homomorphism

Meanings

• noun

  A homomorphism is a structure-preserving map between two algebraic structures. In other words, it is a function between two algebraic objects that preserves their internal algebraic operations.

  - "A homomorphism between two groups preserves the group operation."
  - "The function f(x) = x^2 is a homomorphism from the additive group of real numbers to the multiplicative group of positive real numbers."

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Variants

List of all variants of homomorphism that leads to same result
homomorphic , homomorphism , homomorphisms

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Etymology

origin and the way in which meanings have changed throughout history.

From the Greek words ὁμός (homós), meaning 'same', and μορφή (morphē), meaning 'form' or 'shape'.

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Trivia

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1. The term 'homomorphism' was first introduced by the Norwegian mathematician Marius Sophus Lie in 1888.

2. Homomorphisms have applications in various fields, including cryptography, where they are used to design secure algorithms.

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Related Concepts

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1. Isomorphism: An isomorphism is a special type of homomorphism that is both bijective (one-to-one and onto) and invertible. In other words, an isomorphism establishes a one-to-one correspondence between two algebraic structures.

2. Endomorphism: An endomorphism is a homomorphism from an algebraic structure to itself. In other words, it is a function that maps elements of the algebraic structure back to itself.

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Culture

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Homomorphisms have wide applications in various fields such as mathematics, computer science, physics, and engineering. They play a crucial role in abstract algebra, group theory, ring theory, and category theory.

How to Memorize "homomorphism"

1. visualize
   

   - Imagine homomorphism as a function that maps elements from one algebraic structure to another while preserving their internal operations.
   - Visualize the homomorphism function as a bridge connecting two algebraic structures, ensuring that the operations on the left side have the same result on the right side.

2. associate
   

   - Associate homomorphism with the concept of preserving structure or form.
   - Remember that homomorphisms map elements between algebraic structures while preserving their internal operations.

3. mnemonics
   

   - Use the mnemonic 'Homomorphisms Halt Hurdles' to remember that homomorphisms preserve algebraic operations: H for homomorphism, H for halt, and H for hurdles representing the algebraic operations.

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