  Based on a new generalization of Cauchy-Riemann system presented in this
paper, we introduce a class of quaternion-valued functions of a quaternionic
variable, which are called algebraic regular functions. The set of algebraic
regular functions is not only a real associative algebra, but also respect the
composition of functions. Using algebraic regular functions as transition maps,
we introduce a class of four-manifolds called the regular four-manifolds.
