Homomorphism
As homomorphism (composed of ancient Greek ὁμός homos , same 'or' similar ', and ancient Greek μορφή morphé , shape'; not to be confused with homeomorphism ) are in the math figures indicated that one (often algebraic) mathematical structure obtained or so compatible are. 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
Let and be two algebraic structures of the same type so that for each the number denotes the (matching) arity of the fundamental operations and . A mapping is called homomorphism from to if the following applies for each and for all :
 .
Examples
A classic example of homomorphisms are homomorphisms between groups . Let two groups and one function be given
is called group homomorphism if the following applies to all elements :
From this condition it follows immediately that
for the neutral elements and then
must apply to all 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:
 Group homomorphism
 Ring homomorphism
 Body homomorphism
 Vector Space Homomorphism ( Linear Map )
 Evaluation homomorphism of termalgebra
 Module homomorphism
 Algebra Homomorphism
 Lie algebra homomorphism
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 mapping a homomorphism.
Subgroups, image, archetype, core
If each one homomorphism, then subgroup also
called the image of under , a subset of . The subgroup becomes special
referred to as image of . Furthermore for each subgroup is also
called the archetype of under , a subgroup of . The archetype of the trivial group, i.e. i. the subgroup
is called the core of . It is even a normal divider .
Isomorphisms
If is a bijective homomorphism, then is also a homomorphism. In this case it is said that and are isomorphisms.
Homomorphism theorem
If is a homomorphism then induces an isomorphism
the quotient group on .
Homomorphisms of Relational Structures
Even outside of algebra, structurepreserving maps 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.
definition
Let and two relational structures of the same type be denoted so that for each the arity of the relations and . A mapping is then called a homomorphic mapping , a homomorphism or a homomorphism from to if it has the following compatibility property for each and every one of them :
Notation:
Since every function can be described as a relation , 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
 ,
so one speaks of a 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:
 a complete association homomorphism is compatible with any (also infinite) associations and averages
In some areas of mathematics, the term homomorphism implies that compatibility also includes additional structures:
 a homomorphism of topological groups is a continuous group homomorphism
 a Lie group homomorphism is a smooth group homomorphism between Lie groups
The term is also generalized for heterogeneous algebras, see Heterogeneous Algebra: Homomorphisms .
See also
 Morphism ( category theory )
 Compatibility (mathematics)
 Epimorphism
 Monomorphism
 Isomorphism
 Embedding
 Endomorphism
 Automorphism
 Subquotient
literature
 Serge Lang: Algebra. (= Graduate Texts in Mathematics. 211). 3rd, revised. Edition. SpringerVerlag, New York 2002, ISBN 038795385X .
 Nathan Jacobson: Basic algebra. I. 2nd edition. WH Freeman and Company, New York 1985, ISBN 0716714809 .
 Thomas W. Hungerford: Algebra. (= Graduate Texts in Mathematics. 73). SpringerVerlag, New York / Berlin 1980, ISBN 0387905189 . (Reprint of 1974 edition)
 Garrett Birkhoff : Lattice Theory . 3. Edition. AMS, Providence (RI) 1973, ISBN 0821810251 , pp. 134136 .
 Marcel Erné: Introduction to Order Theory . Bibliographisches Institut, Mannheim / Vienna / Zurich 1982, ISBN 3411016388 , p. 112113 .
 Helmuth Gericke : Theory of Associations . Bibliographisches Institut, Mannheim 1963, p. 5562, 147 .
 George Grätzer: Universal Algebra . 2nd updated edition. Springer, New York 2008, ISBN 9780387774862 , pp. 223224 , doi : 10.1007 / 9780387774879 (first edition: 1979).
 Gunther Schmidt, Thomas Ströhlein: Relations and graphs . Springer, Berlin / Heidelberg / New York 1989, ISBN 3540503048 , pp. 144153 .
 Bartel Leendert van der Waerden : Algebra I (= Heidelberg Pocket Books . Volume 12 ). 8th edition. tape 1 : Modern algebra . Springer, Berlin / Göttingen / Heidelberg / New York 1971, ISBN 3540035613 , pp. 2730 .
 Heinrich Werner: Introduction to general algebra . Bibliographisches Institut, Mannheim 1978, ISBN 3411001208 , p. 48, 19 .
References and comments
 ↑ Every digit operation is a special digit homogeneous relation (function).

↑ This definition is compatible with the one given below if one passes from a function to the relation that is given by the function graph, because then applies
 ,
 ↑ The archetype function , which operates on sets, and the inverse mapping , which operates on elements, are strictly speaking two different functions. If misunderstandings are to be feared, the quantities in the first case are put in square brackets .
 ↑ A general definition was given in the classic textbook Modern Algebra : "If in two sets and certain relations (such as 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 ( see above that, for example, follows from when it is a question of the relation ), then a homomorphic mapping or a homomorphism of in "(BL van der Waerden: Algebra. (= Heidelberger Taschenbücher. Volume 12). Part I, Seventh Edition. SpringerVerlag, Berlin / New York 1966 (introduction to paragraph 10))
 ↑ Some authors ( Wilhelm Klingenberg : Lineare Algebra und Geometrie. Springer, Berlin / Heidelberg 1984, ISBN 3540134271 , p. 7 .; Garrett Birkhoff: Lattice Theory. 1973, p. 134.) also call a homomorphism only briefly “Morphism”, while others (Fritz Reinhardt, Heinrich Sonder: dtvAtlas Mathematik. Volume 1: Fundamentals, Algebra and Geometry. 9th edition. Deutscher Taschenbuchverlag, Munich 1991, ISBN 3423030070 , p. 36–37.) Call any structurally compatible mapping “morphism” and only designate a homomorphism of algebraic structures as “homomorphism”.
 ↑ Philipp Rothmaler: Introduction to the model theory. Spektrum Akademischer Verlag 1995, ISBN 3860254618 , Section 1.3 Homomorphisms. P. 20.
 ^ HeinzDieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to mathematical logic. 3rd, completely revised u. exp. Edition. Bibliographisches Institut, Mannheim 1992, ISBN 3411156031 , p. 225.
 ↑ Every continuous group homomorphism between Lie groups is smooth.