What is homomorphism in mathematics

Homomorphism

 

As Homomorphism (composed of ancient Greek ὁμός (homós) 'Equal to' or 'similar', and μορφή (morphé) ,Shape'; not to be confused with homeomorphism) in mathematics images are referred to that contain a (often algebraic) mathematical structure or are compatible with it. A homomorphism maps the elements from one set into the other set in such a way that their images there behave in terms of structure as their archetypes behave in the original set.

Homomorphisms of algebraic structures

definition

Be there and two algebraic structures of the same type so that for each the arity of the fundamental operations and designated.[1] An illustration is exactly then a Homomorphism of in if for each and for everyone applies:[2]

.

Examples

A classic example of homomorphisms are homomorphisms between groups. Let two groups be given and One function

is called group homomorphism if for all elements applies:

From this condition it follows immediately that

for the neutral elements and then

for all must apply as well as, by means of complete induction, that

holds for any finite number of factors.

The definitions of the homomorphisms of various algebraic structures are based on this example:

properties

In the following we formulate some basic properties of homomorphisms of groups, which also apply analogously to the homomorphisms of the other algebraic structures.

Composition of homomorphisms: If and Are homomorphisms, then that too is through

for all

defined figure a homomorphism.

Subgroups, image, archetype, core: If is a homomorphism then is for each subgroup also

called that picture of under , a subset of . The subgroup becomes special

as picture of designated. Furthermore is for each subgroup also

called that Archetype of under , a subset of . The archetype of the trivial group, i.e. the subgroup

is called core of designated. It is even a normal divider.

Isomorphisms: If is a bijective homomorphism, then is too a homomorphism. In this case it is said that and Are isomorphisms.[3]

Homomorphism theorem: If is a homomorphism then induced an isomorphism

the quotient group on .

Homomorphisms of Relational Structures

Even outside of algebra, structure-preserving mappings are often referred to as homomorphisms. Most of these uses of the term homomorphism, including the algebraic structures listed above, can be subsumed under the following definition.[4]

definition

Be there and two relational structures of the same type so that for each the arity of the relations and designated. An illustration is then called a homomorphic figure, one Homomorphism or a Homomorphism of in if you for each and for everyone the following Compatibility property owns[5]:

Notation:

Because every function as a relation can be described, every algebraic structure can be understood as a relational structure and the special algebraic definition is thus included in this definition.

In the above definition, one even has the equivalence for an injective homomorphism

,

that's how one speaks of one strong homomorphism.

Examples

  • Homomorphisms of algebraic structures (these are also always strong homomorphisms)
  • Order homomorphism
  • Graph homomorphism
  • Homomorphisms in incidence geometry, for example homomorphism of projective spaces
  • Homomorphism between models

Generalizations

Maps that are compatible with structures that have infinite operations are also called homomorphism:

In some sub-areas of mathematics, the term homomorphism implies that compatibility includes other additional structures:

See also

Remarks

  1. ↑ Each digit operation is a special one -digit homogeneous relation (function).
  2. ↑ This definition is compatible with the one given below when speaking of a function to the relation , which is given by the function graph, passes over, because then applies
    ,
    and also for .
  3. ↑ The archetype function , which operates on sets, and the inverse mapping that operates on elements are strictly speaking 2 different functions. If misunderstandings are to be feared, then in the first case the quantities are put in square brackets .
  4. ↑ A general definition was given in the classic textbook Modern Algebra: “If in two sets and certain relations (like or ) are defined and if each element of a picture element is assigned in such a way that all relations between elements of also apply to the picture elements (so that e.g. from follows when it comes to the relation acts), so is called a homomorphic figure or a Homomorphism of in "(Van der Waerden, B. L .: Algebra. Part One. Seventh edition. Heidelberg Pocket Books, Volume 12 Springer-Verlag, Berlin-New York 1966 (Introduction to Paragraph 10))
  5. ↑ Some authors (Wilhelm Klingenberg: Linear algebra and geometry. Springer, Berlin / Heidelberg 1984, ISBN 3-540-13427-1, p. 7 .; Garrett Birkhoff: Lattice Theory. 1973, p. 134.) only briefly mention a homomorphism Morphism, while others (Fritz Reinhardt, Heinrich Sonder: dtv atlas mathematics. Part 1: Fundamentals, algebra and geometry. 9th edition. Deutscher Taschenbuchverlag, Munich 1991, ISBN 3-423-03007-0, pp. 36–37.) Call every structure-compatible figure “morphism” and only call one homomorphism of algebraic structures “homomorphism”.
  6. ↑ Any continuous group homomorphism between Lie groups is smooth.
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Date of the last change: Jena, the: 09.06. 2020