homomorphism love

homomorphism

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.

• n. A structure-preserving 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.

• n. similarity of form

Etymologies

homo- +‎ morphism

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.

Algebra

• 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.

First-order Model Theory

• 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.

First-order Model Theory

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

First-order Model Theory

• 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

First-order Model Theory

• 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 line-complete, with respect to its order.

Dedekind's Contributions to the Foundations of Mathematics

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

Algebra

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

Algebra

• 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.

Category Theory

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

Compositionality