surjection

Etymology

From French surjection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique. Ultimately borrowed from Latin superiectiō (“a throwing over or on; (fig.) an exaggeration, a hyperbole”).

noun

1. (set theory) A function for which every element of the codomain is mapped to by some element of the domain; (formally) Any function f:X→Y for which for every y∈Y, there is at least one x∈X such that f(x)=y.
   In some special cases, however, the number of surjections A→B can be identified. 1992, Rowan Garnier, John Taylor, Discrete Mathematics for New Technology, Institute of Physics Publishing, page 220
   Let J=∩ᵢm_i be the (irredundant) primary decomposition of J. We associate to the pair (J,ω) the element ∑ᵢ(m_i,ωᵢ)∈G, where ωᵢ is the equivalence class of surjections from L/m_iL⊕(A/m_i)ⁿ⁻¹ to m_i/m_i² induced by ω. 1999, M. Pavaman Murthy, “A survey of obstruction theory for projective modules of top rank”, in 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
   In Banach space theory, a mapping u:E→F (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from E/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/ker(u) to F is an isomorphism. 2003, Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press, page 43