homomorphism (plural homomorphisms)

  1. (algebra) A structure-preserving map between two algebraic structures of the same type, such as group, ring, or vector spaces.
    A field homomorphism is a map from one field to another one which is additive, multiplicative, zero-preserving, and unit-preserving.
    • 1954, Kuo-Tsai Chen, Iterated Integrals and Exponential Homomorphisms, Proceedings of the London Mathematical Society, Reprinted in 2001, Philippe Tondeur (editor), Collected Papers of K.-T. Chen, Birkhäuser, page 54 ↗,
      This motivates a generalization, and exponential homomorphisms are now defined, in an algebraic fashion, from certain free products to formal power series rings with non-commutative indeterminates.
    • 1997, Glen E. Bredon, Sheaf Theory, 2nd Edition, Springer, page 8 ↗,
      A homomorphism of presheaves h : A \rightarrow B is a collection of homomorphisms h_U : A(U) \rightarrow B(U) commuting with restrictions.
    • 2003, Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, page 17 ↗,
      Definition 1.15. Let G and H be matrix Lie groups. A map \Phi from G to H is called a Lie group homomorphism if (1) \Phi is a group homomorphism and (2) \Phi is continuous.
  2. (biology) A similar appearance of two unrelated organisms or structures.
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