Set-valued functions and regularity

In this paper, we focus on regularity and set-valued functions. The regularity was first introduced by S.C. Kleene (1952) into the propositional connectives of a ternary logic. Then, M. Mukaidono (1986) expanded the regularity of Kleene into n-variable ternary functions, and a ternary function which is regular is called a regular ternary logic function. Some studies expanded regular ternary logic functions into τ-valued functions, and studied properties of them. In this paper, we propose another extension of the concepts of the regularity in the sense of Kleene and Mukaidono. That is, we introduce regularity into r-valued set-valued functions. Further, we give properties of the set-valued functions with the regularity