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''of an algebraic system''
 
''of an algebraic system''
  
A mapping of an algebraic system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356001.png" /> into itself that is compatible with its structure. Namely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356002.png" /> is an algebraic system with a signature consisting of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356003.png" /> of operation symbols and a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356004.png" /> of predicate symbols, then an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356005.png" /> must satisfy the following conditions:
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A mapping of an algebraic system $  A $
 +
into itself that is compatible with its structure. Namely, if $  A $
 +
is an algebraic system with a signature consisting of a set $  \Omega _ {F} $
 +
of operation symbols and a set $  \Omega _ {P} $
 +
of predicate symbols, then an endomorphism $  \phi : A \rightarrow A $
 +
must satisfy the following conditions:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356006.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356007.png" />-ary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356008.png" /> and any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e0356009.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560010.png" />;
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1) $  \phi ( a _ {1} \dots a _ {n} \omega ) = \phi ( a _ {1} ) \dots \phi ( a _ {n} ) \omega $
 +
for any $  n $-
 +
ary operation $  \omega \in \Omega _ {F} $
 +
and any sequence $  a _ {1} \dots a _ {n} $
 +
of elements of $  A $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560011.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560012.png" />-place predicate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560013.png" /> and any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560014.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560015.png" />.
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2) $  P ( a _ {1} \dots a _ {n} ) \Rightarrow P ( \phi ( a _ {1} ) \dots \phi ( a _ {n} )) $
 +
for any $  n $-
 +
place predicate $  P \in \Omega _ {P} $
 +
and any sequence $  a _ {1} \dots a _ {n} $
 +
of elements of $  A $.
  
 
The concept of an endomorphism is a special case of that of a [[Homomorphism|homomorphism]] of two algebraic systems. The endomorphisms of any algebraic system form a monoid under the operation of composition of mappings, whose unit element is the identity mapping of the underlying set of the system (cf. [[Endomorphism semi-group|Endomorphism semi-group]]).
 
The concept of an endomorphism is a special case of that of a [[Homomorphism|homomorphism]] of two algebraic systems. The endomorphisms of any algebraic system form a monoid under the operation of composition of mappings, whose unit element is the identity mapping of the underlying set of the system (cf. [[Endomorphism semi-group|Endomorphism semi-group]]).
  
 
An endomorphism having an inverse is called an [[Automorphism|automorphism]] of the algebraic system.
 
An endomorphism having an inverse is called an [[Automorphism|automorphism]] of the algebraic system.
 
 
  
 
====Comments====
 
====Comments====
Thus, by way of one of the simplest examples, an endomorphism of an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560016.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560019.png" /> for all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560023.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560024.png" />. For an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560025.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560026.png" /> with a unit 1, the requirements are that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560027.png" /> be an endomorphism of the underlying commutative group and that, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035600/e03560030.png" />.
+
Thus, by way of one of the simplest examples, an endomorphism of an Abelian group $  A $
 +
is a mapping $  \phi : A \rightarrow A $
 +
such that $  \phi ( 0) = 0 $,  
 +
$  \phi ( a + b ) = \phi ( a) + \phi ( b) $
 +
for all elements $  a $
 +
and $  b $
 +
in $  A $
 +
and $  \phi (- a) = \phi ( a) $
 +
for all $  a \in A $.  
 +
For an endomorphism $  \phi $
 +
of a ring $  R $
 +
with a unit 1, the requirements are that $  \phi $
 +
be an endomorphism of the underlying commutative group and that, moreover, $  \phi ( 1) = 1 $
 +
and $  \phi ( a b ) = \phi ( a) \phi ( b) $
 +
for all $  a , b \in R $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR></table>

Latest revision as of 19:37, 5 June 2020


of an algebraic system

A mapping of an algebraic system $ A $ into itself that is compatible with its structure. Namely, if $ A $ is an algebraic system with a signature consisting of a set $ \Omega _ {F} $ of operation symbols and a set $ \Omega _ {P} $ of predicate symbols, then an endomorphism $ \phi : A \rightarrow A $ must satisfy the following conditions:

1) $ \phi ( a _ {1} \dots a _ {n} \omega ) = \phi ( a _ {1} ) \dots \phi ( a _ {n} ) \omega $ for any $ n $- ary operation $ \omega \in \Omega _ {F} $ and any sequence $ a _ {1} \dots a _ {n} $ of elements of $ A $;

2) $ P ( a _ {1} \dots a _ {n} ) \Rightarrow P ( \phi ( a _ {1} ) \dots \phi ( a _ {n} )) $ for any $ n $- place predicate $ P \in \Omega _ {P} $ and any sequence $ a _ {1} \dots a _ {n} $ of elements of $ A $.

The concept of an endomorphism is a special case of that of a homomorphism of two algebraic systems. The endomorphisms of any algebraic system form a monoid under the operation of composition of mappings, whose unit element is the identity mapping of the underlying set of the system (cf. Endomorphism semi-group).

An endomorphism having an inverse is called an automorphism of the algebraic system.

Comments

Thus, by way of one of the simplest examples, an endomorphism of an Abelian group $ A $ is a mapping $ \phi : A \rightarrow A $ such that $ \phi ( 0) = 0 $, $ \phi ( a + b ) = \phi ( a) + \phi ( b) $ for all elements $ a $ and $ b $ in $ A $ and $ \phi (- a) = \phi ( a) $ for all $ a \in A $. For an endomorphism $ \phi $ of a ring $ R $ with a unit 1, the requirements are that $ \phi $ be an endomorphism of the underlying commutative group and that, moreover, $ \phi ( 1) = 1 $ and $ \phi ( a b ) = \phi ( a) \phi ( b) $ for all $ a , b \in R $.

References

[a1] P.M. Cohn, "Universal algebra" , Reidel (1981)
How to Cite This Entry:
Endomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Endomorphism&oldid=46820
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article