# Endomorphism

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.

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 $a , b \in R$.