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.

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)