# 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:

### 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}$

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

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}$

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

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}$

${\ displaystyle G / \ operatorname {core} (\ phi) \ cong \ operatorname {image} (\ phi)}$

the quotient group on . ${\ displaystyle G / \ operatorname {core} (\ phi)}$${\ displaystyle \ operatorname {image} (\ 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 {\ varvec {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:

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

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 .

## 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:

The term is also generalized for heterogeneous algebras, see Heterogeneous Algebra: Homomorphisms .

## 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 .

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 ${\ displaystyle ab = c}$${\ displaystyle a}$${\ displaystyle {\ mathfrak {M}}}$${\ displaystyle {\ bar {a}} = \ varphi a}$${\ displaystyle {\ mathfrak {M}}}$${\ displaystyle a ${\ displaystyle {\ bar {a}} <{\ bar {b}},}$${\ 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.