Homomorphism

A morphism in a category of algebraic systems (cf. Algebraic system). It is a mapping of an algebraic system $ \mathbf A $ that preserves the basic operations and the basic relations. More exactly, let $ \mathbf A = \langle A, \{ o _ {i} : i \in I \} , \{ r _ {j} : j \in J \}\rangle $ be an algebraic system with basic operations $ o _ {i} $, $ i \in I $, and with basic relations $ r _ {j} $, $ j \in J $. A homomorphism from $ \mathbf A $ into a system $ \mathbf A ^ \prime = \langle A ^ \prime , \{ o _ {i} ^ \prime : i \in I \} , \{ r _ {j} ^ \prime : j \in J \} \rangle $ of the same type is a mapping $ \phi : A \rightarrow A ^ \prime $ that satisfies the following two conditions:

$$ \tag{1 } \phi ( o _ {i} ( a _ {1} \dots a _ {n _ {i} } )) = \ o _ {i} ^ \prime ( \phi ( a _ {1} ) \dots \phi ( a _ {n _ {i} } )), $$

$$ \tag{2 } ( a _ {1} \dots a _ {m} ) \in r _ {j} \Rightarrow ( \phi ( a _ {1} ) \dots \phi ( a _ {m} )) \in r _ {j} ^ \prime , $$

for all elements $ a _ {1} , a _ {2} \dots $ from $ A $ and all $ i \in I $, $ j \in J $.

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

Suppose that each element $ i $ from $ I $ is brought into correspondence with some $ n _ {i} $- ary function symbol $ F _ {i} $, while each element $ j $ from $ J $ is brought into correspondence with an $ m _ {j} $- place predicate symbol $ P _ {j} $, and suppose that in each system $ \mathbf A ^ \prime $ of the same type as $ \mathbf A $ the result of the $ i $- th basic operation $ o _ {i} ^ \prime $, applied to the elements $ x _ {1} \dots x _ {n _ {i} } $ from $ A ^ \prime $, is written as $ F _ {i} ( x _ {1} \dots x _ {n _ {i} } ) $, while $ ( x _ {1} \dots x _ {m _ {j} } ) \in r _ {j} ^ \prime $ is denoted by $ P _ {j} ( x _ {1} \dots x _ {m _ {j} } ) $. Conditions (1), (2) are then simplified and take the form

$$ \phi ( F _ {i} ( a _ {1} \dots a _ {n _ {i} } )) = \ F _ {i} ( \phi ( a _ {1} ) \dots \phi ( a _ {n _ {i} } )), $$

$$ P _ {j} ( a _ {1} \dots a _ {m _ {j} } ) \Rightarrow P _ {j} ( \phi ( a _ {1} ) \dots \phi ( a _ {m _ {j} } )). $$

A homomorphism $ \phi : \mathbf A \rightarrow \mathbf A ^ \prime $ is called strong if for any elements $ a _ {1} ^ \prime \dots a _ {m _ {j} } ^ \prime $ from $ A ^ \prime $ and for any predicate symbol $ P _ {j} $, $ j \in J $, the condition $ P _ {j} ( a _ {1} ^ \prime \dots a _ {m _ {j} } ^ \prime ) $ implies that there exist elements $ a _ {1} \dots a _ {m _ {j} } $ in $ A $ such that $ a _ {1} ^ \prime = \phi ( a _ {1} ) \dots a _ {m _ {j} } ^ \prime = \phi ( a _ {m _ {j} } ) $, and such that the relation $ P _ {j} ( a _ {1} \dots a _ {m _ {j} } ) $ 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 $ \phi $ is a homomorphism of an algebraic system $ \mathbf A $ into an algebraic system $ \mathbf A ^ \prime $ and $ \theta $ is the kernel congruence of $ \phi $, then the mapping $ \psi : ( A/ \theta ) \rightarrow A ^ \prime $ defined by the formula $ \psi ( a/ \theta ) = \phi ( a) $ is a homomorphism of the quotient system $ \mathbf A / \theta $ into $ \mathbf A ^ \prime $. If, in addition, $ \phi $ is a strong homomorphism, then $ \psi $ 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).

References

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

Comments

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

$$ \phi ( g _ {1} g _ {2} ) = \ \phi ( g _ {1} ) \phi ( g _ {2} ) ,\ \ \phi ( g ^ {-} 1 ) = ( \phi ( g ) ) ^ {-} 1 ,\ \ \phi ( e _ {G} ) = e _ {H} . $$

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.