The purpose of this correspondence is to survey the literature concerning the iterative properties of multivalued switching functions. These properties are important for the synthesis of switching circuits by cascades of simpler elements. Our presentation evolves around the graphs of transformations of finite sets. We discuss such topics as limitations of the computational capabilities of cascades, the existence of roots of given functions with respect to iteration powers, etc.

We introduce and characterize those Boolean functions (graph functions) which can be regarded as characteristic functions of graphs of other Boolean functions. An algorithm for detecting these functions is also presented. Finally, we discuss the complexity of computing a Boolean function which can be regarded as a graph function.

The solution of the problem of the existence of an analog with periodic-variable structure of a given fixed-structure connected automaton is presented. Using a Boolean approach, a necessary and sufficient condition for the existence of the periodic analog is proven, and an algorithm that enables us to actually find this analog is given. The saving of the memory capacity needed for the construction of a given automaton is the main interest of the existence of a periodic analog.

A class of linear algebraic structures over the set of Boolean functions is presented. Afterwards, from these structures a new one is studied—the right-weak algebra. By using the properties of this algebra, some new results are established about the commutativity of the superposition, the ideals, the annulators, and the properties of the iteration.