# 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 ?
Recall that pV denotes the natural continuous homomorphism from [[bar.
Definition 2: Partial Homomorphic Encryption (PHE) is either an additive homomorphism that supports only additive operations, or multiplicative homomorphism that supports only multiplicative operations.
Let A, B be sup-algebras, and let f : A [right arrow] B be a homomorphism of sup-algebras.
Based on such kind of homomorphism property, we can verify the validity of secret shares without revealing them.
Let f be a homomorphism from BF-algebras X onto Y and A be an ILFC-p-ideal of X with Sup-Inf property.
A clear motivation for studying data types by promoting homomorphism is to ease the development and maintenance of business applications because data types support better business data organization, error checking, and built-in operator utilizations [8].
10: If S and T are inverse semigroups and f: S [right arrow] T is homomorphism and T is an inverse sub semigroup of T such that [f.
Thus f is an R[G] homomorphism (REN & Shum, 2004) as required.
A model is an isomorphism or homomorphism representation of an object in the form of a selected objective set transferring object characteristics adequately to a representation.
It was then discovered (Wong and Masaro (22) and Masaro and Wong (12)) that these conditions reflected the fact that the Wishartness of Y'WY depended on whether or not a certain linear transformation was a Jordan algebra homomorphism.
2] is a homomorphism between graded rings, then we do not necessarily get a corresponding morphism Proj([D.
He reviews linear algebras, then describes the group and its subsets, including homomorphism of two groups and the proper symmetric group of a regular polyhedron, the theory of linear representations of groups, the three-dimensional rotation group, permutation groups, Lie groups and Lie algebras, unitary groups, real orthogonal groups and symplectic groups.

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