Endomorphism
of an algebraic system
A mapping of an algebraic system into itself that is compatible with its structure. Namely, if
is an algebraic system with a signature consisting of a set
of operation symbols and a set
of predicate symbols, then an endomorphism
must satisfy the following conditions:
1) for any
-ary operation
and any sequence
of elements of
;
2) for any
-place predicate
and any sequence
of elements of
.
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 is a mapping
such that
,
for all elements
and
in
and
for all
. For an endomorphism
of a ring
with a unit 1, the requirements are that
be an endomorphism of the underlying commutative group and that, moreover,
and
for all
.
References
[a1] | P.M. Cohn, "Universal algebra" , Reidel (1981) |
Endomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Endomorphism&oldid=17409