homomorphism

/,hɔmə'mɔ:fizm/
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Thân thiện
homomorphism

A homomorphism preserves the structure between two algebraic sets.

Definition
  1. Noun:
    • A similarity of form or structure between different entities: In mathematics, particularly in abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as groups, rings, or vector spaces). It is a function that respects the operations defined on the structures.
Usage
  • A homomorphism is a fundamental concept for comparing algebraic structures. It describes how one structure can be mapped onto another in a way that the essential operations (like addition or multiplication) are maintained. The concept is central to understanding isomorphisms, kernels, and quotients.
Examples
  • Noun:
    • The function f(x) = e^x is a group homomorphism from the additive group of real numbers to the multiplicative group of positive real numbers.
    • Studying the homomorphism between two rings reveals their underlying similarities.
Advanced Usage
  • "Kernel of a homomorphism": The set of elements in the source structure that map to the identity element in the target structure. It measures how much the homomorphism fails to be one-to-one.
    • The kernel of the homomorphism is a normal subgroup.
  • "Natural homomorphism": The canonical map from a group to its quotient group.
    • The natural homomorphism sends each element to its coset.
Variants and Related Words
  • Homomorphic (adj): Pertaining to or characterized by homomorphism.
    • The two structures are homomorphic.
  • Homomorphy (n): Another term for homomorphism.
  • Endomorphism (n): A homomorphism from a mathematical structure to itself.
  • Isomorphism (n): A bijective homomorphism (a perfect, reversible structure-preserving map).
Synonyms
  • Structure-preserving map: A direct description of its function.
  • Morphism: A more general term used in category theory that includes homomorphisms.
Related Phrases
  • Group homomorphism: A homomorphism between groups.
  • Ring homomorphism: A homomorphism between rings.
  • Module homomorphism: A homomorphism between modules.
homomorphism

A homomorphism preserves the structure between two algebraic sets.

Noun
  1. similarity of form

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