A theory for generalized morphisms and beyond

TL;DR

This paper develops a unified theory of generalized morphisms encompassing various mathematical structures, explores their properties, and applies the framework to classical problems like solvability and algebraic extensions.

Contribution

It introduces a comprehensive framework for generalized morphisms, extending classical concepts and establishing new Galois correspondences and isomorphism theorems.

Findings

Inverse of bijective generalized morphisms is also a generalized morphism

Galois correspondences for automorphisms and endomorphisms are established

Generalizations of algebraic and differential concepts are presented

Abstract

Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring homomorphisms, module homomorphisms, group homomorphisms, and covariant functors between categories can be characterized in terms of the generalized morphisms. We show that the inverse of any bijective generalized morphism is also a generalized morphism (of the same kind), and hence a generalized isomorphism can be defined as a bijective generalized morphism. Galois correspondences are established and studied, not only for the Galois groups of the generalized automorphisms, but also for the "Galois monoids" of the generalized endomorphisms. Ways to construct the generalized morphisms and the generalized isomorphisms are studied. New interpretations on solvability of polynomials and solvability of homogeneous linear differential equations are introduced, and these ideas are roughly generalized for "general" equation solving in terms of our theory for the generalized morphisms. Some more results are presented. For example, we generalize the algebraic notions of transcendental elements over a field and purely transcendental field extensions, we obtain an isomorphism theorem that generalizes the first isomorphism theorems (for groups, rings, and modules), and we show that a part of our theory is closely related to dynamical systems.