# homomorphism

Also found in: Dictionary, Thesaurus, Medical, Acronyms, Wikipedia.
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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 ?
In order to achieve multiplicative homomorphism, the decryption structure of the product of [C.sub.1] and [C.sub.2] need to has the same structure as the expected decryption structure x([m.sub.1][m.sub.2]) + [e.sup.x].
Then [omega] is an algebra homomorphism, because, by the definition of [omega], [omega](s + t) = [omega](s) + [omega](t), [omega](st) = [omega](s)[omega](t),[omega]([lambda]t) = [lambda][omega](t) for every s, t [member of] T and [lambda] [member of] K.
Let [sigma]: M [right arrow] M be a homomorphism; in view of , a sequence {[d.sub.n]} of linear mappings on A is called a [sigma]-higher derivation if it satifies
Equivalently, there exist elements ([phi] [member of] [A.sup.*] (called Frobenius homomorphism), and [a.sub.i] [cross product] [b.sub.i] [member of] A [cross product] A such that [phi][alpha] = [phi], [alpha]([a.sub.i]) [cross product] [alpha]([b.sub.i]) = [a.sub.i] [cross product] [b.sub.i] and [phi]([xa.sub.i])[b.sub.i] = [a.sub.i][phi]([b.sub.i]x) = x, for any x [member of] A.
Tardif, "Graph homomorphisms: structure and symmetry," in Graph symmetry, vol.
From here onwards by SV-Hom, we will mean the set-valued homomorphism. For strong set-valued homomorphism, we will use SSV-Hom.
An important property of the above encryption is the addition homomorphism in the plaintext domain.
is the extension of the induced homomorphism f# : [intersection] [right arrow] [pi] of fundamental groups.
Let f : R [right arrow] S be a surjective ring homomorphism. If to is a 2-absorbing primary fuzzy ideal of R which is constant on Ker f, then f([mu]) is a 2-absorbing primary fuzzy ideal of S.
It is obvious that each n-homomorphism is an n-Jordan homomorphism, but in general the converse is false.
Note that the semigroup homomorphism [phi]: [S.sub.m,p] [right arrow] [S.sub.dm,dp] mapping a to [a.sup.d] is injective.

Site: Follow: Share:
Open / Close