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''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|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.

====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR></table>