Existence theorems for commutative diagrams

V. Rechnoi

Lobachevskii journal of mathematics, Tome 17 (2005), pp. 213-230

Résumé

Given a relation $f\subset A\times B$, there exist two symmetric relations (see [1], Chapter 2) $f^{-1}f\subset A^2$, $ff^{-1}\subset B^2$. These relations make it possible to formalize definitions and proofs of existence theorems. For example, the equation $h=gf$, where $h$ and $g$ (or $h$ and $f$) are given maps, admits a solution $f$ ($g$, respectively) if and only if $hh^{-1}\subset gg^{-1}$ $(h^{-1}h\supset f^{-1}f)$. 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 [2] and consider it as an application in category of vector spaces and linear maps.

Détail

Keywords: existence of absent maps in commutative diagrams, map iterations.

Bibliographie

[1] N. Bourbaki, Éléments de mathématique, Fasc. XVII, $1^{\underline{\text{\`{e}re}}}$ partie, livre I, Théorie des ensembles, Hermann, Paris, 1960

[2] M. Rahula, “About Theory of Maps”, Webs and Quasigroups, University of Kalinin, 1981, 136–153 (Russian)

[3] M. Rahula, Vector Fields and Symmetries, Tartu University press, Tartu, 2004 (Russian) | MR | Zbl