Information About

Homomorphism
  CATEGORIES ABOUT HOMOMORPHISM
   
functions and mappings
   
abstract algebra
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



In Abstract Algebra , a homomorphism is a structure-preserving Map between two Algebraic Structure s (such as Group s, Ring s, or Vector Space s). The word ''homomorphism'' comes from the Greek Language : ''homo'' meaning "same" and ''morphos'' meaning "shape".


INFORMAL DISCUSSION


Because abstract algebra studies Set s with Operation s that generate interesting structure or properties on the set, the most interesting Function s are those which preserve the operations. These functions are known as ''homomorphisms''.

For example, consider the Natural Number s with addition as the operation. A function which preserves addition should have this property: ''f''(''a'' + ''b'') = ''f''(''a'') + ''f''(''b''). Note that ''f''(''x'') = 3''x'' is a homomorphism, since ''f''(''a'' + ''b'') = 3(''a'' + ''b'') = 3''a'' + 3''b'' = ''f''(''a'') + ''f''(''b''). Note that this homomorphism maps the natural numbers back onto themselves.

  • ''f''(''b''), since addition is the operation in the first set and multiplication is the operation in the second. Given the laws of Exponent s, ''f''(''x'') = e''x'' satisfies this condition.


A particularly important property of homomorphisms is that if an Identity Element is present, it is always preserved, that is, mapped to the identity. Note in the first example ''f''(0) = 0, and 0 is the additive identity. In the second example, ''f''(0) = 1, since 0 is the additive identity, and 1 is the multiplicative identity.

If we are considering multiple operations on a set, then all operations must be preserved for a function to be a considered a homomorphism. Even though the set may be the same, the same function might be a homomorphism, say, in Group Theory (sets with a single operation) but not in Ring Theory (sets with two related operations), because it fails to preserve the additional operation that ring theory considers.


FORMAL DEFINITION


A homomorphism is a Map from one Algebraic Structure to another of the same type that preserves all the relevant structure; i.e. properties like Identity Element s, Inverse Element s, and Binary Operation s.

: N.B. Some authors use the word ''homomorphism'' in a larger context than that of algebra. Some take it to mean any kind of structure preserving map (such as Continuous Map s in Topology ), or even a more abstract kind of map—what we term a '' Morphism ''—used in Category Theory . This article only treats the algebraic context. For more general usage see the Morphism article.

For example; if one considers Set s with a single Binary Operation defined on them (an algebraic structure known as a Magma ), a homomorphism is a map \phi: X ightarrow Y such that
:\phi(u \cdot v) = \phi(u) \circ \phi(v)
where \cdot is the operation on X and \circ is the operation on Y.

Each type of algebraic structure has its own type of homomorphism. For specific definitions see:

The notion of a homomorphism can be given a formal definition in the context of Universal Algebra , a field which studies ideas common to all Algebraic Structure s. In this setting, a homomorphism \phi: A ightarrow B is a map between two algebraic structures of the same type such that
:\phi(f_A(x_1, \ldots, x_n)) = f_B(\phi(x_1), \ldots, \phi(x_n))\,
for each ''n''-ary operation f and for all x_i in A.


TYPES OF HOMOMORPHISMS


  • An Isomorphism is a Bijective homomorphism. Two objects are said to be isomorphic if there is an isomorphism between them. Isomorphic objects are completely indistinguishable as far as the structure in question is concerned.




  • A homomorphism from an object to itself is called an Endomorphism .


  • An endomorphism which is also an isomorphism is called an Automorphism .


The above terms are used in an analogous fashion in Category Theory , however, the definitions in Category Theory are more subtle; see the article on Morphism for more details.

Note that in the larger context of structure preserving maps, it is generally insufficient to define an isomorphism as a bijective morphism. One must also require that the inverse is a morphism of the same type. In the algebraic setting (at least within the context of Universal Algebra ) this extra condition is automatically satisfied.

:
Relationships between different kinds of homomorphisms.
H = set of Homomorphisms, M = set of Monomorphisms,
P = set of ePimorphisms, S = set of iSomorphisms,
N = set of eNdomorphisms, A = set of Automorphisms.
Notice that: M ∩ P = S, S ∩ N = A, P ∩ N = A,
M ∩ N \ A contains only infinite homomorphisms, and
P ∩ N \ A is empty.



KERNEL OF A HOMOMORPHISM

See Also: Kernel (algebra)



  • [''y'' = --- ''y'' . In that case the image of ''X'' in ''Y'' under the homomorphism ''f'' is necessarily Isomorphic to ''X''/~; this fact is one of the Isomorphism Theorem s. Note in some cases (e.g. Group s or Ring s), a single Equivalence Class ''K'' suffices to specify the structure of the quotient, so we write it ''X''/''K''. (''X''/''K'' is usually read as ''X'' Mod ''K''.) Also in these cases, it is ''K'', rather than ~, that is called the Kernel of ''f'' (cf. Normal Subgroup , Ideal ).



SEE ALSO



REFERENCE


A monograph available free online: