surjection

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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

Related terms

• injection
• bijection
• surject

Translations

• French: surjection
• German: Surjektion
• Italian: suriezione
• Portuguese: sobrejeção
• Russian: сюръе́кция
• Spanish: función sobreyectiva

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