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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
r0771001.png
 
$#A+1 = 67 n = 1
 
$#C+1 = 67 : ~/encyclopedia/old_files/data/R077/R.0707100 Radical in a class of semi\AAhgroups
 
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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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A function  $  \rho $
+
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.
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 )
 
$$
 
  
for every congruence  $  \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
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.
 
  
If  $  \mathfrak K $
+
<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>
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]]).
 
  
Example. Let  $  \mathfrak K $
+
<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>
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 $  \rho $
+
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
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 14:53, 7 June 2020

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

for every congruence on . A semi-group is called -semi-simple if . The class of -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 -semi-simple semi-groups for some radical . If , then is called -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 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 the intersection of all congruences such that turns out to be a radical, called . The class coincides with the class of -semi-simple semi-groups if and only if it is closed relative to subdirect products. In this case is the largest quotient semi-group of that lies in (see Replica).

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

where

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

Related to each radical is the class of left polygons (cf. Polygon (over a monoid)). Namely, if is a left -polygon, then a congruence on is called -annihilating if implies for all . The least upper bound of all -annihilating congruences turns out to be an -annihilating congruence, and is denoted by . The class , by definition, consists of all left -polygons such that , where runs through the class of all semi-groups. If is a congruence on , then a left -polygon lies in if and only if it lies in when considered as a left -polygon. Conversely, if one is given a class of left polygons with these properties and if is the class of all left -polygons in , then the function

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=48413
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article