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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771001.png" /> associating to each [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771002.png" /> a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771003.png" /> (cf. [[Congruence (in algebra)|Congruence (in algebra)]]) and having the following properties: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771004.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771006.png" /> (0 denotes the equality relation), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771007.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771008.png" /> is a congruence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710011.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710012.png" />. If 1) and 3) are satisfied, then 2) is equivalent to
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<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/r/r077/r077100/r07710013.png" /></td> </tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
for every congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710015.png" />. A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710016.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710018.png" />-semi-simple if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710019.png" />. The class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710020.png" />-semi-simple semi-groups contains the one-element semi-group and is closed relative to isomorphism and subdirect products. Conversely, each class of semi-groups having these properties is the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710021.png" />-semi-simple semi-groups for some radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710024.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710026.png" />-radical. In contrast to rings, in semi-groups the radical is not determined by the corresponding radical class. If in the definition of a radical the discussion is limited to congruences defined by ideals, then another concept of a radical arises, where the corresponding function chooses an [[Ideal|ideal]] in each semi-group.
+
A function  $  \rho $
 +
associating to each [[Semi-group|semi-group]]  $  S $
 +
a congruence  $  \rho ( S) $(
 +
cf. [[Congruence (in algebra)|Congruence (in algebra)]]) and having the following properties: 1) if $  S $
 +
is isomorphic to  $  T $
 +
and  $  \rho ( S) = 0 $(
 +
0 denotes the equality relation), then  $  \rho ( T) = 0 $;
 +
2) if  $  \theta $
 +
is a congruence on  $  S $
 +
and  $  \rho ( S / \theta ) = 0 $,  
 +
then $  \rho ( S) \leq  \theta $;
 +
and 3)  $  \rho ( S / \rho ( S) ) = 0 $.  
 +
If 1) and 3) are satisfied, then 2) is equivalent to
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710027.png" /> is a class of semi-groups that is closed relative to isomorphisms and that contains the one-element semi-group, then the function that associates to each semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710028.png" /> the intersection of all congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710030.png" /> turns out to be a radical, called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710031.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710032.png" /> coincides with the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710033.png" />-semi-simple semi-groups if and only if it is closed relative to subdirect products. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710034.png" /> is the largest quotient semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710035.png" /> that lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710036.png" /> (see [[Replica|Replica]]).
+
$$
 +
\sup \{ \rho ( S) , \theta \} / \theta  \leq  \rho ( S / \theta )
 +
$$
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710037.png" /> be the class of semi-groups admitting a faithful irreducible representation (cf. [[Representation of a semi-group|Representation of a semi-group]]). Then
+
for every congruence  $  \theta $
 +
on  $  S $.  
 +
A semi-group  $  S $
 +
is called  $  \rho $-
 +
semi-simple if  $  \rho ( S) = 0 $.  
 +
The class of  $  \rho $-
 +
semi-simple semi-groups contains the one-element semi-group and is closed relative to isomorphism and subdirect products. Conversely, each class of semi-groups having these properties is the class of $  \rho $-
 +
semi-simple semi-groups for some radical  $  \rho $.
 +
If  $  \rho ( S) = S \times S $,
 +
then  $  S $
 +
is called  $  \rho $-
 +
radical. In contrast to rings, in semi-groups the radical is not determined by the corresponding radical class. If in the definition of a radical the discussion is limited to congruences defined by ideals, then another concept of a radical arises, where the corresponding function chooses an [[Ideal|ideal]] in each semi-group.
  
<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/r/r077/r077100/r07710038.png" /></td> </tr></table>
+
If  $  \mathfrak K $
 +
is a class of semi-groups that is closed relative to isomorphisms and that contains the one-element semi-group, then the function that associates to each semi-group  $  S $
 +
the intersection of all congruences  $  \theta $
 +
such that  $  S / \theta \in \mathfrak K $
 +
turns out to be a radical, called  $  \rho _ {\mathfrak K }  $.  
 +
The class  $  \mathfrak K $
 +
coincides with the class of  $  \rho _ {\mathfrak K }  $-
 +
semi-simple semi-groups if and only if it is closed relative to subdirect products. In this case  $  S / \rho _ {\mathfrak K }  ( S) $
 +
is the largest quotient semi-group of  $  S $
 +
that lies in  $  \mathfrak K $(
 +
see [[Replica|Replica]]).
  
