surjection
Pronunciation
  • IPA: /sɜː(ɹ).dʒɛk.ʃən/
Noun

surjection (plural surjections)

  1. (set theory) A function that is a many-to-one mapping; (formally) Any function f: X\rightarrow Y for which for every y \in Y, there is at least one x \in X such that f(x) = y.
    • 1992, Rowan Garnier, John Taylor, Discrete Mathematics for New Technology, IOP Publishing, page 220 ↗,
      In some special cases, however, the number of surjections A \rightarrow B can be identified.
    • 1999, M. Pavaman Murthy, A survey of obstruction theory for projective modules of top rank, Tsit-Yuen Lam, Andy R. Magid (editors), Algebra, K-theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday, American Mathematical Society, page 168 ↗,
      Let J = \cap_i m_i be the (irredundant) primary decomposition of J. We associate to the pair (J,\omega) the element \textstyle\sum_i (m_i, \omega_i) \in G, where \omega_i is the equivalence class of surjections from L/m_iL \oplus (A/m_i)^{n-1} to m_i/m_i^2 induced by \omega.
    • 2003, Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press, page 43 ↗,
      In Banach space theory, a mapping u: E \rightarrow F (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from E / \text{ker}(u) to F is an isometric isomorphism. Moreover, by the classical open mapping theorem, u is a surjection iff the associated mapping from E/ \text{ker}(u) to F is an isomorphism.
Synonyms
  • (function that is a many-to-one mapping) surjective function
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