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Monoids and Homomorphisms

by gpt-5.2-codex
Learning objectives
  • Monoids and Homomorphisms
Prerequisites
  • /mathematics/objects/prealgebra/curricula/magmas-semigroups.md
  • /mathematics/objects/prealgebra/terms/index.md

Monoids add an identity element, and homomorphisms describe structure-preserving maps between monoids.

Monoid

A monoid is a semigroup with an identity element e such that e * a = a and a * e = a for all a in the set.

Example: The natural numbers with addition form a monoid, with identity 0.

Homomorphism

Given monoids (M, *) and (N, o), a homomorphism f : M -> N preserves the operation: f(a * b) = f(a) o f(b) for all a, b in M.

Homomorphisms let us compare algebraic structures while respecting their operations.

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Authors
Date created
Requires
  • Mathematics objects prealgebra curricula magmas semigroups.md
  • Mathematics objects prealgebra terms index.md
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Cite

@misc{gpt-5.2-codex2025-monoids-homomorphisms,
  author    = {gpt-5.2-codex},
  title     = {Monoids and Homomorphisms},
  year      = {2025},
  url       = {https://emsenn.net/library/math/domains/algebra/texts/monoids-homomorphisms/},
  publisher = {emsenn.net},
  license   = {CC BY-SA 4.0}
}