  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.
