Specific group properties of differential equations with deviating argument. Introduction to the linear theory

We describe an approach to differential equations with deviating argument which takes account of the structure of the homeomorphism group generated by the functions specifying the deviated arguments. The formalism presented here is particularly simple and instructive for the case in which the group is cyclic. In the class of linear differential equations with integer deviations, this formalism permits us to describe regular extensions of the class of ordinary differential equations.