Surjection

In some special cases, however, the number of surjections A#92;rightarrowB can be identified.

Let J#61;#92;cap#95;im#95;i be the (irredundant) primary decomposition of J. We associate to the pair (J,#92;omega) the element #92;textstyle#92;sum#95;i(m#95;i,#92;omega#95;i)#92;inG, where #92;omega#95;i is the equivalence class of surjections from L#47;m#95;iL#92;oplus(A#47;m#95;i)#123;n-1#125; to m#95;i#47;m#95;i² induced by #92;omega.

In Banach space theory, a mapping u#58;E#92;rightarrowF (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from E#47;#92;text#123;ker#125;(u) to F is an isometric isomorphism. Moreover, by the classical open mapping theorem, u is a surjection iff the associated mapping from E#47;#92;text#123;ker#125;(u) to F is an isomorphism.