homomorphism


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Related to homomorphism: homeomorphism, Automorphism

homomorphism

[‚hä·mə′mȯr‚fiz·əm]
(botany)
Having perfect flowers consisting of only one type.
(mathematics)
A function between two algebraic systems of the same type which preserves the algebraic operations.

Homomorphism

 

a concept of mathematics and logic that first appeared in algebra but proved to be very important in understanding the structure and the area of possible applications of other branches of mathematics. The concept of homomorphism applies to a set of objects with prescribed operations (or relations). Thus, a homomorphism (homomorphic mapping) of a group G onto a group H is a mapping that associates to every element G∈G a definite element h∈H (the image of g) and satisfies the requirements that every element of H is the image of some element in G, and the image of the product (sum) of two elements in G is the product (sum) of their images. For example, the mapping that associates to an integer a the remainder when a is divided by a fixed positive integer m is a homomorphism of the group of integers (under addition) onto the group of residues modulo m. (The latter consists of m elements represented by the remainders 0, 1, . . . , m - 1.) The sum of two elements is represented by the sum of the corresponding remainders possibly diminished by m.

homomorphism

A map f between groups A and B is a homomorphism of A into B if f(a1 * a2) = f(a1) * f(a2) for all a1,a2 in A.

where the *s are the respective group operations.
References in periodicals archive ?
Let A, B be sup-algebras, and let f : A [right arrow] B be a homomorphism of sup-algebras.
Chandramouleeswaran, Homomorphism on Intuitionistic L-Fuzzy BF/BG-Subalgebras, Advances in Fuzzy Mathematics, 5(2010), No.
To distinguish a data model from a data type, we say that a data model has decidable homomorphism.
A mapping [phi] from V to a Jordan algebra (W, #) is a homomorphism if [phi] is linear and [phi](ab) = [phi](a) # [phi](b) for all a, b in V.
Structural homomorphism between the DSyntS and SSyntS of the same sentence
Then G/Ker z is Abelian, G' [subset] Ker z and z = [epsilon]y for a homomorphism y : G/G' [right arrow] &lt;b&gt; x <d>.
ii) The map [phi] : X [right arrow] D [disjunction] X is a homomorphism of I(M) onto (D] = {X [member of] I(M)/X [less than or equal to] D}
We called f is a homomorphism if and only if f (xy) = f (x)f (y) for all x,y [member of] X.
The breakthrough, called "privacy homomorphism," or "fully homomorphic encryption," makes possible the deep and unlimited analysis of encrypted information -- data that has been intentionally scrambled -- without sacrificing confidentiality.
psi],[gamma]](G, E), then [phi] has presentation of the form [phi](x) = q([tau](x)), where [tau] : G [right arrow] K x K is a homomorphism defined by the formula
The five here consider such topics as formulation of conjectures on p-adic zeta functions in noncummutative Iwasawa theory, and the norm residue of homomorphism of degree two.