and if the quotient semi-group  $  S/A $
 +
is isomorphic to  $  Q $,
 +
then  $  S $
 +
is called an extension of  $  A $
 +
by  $  Q $.
 +
The following cases have been studied extensively: ideal extensions of completely-simple semi-groups, of a group by a completely  $  O $-
 +
simple semi-group, of a commutative semi-group with cancellation by a group with added zero, etc. In general, the problem of describing all ideal extensions of a semi-group  $  A $
 +
by  $  Q $
 +
is far from being solved.
 +
 
 +
Among other types of extensions of  $  A $
 +
one can mention semi-groups that have a congruence with  $  A $
 +
as one of its classes, and in particular the so-called Schreier extensions of a semi-group with identity [[#References|[1]]], which are analogues of Schreier extensions of groups. In studying the various forms of extensions of a semi-group (in particular, for inverse semi-groups), one uses cohomology of semi-groups.
 +
 
 +
Another broad area in the theory of extensions of semi-groups is concerned with various problems on the existence of extensions of a semi-group $  A $
 +
that belong to a given class. Thus, any semi-group $  A $
 +
can be imbedded in a complete semi-group, in a simple semi-group (relative to congruences), or in a bi-simple semi-group with zero and an identity (see [[Simple semi-group|Simple semi-group]]), and any finite or countable semi-group can be imbedded in a semi-group with two generators. Conditions are known under which a semi-group $  A $
 +
can be imbedded in a semi-group without proper left ideals, in an inverse semi-group (cf. [[Inversion semi-group|Inversion semi-group]]), in a group (see [[Imbedding of semi-groups|Imbedding of semi-groups]]), etc.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1''' , Amer. Math. Soc.  (1961)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Petrich,  "Introduction to semigroups" , C.E. Merrill  (1973)</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''' , Amer. Math. Soc.  (1961)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Petrich,  "Introduction to semigroups" , C.E. Merrill  (1973)</TD></TR></table>

Latest revision as of 19:38, 5 June 2020


A semi-group $ S $ containing the given semi-group $ A $ as a sub-semi-group. One is usually concerned with extensions that are in some way related to the given semi-group $ A $. The most well-developed theory is that of ideal extensions (those semi-groups containing $ A $ as an ideal). To each element $ s $ of an ideal extension $ S $ of a semi-group $ A $ are assigned its left and right translations $ \lambda _ {s} $, $ \rho _ {s} $: $ \lambda _ {s} x = sx $, $ x \rho _ {s} = xs $( $ x \in A $); let $ \tau = \tau _ {s} = ( \lambda _ {s} , \rho _ {s} ) $. The mapping $ \tau $ is a homomorphism of $ S $ into the translation hull $ T ( A) $ of $ A $, and is an isomorphism in the case when $ A $ is weakly reductive (see Translations of semi-groups). The semi-group $ \tau S $ is called the type of the ideal extension $ S $. Among the ideal extensions $ S $ of $ A $, one can distinguish strong extensions, for which $ \tau S = \tau A $, and pure extensions, for which $ \tau ^ {-} 1 \tau A = A $. Every ideal extension of $ A $ is a pure extension of one of its strong extensions.

An ideal extension $ S $ of $ A $ is called dense (or essential) if every homomorphism of $ S $ that is injective on $ A $ is an isomorphism. $ A $ has a maximal dense ideal extension $ D $ if and only if $ A $ is weakly reductive. In this case, $ D $ is unique up to an isomorphism and is isomorphic to $ T ( A) $. Also, in this case, $ A $ is called a densely-imbedded ideal in $ D $. The sub-semi-groups of $ T ( A) $ containing $ \tau A $, and only these, are isomorphic to dense ideal extensions of a weakly reductive semi-group $ A $.

If $ S $ is an ideal extension of $ A $ and if the quotient semi-group $ S/A $ is isomorphic to $ Q $, then $ S $ is called an extension of $ A $ by $ Q $. The following cases have been studied extensively: ideal extensions of completely-simple semi-groups, of a group by a completely $ O $- simple semi-group, of a commutative semi-group with cancellation by a group with added zero, etc. In general, the problem of describing all ideal extensions of a semi-group $ A $ by $ Q $ is far from being solved.

Among other types of extensions of $ A $ one can mention semi-groups that have a congruence with $ A $ as one of its classes, and in particular the so-called Schreier extensions of a semi-group with identity [1], which are analogues of Schreier extensions of groups. In studying the various forms of extensions of a semi-group (in particular, for inverse semi-groups), one uses cohomology of semi-groups.

Another broad area in the theory of extensions of semi-groups is concerned with various problems on the existence of extensions of a semi-group $ A $ that belong to a given class. Thus, any semi-group $ A $ can be imbedded in a complete semi-group, in a simple semi-group (relative to congruences), or in a bi-simple semi-group with zero and an identity (see Simple semi-group), and any finite or countable semi-group can be imbedded in a semi-group with two generators. Conditions are known under which a semi-group $ A $ can be imbedded in a semi-group without proper left ideals, in an inverse semi-group (cf. Inversion semi-group), in a group (see Imbedding of semi-groups), etc.

References

[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1 , Amer. Math. Soc. (1961)
[2] M. Petrich, "Introduction to semigroups" , C.E. Merrill (1973)
How to Cite This Entry:
Extension of a semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_semi-group&oldid=46881
This article was adapted from an original article by L.M. Gluskin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article