Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. Biology Similarity of external form or appearance but not of structure or origin.
 n. Zoology A resemblance in form between the immature and adult stages of an animal.
 n. Mathematics A transformation of one set into another that preserves in the second set the operations between the members of the first set.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A structurepreserving map between two algebraic structures, such as groups, rings, or vector spaces.
 n. A similar appearance of two unrelated organisms or structures
from the GNU version of the Collaborative International Dictionary of English
 n. Same as homomorphy.
 n. The possession, in one species of plants, of only one kind of flowers;  opposed to heteromorphism, dimorphism, and trimorphism.
 n. The possession of but one kind of larvæ or young, as in most insects.
from The Century Dictionary and Cyclopedia
 n. Mimicry or imitation of one thing by another; adaptive or analogical resemblance, without true homological or morphological similarity; superficial likeness without structural affinity or relationship. Also homomorphy.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. similarity of form
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

However F also maps functions to homomorphisms, mapping f to its unique extension as a homomorphism, while U maps homomorphisms to functions, namely the homomorphism itself as a function.

We can define an equivalence relation on the domain of a product C, and then take a structure D whose elements are the equivalence classes; the predicate symbols are interpreted in D so as to make the natural map from dom (C) to dom (D) a homomorphism.

If it is onto, then the inverse map from dom (B) to dom (A) is also a homomorphism, and both the embedding and its inverse are said to be isomorphisms.

Then T has a model A with the property that for every model B of T there is a unique homomorphism from

A homomorphism from structure A to structure B is a function f from dom (A) to dom (B) with the property that for every atomic formula

Next, two properties of that system are established: The rational numbers can be embedded into it, in a way that respects the order and the arithmetic operations defined on those numbers (a corresponding field homomorphism exists); and the new system is continuous, or linecomplete, with respect to its order.

Each of these functions is from a generator set to an algebra and therefore has a unique extension to a homomorphism.

Likewise the homomorphism from G to G is an identity function.

Moreover, this correspondence is functorial: any Boolean homomorphism is sent to a continuous map of topological spaces, and, conversely, any continuous map between the spaces is sent to a Boolean homomorphism.

The key idea is that compositionality requires the existence of a homomorphism between the expressions of a language and the meanings of those expressions.
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