# homomorphism

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Related to Homomorphisms: isomorphism

## 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 It might be outdated or ideologically biased.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 ?
To achieve additive homomorphism, the decryption structure with respect to adding [c.sub.1] and [c.sub.2] is required to keep the structure as x([m.sub.1]+[m.sub.2]) + [e.sup.+], where [e.sup.+] is the noise in the sum and x is an unknown variable.
As f, g, [pr.sub.A], [pr.sub.C] are all continuous algebra homomorphisms, [pr.sub.A]|D, [pr.sub.C]|D, and h|D are also continuous algebra homomorphisms.
Now in what follows, assume that [sigma], [tau] be two homomorphism on A.
Let f : [G.sub.1] [right arrow] [G.sub.2] be a homomorphism. Since [G.sub.2] is subgraph of [G.sub.1][bar.V][G.sub.2], then there exists a homomorphism r : [G.sub.1][bar.V][G.sub.2] [right arrow] [G.sub.2] with r(x) = x, for any vertex x of [G.sub.2] and so [G.sub.2] is a retract of [G.sub.1][bar.V][G.sub.2].
We set Q = [e.sub.i] [cross product] [e.sup.i] [member of] M [cross product] M, and define a homomorphism in [??]([M.sub.k])
where coin denotes the coincidence subgroup of a pair of homomorphisms. Here, [i.sup.[beta].sub.1] and [u.sub.1.sup.[beta]] are (co)restrictions of [i.sub.1] and [u.sub.1], respectively.
Let f : M [right arrow] N be a homomorphism of BCK-modules and [PSI] = (N; [[PSI].sup.+], [[PSI].sup.-]) a bipolar fuzzy set of N.
A 2-homomorphism is then just a homomorphism, in the usual sense.
Note that the semigroup homomorphism [phi]: [S.sub.m,p] [right arrow] [S.sub.dm,dp] mapping a to [a.sup.d] is injective.
Then there are homomorphisms [C.sub.l] [right arrow] G [right arrow] [K.sub.n], so it remains to use Lemma 1.2 and Corollary 2.2.

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