<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/r/r077/r077100/r07710039.png" /></td> </tr></table>
+
Example. Let  $  \mathfrak K $
 +
be the class of semi-groups admitting a faithful irreducible representation (cf. [[Representation of a semi-group|Representation of a semi-group]]). Then
 +
 
 +
$$
 +
\rho _ {\mathfrak K }  ( s) =
 +
$$
 +
 
 +
$$
 +
= \
 +
\{ ( a , b ) : a , b \in S , ( a , b )
 +
\in \mu ( as ) \cap \mu ( b s )  \textrm{ for  all  }  s \in S \cup \emptyset \} ,
 +
$$
  
 
where
 
where
  
<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/r/r077/r077100/r07710040.png" /></td> </tr></table>
+
$$
 +
\mu ( a)  = \{ {( x , y ) } : {
 +
x , y \in S , a  ^ {m} x = a  ^ {n} y  \textrm{ for  some  } \
 +
m , n \geq  0 } \}
 +
.
 +
$$
  
 
Radicals defined on a given class of semi-groups that is closed relative to homomorphic images have been studied.
 
Radicals defined on a given class of semi-groups that is closed relative to homomorphic images have been studied.
  
Related to each radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710041.png" /> is the class of left polygons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710042.png" /> (cf. [[Polygon (over a monoid)|Polygon (over a monoid)]]). Namely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710043.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710044.png" />-polygon, then a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710046.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710048.png" />-annihilating if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710049.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710050.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710051.png" />. The least upper bound of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710052.png" />-annihilating congruences turns out to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710053.png" />-annihilating congruence, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710054.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710055.png" />, by definition, consists of all left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710056.png" />-polygons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710059.png" /> runs through the class of all semi-groups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710060.png" /> is a congruence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710061.png" />, then a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710062.png" />-polygon lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710063.png" /> if and only if it lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710064.png" /> when considered as a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710065.png" />-polygon. Conversely, if one is given a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710066.png" /> of left polygons with these properties and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710067.png" /> is the class of all left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710068.png" />-polygons in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710069.png" />, then the function
+
Related to each radical $  \rho $
 +
is the class of left polygons $  \Sigma ( \rho ) $(
 +
cf. [[Polygon (over a monoid)|Polygon (over a monoid)]]). Namely, if $  A $
 +
is a left $  S $-
 +
polygon, then a congruence $  \theta $
 +
on $  S $
 +
is called $  A $-
 +
annihilating if $  ( \lambda , \mu ) \in \theta $
 +
implies $  \lambda a = \mu a $
 +
for all $  a \in A $.  
 +
The least upper bound of all $  A $-
 +
annihilating congruences turns out to be an $  A $-
 +
annihilating congruence, and is denoted by $  \mathop{\rm Ann}  A $.  
 +
The class $  \Sigma ( \rho ) $,  
 +
by definition, consists of all left $  S $-
 +
polygons $  A $
 +
such that $  \rho ( S /  \mathop{\rm Ann}  A ) = 0 $,  
 +
where $  S $
 +
runs through the class of all semi-groups. If $  \theta $
 +
is a congruence on $  S $,  
 +
then a left $  ( S / \theta ) $-
 +
polygon lies in $  \Sigma ( \rho ) $
 +
if and only if it lies in $  \Sigma ( \rho ) $
 +
when considered as a left $  S $-
 +
polygon. Conversely, if one is given a class $  \Sigma $
 +
of left polygons with these properties and if $  \Sigma ( s) $
 +
is the class of all left $  S $-
 +
polygons in $  \Sigma $,  
 +
then the function
  
<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/r/r077/r077100/r07710070.png" /></td> </tr></table>
+
$$
 +
\rho ( S)  = \
 +
\left \{
  
 
is a radical.
 
is a radical.

