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 : ${\displaystyle {\boldsymbol {A}}=(A,(f_{i})_{i\in I})}$ ${\displaystyle {\boldsymbol {B}}=(B,(g_{i})_{i\in I})}$ ${\displaystyle \sigma =(m_{i})_{i\in I},}$ ${\displaystyle i}$ ${\displaystyle m_{i}\in \mathbb {N} _{0}}$ ${\displaystyle f_{i}}$ ${\displaystyle g_{i}}$ ${\displaystyle \varphi \colon A\to B}$ ${\displaystyle {\boldsymbol {A}}}$ ${\displaystyle {\boldsymbol {B}},}$ ${\displaystyle i}$ ${\displaystyle a_{1},\ldots ,a_{m_{i}}\in A}$

${\displaystyle \varphi (f_{i}(a_{1},\ldots ,a_{m_{i}}))=g_{i}(\varphi (a_{1}),\ldots ,\varphi (a_{m_{i}}))}$.

Examples

A classic example of homomorphisms are homomorphisms between groups . Let two groups and one function be given ${\displaystyle (G,*)}$ ${\displaystyle (H,\star ).}$

${\displaystyle \phi \colon G\to H}$

is called group homomorphism if the following applies to all elements : ${\displaystyle g_{1},g_{2}\in G}$

${\displaystyle \phi (g_{1}*g_{2})=\phi (g_{1})\star \phi (g_{2}).}$

From this condition it follows immediately that

${\displaystyle \phi (e_{G})=e_{H}}$

for the neutral elements and then ${\displaystyle e_{G}\in G,e_{H}\in H}$

${\displaystyle \phi (g^{-1})=\phi (g)^{-1}}$

must apply to all as well as, by means of complete induction, that ${\displaystyle g\in G}$

${\displaystyle \phi (g_{1}*\ldots *g_{n})=\phi (g_{1})\star \ldots \star \phi (g_{n})}$

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 ${\displaystyle \phi \colon G\to H}$ ${\displaystyle \psi \colon H\to J}$

${\displaystyle (\psi \circ \phi )(g):=\psi (\phi (g))}$ for all ${\displaystyle g\in G}$

defined mapping a homomorphism. ${\displaystyle \psi \circ \phi \colon G\to J}$

Subgroups, image, archetype, core

If each one homomorphism, then subgroup also ${\displaystyle \phi \colon G\to H}$ ${\displaystyle U\subseteq G}$

${\displaystyle \phi (U):=\left\{\phi (g)\mid g\in U\right\},}$

called the image of under , a subset of . The subgroup becomes special ${\displaystyle U}$ ${\displaystyle \phi }$ ${\displaystyle H}$

${\displaystyle \operatorname {Bild} (\phi ):=\phi (G)\subseteq H}$

referred to as image of . Furthermore for each subgroup is also ${\displaystyle \phi }$ ${\displaystyle V\subseteq H}$

${\displaystyle \phi ^{-1}[V]:=\phi ^{-1}(V):=\left\{g\in G\mid \phi (g)\in V\right\},}$

called the archetype of under , a subgroup of . The archetype of the trivial group, i.e. i. the subgroup ${\displaystyle V}$ ${\displaystyle \phi }$ ${\displaystyle G}$

${\displaystyle \operatorname {Kern} (\phi ):=\phi ^{-1}(e_{H}):=\phi ^{-1}[\{e_{H}\}]\subseteq G,}$

is called the core of . It is even a normal divider . ${\displaystyle \phi }$

Isomorphisms

If is a bijective homomorphism, then is also a homomorphism. In this case it is said that and are isomorphisms. ${\displaystyle \phi \colon G\to H}$ ${\displaystyle \phi ^{-1}\colon H\to G}$ ${\displaystyle \phi }$ ${\displaystyle \phi ^{-1}}$

Homomorphism theorem

If is a homomorphism then induces an isomorphism ${\displaystyle \phi \colon G\to H}$ ${\displaystyle \phi }$

${\displaystyle G/\operatorname {Kern} (\phi )\cong \operatorname {Bild} (\phi )}$

the quotient group on . ${\displaystyle G/\operatorname {Kern} (\phi )}$ ${\displaystyle \operatorname {Bild} (\phi )}$

Homomorphisms of Relational Structures

Even outside of algebra, structure-preserving 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 : ${\displaystyle {\boldsymbol {A}}=(A,(R_{i}))}$ ${\displaystyle {\boldsymbol {B}}=(B,(S_{i}))}$ ${\displaystyle (n_{i}),}$ ${\displaystyle n_{i}\in \mathbb {N} }$ ${\displaystyle i}$ ${\displaystyle R_{i}}$ ${\displaystyle S_{i}}$ ${\displaystyle \varphi \colon A\to B}$ ${\displaystyle {\boldsymbol {A}}}$ ${\displaystyle {\boldsymbol {B}},}$ ${\displaystyle i}$ ${\displaystyle a_{1},\ldots ,a_{n_{i}}\in A}$

