A characterization of r-valued functions monotonic in an order based on regularity

In this paper, we focus on r-valued functions for treating ambiguities. These functions are defined as r-valued functions monotonic in an order (denoted by <) which is a natural extension of the Kleene´s regularity in his ternary logic. The order < is suitable for treating ambiguities. The functions were first introduced in Mukaidono (1986), and the authors have clarified some of their interesting mathematical properties. In this paper, we will show two different necessary and sufficient condition for an r-valued function to be monotonic in <. Next, we will discuss functions monotonic in < whose information loss are the minimal and the maximal. Then, we will give the representations of these two types of functions