Revision as of 08:09, 6 June 2020


A function $ \rho $ associating to each semi-group $ S $ a congruence $ \rho ( S) $( cf. Congruence (in algebra)) and having the following properties: 1) if $ S $ is isomorphic to $ T $ and $ \rho ( S) = 0 $( 0 denotes the equality relation), then $ \rho ( T) = 0 $; 2) if $ \theta $ is a congruence on $ S $ and $ \rho ( S / \theta ) = 0 $, then $ \rho ( S) \leq \theta $; and 3) $ \rho ( S / \rho ( S) ) = 0 $. If 1) and 3) are satisfied, then 2) is equivalent to

$$ \sup \{ \rho ( S) , \theta \} / \theta \leq \rho ( S / \theta ) $$

for every congruence $ \theta $ on $ S $. A semi-group $ S $ is called $ \rho $- semi-simple if $ \rho ( S) = 0 $. The class of $ \rho $- semi-simple semi-groups contains the one-element semi-group and is closed relative to isomorphism and subdirect products. Conversely, each class of semi-groups having these properties is the class of $ \rho $- semi-simple semi-groups for some radical $ \rho $. If $ \rho ( S) = S \times S $, then $ S $ is called $ \rho $- radical. In contrast to rings, in semi-groups the radical is not determined by the corresponding radical class. If in the definition of a radical the discussion is limited to congruences defined by ideals, then another concept of a radical arises, where the corresponding function chooses an ideal in each semi-group.

If $ \mathfrak K $ is a class of semi-groups that is closed relative to isomorphisms and that contains the one-element semi-group, then the function that associates to each semi-group $ S $ the intersection of all congruences $ \theta $ such that $ S / \theta \in \mathfrak K $ turns out to be a radical, called $ \rho _ {\mathfrak K } $. The class $ \mathfrak K $ coincides with the class of $ \rho _ {\mathfrak K } $- semi-simple semi-groups if and only if it is closed relative to subdirect products. In this case $ S / \rho _ {\mathfrak K } ( S) $ is the largest quotient semi-group of $ S $ that lies in $ \mathfrak K $( see Replica).

Example. Let $ \mathfrak K $ be the class of semi-groups admitting a faithful irreducible representation (cf. Representation of a semi-group). Then

$$ \rho _ {\mathfrak K } ( s) = $$

$$ = \ \{ ( a , b ) : a , b \in S , ( a , b ) \in \mu ( as ) \cap \mu ( b s ) \textrm{ for all } s \in S \cup \emptyset \} , $$

where

$$ \mu ( a) = \{ {( x , y ) } : { x , y \in S , a ^ {m} x = a ^ {n} y \textrm{ for some } \ m , n \geq 0 } \} . $$

Radicals defined on a given class of semi-groups that is closed relative to homomorphic images have been studied.

Related to each radical $ \rho $ is the class of left polygons $ \Sigma ( \rho ) $( cf. Polygon (over a monoid)). Namely, if $ A $ is a left $ S $- polygon, then a congruence $ \theta $ on $ S $ is called $ A $- annihilating if $ ( \lambda , \mu ) \in \theta $ implies $ \lambda a = \mu a $ for all $ a \in A $. The least upper bound of all $ A $- annihilating congruences turns out to be an $ A $- annihilating congruence, and is denoted by $ \mathop{\rm Ann} A $. The class $ \Sigma ( \rho ) $, by definition, consists of all left $ S $- polygons $ A $ such that $ \rho ( S / \mathop{\rm Ann} A ) = 0 $, where $ S $ runs through the class of all semi-groups. If $ \theta $ is a congruence on $ S $, then a left $ ( S / \theta ) $- polygon lies in $ \Sigma ( \rho ) $ if and only if it lies in $ \Sigma ( \rho ) $ when considered as a left $ S $- polygon. Conversely, if one is given a class $ \Sigma $ of left polygons with these properties and if $ \Sigma ( s) $ is the class of all left $ S $- polygons in $ \Sigma $, then the function

$$ \rho ( S) = \ \left \{

is a radical.

References

[1] A.H. Clifford, G.B. Preston, "The algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967)
[2] L.A. Skornyakov, "Radicals of -rings" , Selected problems in algebra and logic , Novosibirsk (1973) pp. 283–299 (In Russian)
[3] A.H. Clifford, "Radicals in semigroups" Semigroup Forum , 1 : 2 (1970) pp. 103–127
[4] E.N. Roiz, B.M. Schein, "Radicals of semigroups" Semigroup Forum , 16 : 3 (1978) pp. 299–344
How to Cite This Entry:
Radical in a class of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radical_in_a_class_of_semi-groups&oldid=17321
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article