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A morphism in a category of algebraic systems (cf. Algebraic system). It is a mapping of an algebraic system that preserves the basic operations and the basic relations. More exactly, let be an algebraic system with basic operations , , and with basic relations , . A homomorphism from into a system of the same type is a mapping that satisfies the following two conditions:


for all elements from and all , .

E.g., if is a group and is a normal subgroup of it, then by assigning to each element its coset one obtains a homomorphism from onto the quotient group .

Suppose that each element from is brought into correspondence with some -ary function symbol , while each element from is brought into correspondence with an -place predicate symbol , and suppose that in each system of the same type as the result of the -th basic operation , applied to the elements from , is written as , while is denoted by . Conditions (1), (2) are then simplified and take the form

A homomorphism is called strong if for any elements from and for any predicate symbol , , the condition implies that there exist elements in such that , and such that the relation holds.

In the case of algebras the concepts of a homomorphism and a strong homomorphism coincide. For models there exist homomorphisms that are not strong, and one-to-one homomorphisms that are not isomorphisms (cf. Isomorphism).

If is a homomorphism of an algebraic system into an algebraic system and is the kernel congruence of , then the mapping defined by the formula is a homomorphism of the quotient system into . If, in addition, is a strong homomorphism, then is an isomorphism. This is one of the most general formulations of the homomorphism theorem.

It should be noted that the name "homomorphism" is sometimes applied to morphisms in categories other than categories of algebraic systems (homomorphisms of graphs, sheaves, Lie groups).


[1] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)
[2] C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973)


For example, a homomorphism between two groups (cf. Group) is a mapping which commutes with the basic group-theoretic operations of multiplication, inversion and identity:

In this particular case it is well-known that the first of these conditions implies the other two, but in general the definition cannot be simplified in this way.

How to Cite This Entry:
Homomorphism. D.M. Smirnov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098