# Existence Theorems for Commutative Diagrams

## (*Lobachevskii Journal of Mathematics, Volume XVII*)

Given a relation *f ⊂ A × B*, there exist two symmetric relations (see [Bourbaki], Chapter 2) *f*^{-1} f ⊂ A^{2}, f f^{-1} ⊂ B^{2}. These relations make it possible to formalize definitions and proofs of existence theorems. For example, the equation *h = g f*, where *h* and *g* (or *h* and *f*) are given maps, admits a solution *f* (*g*, respectively.) if and only if *hh*^{-1}⊂ gg^{-1} (h^{-1}h ⊂ f^{-1}f). Well-known ,,homomorphism theorems'' get more general interpretation. Namely, any map can be represented up to bijection as a composition of surjection and injection, and any morphism of diagrams can be represented up to isomorphism as a composition of epimorphism and monomorphism. In this paper we further develop the scheme from [MR1] and consider it as an application in category of vector spaces and linear maps.