# 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.

## 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 ?
n] is a homomorphism can be used to garner information about certain polynomials.
Also [mu] is order preserving, and hence a homomorphism of pomonoids.
To see that f is a homomorphism, take s, s' [member of] S and let u = { p, q) [member of] S be such that s = su = us and s' = s'u = us'.
A homomorphism is a map f : G [right arrow] H satisfying f (x x y) = f (x) * f (y) for all x,y [member of] G.
The graph homomorphisms from G to H correspond to the looped vertices of [G, H] in accordance with G(G, H) [congruent to] G(1 x G, H) [congruent to] G(1, [G, H]).
The set of all homomorphisms of E into U has the structure of the dual module of the left U-module E, and is denoted by [E.
It can be shown that kernels of sup-algebra homomorphisms are sup-algebra congruences.
They cover the definition and first properties of homology and co-homology modules, formally smooth homomorphisms, the structure of complete noetherian local rings, complete intersections, Popescu's theorem of regular homomorphisms, and the localization of formal smoothness.
We shall exploit the connection between Jordan algebras and the Wishart distribution, and characterize the Wishartness of a quadratic form Q(Y) in terms of Jordan algebra homomorphisms or more precisely, Jordan algebra representations.
The first stability problem concerning group homomorphisms was raised by Ulam (29) in 1940 and solved in the next year by Hyers (12).
Other topics include quotients and homomorphisms of relational systems, power automorphisms and induced automorphisms in finite groups, the relation between weak entwining structures and weak coring, sheaves over Boolean spaces, and cotorsion pairs of complexes.

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