${\displaystyle (a_{1},\ldots ,a_{n_{i}})\in R_{i}\Rightarrow (\varphi (a_{1}),\ldots ,\varphi (a_{n_{i}}))\in S_{i}.}$

Notation:

${\displaystyle \varphi \colon {\boldsymbol {A}}\to {\boldsymbol {B}}.}$

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. ${\displaystyle f\colon A^{n}\to A}$ ${\displaystyle f\subset A^{n+1}}$

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

${\displaystyle (a_{1},\ldots ,a_{n_{i}})\in R_{i}\Leftrightarrow (\varphi (a_{1}),\ldots ,\varphi (a_{n_{i}}))\in S_{i}}$,

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. Springer-Verlag, New York 2002, ISBN 0-387-95385-X .
• Nathan Jacobson: Basic algebra. I. 2nd edition. WH Freeman and Company, New York 1985, ISBN 0-7167-1480-9 .
• Thomas W. Hungerford: Algebra. (= Graduate Texts in Mathematics. 73). Springer-Verlag, New York / Berlin 1980, ISBN 0-387-90518-9 . (Reprint of 1974 edition)
• Garrett Birkhoff : Lattice Theory . 3. Edition. AMS, Providence (RI) 1973, ISBN 0-8218-1025-1 , pp. 134-136 .
• Marcel Erné: Introduction to Order Theory . Bibliographisches Institut, Mannheim / Vienna / Zurich 1982, ISBN 3-411-01638-8 , p. 112-113 .
• Helmuth Gericke : Theory of Associations . Bibliographisches Institut, Mannheim 1963, p. 55-62, 147 .
• George Grätzer: Universal Algebra . 2nd updated edition. Springer, New York 2008, ISBN 978-0-387-77486-2 , pp. 223-224 , doi : 10.1007 / 978-0-387-77487-9 (first edition: 1979).
• Gunther Schmidt, Thomas Ströhlein: Relations and graphs . Springer, Berlin / Heidelberg / New York 1989, ISBN 3-540-50304-8 , pp. 144-153 .
• 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 3-540-03561-3 , pp. 27-30 .
• Heinrich Werner: Introduction to general algebra . Bibliographisches Institut, Mannheim 1978, ISBN 3-411-00120-8 , p. 48, 19 .

References and comments

1. ↑ Every -digit operation is a special -digit homogeneous relation (function). ${\displaystyle m}$ ${\displaystyle m+1}$
2. ↑ 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 ${\displaystyle f_{i}}$ ${\displaystyle R_{i}}$

   ${\displaystyle f_{i}(a_{1},\ldots ,a_{m_{i}})=a\Leftrightarrow (a_{1},\ldots ,a_{m_{i}},a)\in R_{i}}$,

   and also for . ${\displaystyle (B,(g_{i}))}$
3. ↑ 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 . ${\displaystyle \phi ^{-1}}$ ${\displaystyle \phi ^{-1}}$ ${\displaystyle [\;]}$
4. ↑ 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. Springer-Verlag, Berlin / New York 1966 (introduction to paragraph 10)) ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle {\mathfrak {N}}}$ ${\displaystyle a<b}$ ${\displaystyle ab=c}$ ${\displaystyle a}$ ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle {\bar {a}}=\varphi a}$ ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle a<b}$ ${\displaystyle {\bar {a}}<{\bar {b}},}$ ${\displaystyle <}$ ${\displaystyle \varphi }$ ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle {\mathfrak {N}}.}$
5. ↑ Some authors ( Wilhelm Klingenberg : Lineare Algebra und Geometrie. Springer, Berlin / Heidelberg 1984, ISBN 3-540-13427-1 , p. 7 .; Garrett Birkhoff: Lattice Theory. 1973, p. 134.) also call a homomorphism only briefly “Morphism”, while others (Fritz Reinhardt, Heinrich Sonder: dtv-Atlas Mathematik. Volume 1: Fundamentals, Algebra and Geometry. 9th edition. Deutscher Taschenbuchverlag, Munich 1991, ISBN 3-423-03007-0 , p. 36–37.) Call any structurally compatible mapping “morphism” and only designate a homomorphism of algebraic structures as “homomorphism”.
6. ↑ Philipp Rothmaler: Introduction to the model theory. Spektrum Akademischer Verlag 1995, ISBN 3-86025-461-8 , Section 1.3 Homomorphisms. P. 20.
7. ^ Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to mathematical logic. 3rd, completely revised u. exp. Edition. Bibliographisches Institut, Mannheim 1992, ISBN 3-411-15603-1 , p. 225.
8. ↑ Every continuous group homomorphism between Lie groups is smooth.

Web links

Wiktionary: Homomorphism  - explanations of meanings, word origins, synonyms, translations