As homomorphism (composed of ancient Greek ὁμός ( homos ), same 'or' similar ', and μορφή ( morphé ), form ', . Should not be confused with homeomorphism ) are in mathematics drawings indicates that receive a (often algebraic ) mathematical structure and thus are compatible. A homomorphism maps the elements of the a lot from so in the other quantity that behave their pictures there in terms of structure as well as to their archetypes behave in the initial amount.
- 2.1 Definition
- 2.2 Examples
- 2.3 generalizations
Homomorphisms of algebraic structures
There are two algebraic structures and of the same type so that for each the arity of the fundamental operations and inscribed. A picture is exactly then a homomorphism of into if for all and for all:
A classic example of homomorphisms are homomorphisms between groups. Given two groups A and function
Is called homomorphism if for all elements:
From this condition follows immediately that
For the neutral elements and then
Must apply to all, and, by induction, that
Is true for any finite number of factors.
In this example, the definitions of the homomorphisms of various algebraic structures are based:
- Group homomorphism
- Ring homomorphism
- Homomorphism of vector (linear mapping )
- Homomorphism of associative algebras
- Lie algebra homomorphism
We formulate the following are some basic properties of homomorphisms of groups that apply similarly to the homomorphisms of other algebraic structures.
Composition of homomorphisms: When and homomorphisms, then so is by
Defined mapping is a homomorphism.
Subgroups image core, if a homomorphism is, then, for each sub-group and
A subgroup of. The subgroup
Is called the image of. Furthermore, for every sub-group and
A subgroup of. The subgroup
Is called the core of, it is even a normal subgroup.
Isomorphisms: If a bijective homomorphism, then is also a homomorphism. One says in this case that and are isomorphisms.
Homomorphism theorem: If a homomorphism, then induces an isomorphism
The quotient group.
Homomorphisms of relational structures
Even outside the algebra structure preserving mappings are often called homomorphisms. Most of these uses of the term homomorphism, including those listed above algebraic structures, can be subsumed under the following definition.
There were two and relational structures of the same type so that for each the arity of the relations and inscribed. An illustration is then called a homomorphic mapping, a homomorphism or a homomorphism of into if it has the following compatibility property for each and for all:
Since each function can be described as a relation, can be any algebraic structure conceived as a relational structure and the special algebraic definition is therefore included in this definition.
If one has in the above definition, an injective homomorphism even the equivalence
This is called a strong homomorphism.
- Homomorphisms of algebraic structures (these are always strong homomorphisms )
- Homomorphisms in the incidence geometry, for example homomorphism of projective spaces
- Homomorphism between models
Even images that are compatible with structures that have unendlichstellige operations homomorphism are called:
- A complete Verbandshomomorphismus is compatible with any (even infinite) unions and intersections
In some areas of mathematics, the notion of homomorphism implies that the compatibility still includes additional structures:
- A homomorphism of topological groups is a continuous group homomorphism
- A Lie group homomorphism is a smooth homomorphism between Lie groups