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''of an algebraic system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813201.png" /> in a given class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813202.png" /> of algebraic systems of the same signature''
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An [[Algebraic system|algebraic system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813203.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813204.png" /> possessing the following properties: 1) there is a surjective homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813205.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813206.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813207.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813208.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r0813209.png" /> is a homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132012.png" /> for some homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132013.png" /> from the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132014.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132015.png" />. The replica of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132016.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132017.png" /> (if it exists) is uniquely defined up to an isomorphism. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132018.png" /> is called replica full if it contains a replica for any algebraic system of the same signature. A class of algebraic systems of a fixed signature is replica full if and only if it contains a one-element system and is closed with respect to taking subsystems and direct products. The axiomatizable replica-full classes (and only these) are quasi-varieties (cf. [[Quasi-variety|Quasi-variety]]).
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''of an algebraic system  $  A $
 +
in a given class  $  \mathfrak K $
 +
of algebraic systems of the same signature''
 +
 
 +
An [[Algebraic system|algebraic system]] $  K _ {0} $
 +
from $  \mathfrak K $
 +
possessing the following properties: 1) there is a surjective homomorphism $  \phi _ {0} $
 +
from $  A $
 +
onto $  K _ {0} $;  
 +
2) if $  K \in \mathfrak K $
 +
and if $  \phi $
 +
is a homomorphism from $  A $
 +
to $  K $,  
 +
then $  \phi = \phi _ {0} \psi $
 +
for some homomorphism $  \psi $
 +
from the system $  K _ {0} $
 +
to $  K $.  
 +
The replica of the system $  A $
 +
in the class $  \mathfrak K $(
 +
if it exists) is uniquely defined up to an isomorphism. The class $  \mathfrak K $
 +
is called replica full if it contains a replica for any algebraic system of the same signature. A class of algebraic systems of a fixed signature is replica full if and only if it contains a one-element system and is closed with respect to taking subsystems and direct products. The axiomatizable replica-full classes (and only these) are quasi-varieties (cf. [[Quasi-variety|Quasi-variety]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The concept of a replica is closely related to that of a universal problem (cf. [[Universal problems|Universal problems]]).
 
The concept of a replica is closely related to that of a universal problem (cf. [[Universal problems|Universal problems]]).
  
A second notion of replica occurs in the theory of algebraic Lie algebras, the Lie algebras of algebraic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132019.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132021.png" /> is a finite-dimensional vector space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132022.png" /> be the smallest algebraic Lie subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132023.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132024.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132025.png" /> are called the replicas of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132026.png" />. One has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132027.png" /> is nilpotent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132028.png" /> for all replicas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081320/r08132030.png" />.
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A second notion of replica occurs in the theory of algebraic Lie algebras, the Lie algebras of algebraic subgroups of $  \mathop{\rm GL} ( V) $.  
 +
Let $  X \in  \mathop{\rm End} ( V) $,  
 +
where $  V $
 +
is a finite-dimensional vector space, and let $  \mathfrak g ( X) $
 +
be the smallest algebraic Lie subalgebra of $  \mathfrak g \mathfrak l ( V) $
 +
that contains $  X $.  
 +
The elements of $  \mathfrak g ( X) $
 +
are called the replicas of $  X $.  
 +
One has that $  X $
 +
is nilpotent if and only if $  \mathop{\rm Tr} ( XX  ^  \prime  ) = 0 $
 +
for all replicas $  X  ^  \prime  $
 +
of $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Chevalley,  "Théorie des groupes de Lie" , '''2''' , Hermann  (1951)  pp. Chapt. II, §14</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Chevalley,  "Théorie des groupes de Lie" , '''2''' , Hermann  (1951)  pp. Chapt. II, §14</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


of an algebraic system $ A $ in a given class $ \mathfrak K $ of algebraic systems of the same signature

An algebraic system $ K _ {0} $ from $ \mathfrak K $ possessing the following properties: 1) there is a surjective homomorphism $ \phi _ {0} $ from $ A $ onto $ K _ {0} $; 2) if $ K \in \mathfrak K $ and if $ \phi $ is a homomorphism from $ A $ to $ K $, then $ \phi = \phi _ {0} \psi $ for some homomorphism $ \psi $ from the system $ K _ {0} $ to $ K $. The replica of the system $ A $ in the class $ \mathfrak K $( if it exists) is uniquely defined up to an isomorphism. The class $ \mathfrak K $ is called replica full if it contains a replica for any algebraic system of the same signature. A class of algebraic systems of a fixed signature is replica full if and only if it contains a one-element system and is closed with respect to taking subsystems and direct products. The axiomatizable replica-full classes (and only these) are quasi-varieties (cf. Quasi-variety).

References

[1] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)

Comments

The concept of a replica is closely related to that of a universal problem (cf. Universal problems).

A second notion of replica occurs in the theory of algebraic Lie algebras, the Lie algebras of algebraic subgroups of $ \mathop{\rm GL} ( V) $. Let $ X \in \mathop{\rm End} ( V) $, where $ V $ is a finite-dimensional vector space, and let $ \mathfrak g ( X) $ be the smallest algebraic Lie subalgebra of $ \mathfrak g \mathfrak l ( V) $ that contains $ X $. The elements of $ \mathfrak g ( X) $ are called the replicas of $ X $. One has that $ X $ is nilpotent if and only if $ \mathop{\rm Tr} ( XX ^ \prime ) = 0 $ for all replicas $ X ^ \prime $ of $ X $.

References

[a1] C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951) pp. Chapt. II, §14
How to Cite This Entry:
Replica. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Replica&oldid=48514
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article