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.

Noun

1. similarity of form