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 ⊂ A2, f f-1 ⊂ B2. 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-1h ⊂ f-1f). 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.