<p><b>Abstract</b>—In this paper, we focus on functions defined on a special subset of the power set of <tmath>$\left\{0, 1, \ldots, r-1\right\}$</tmath> (the elements in the subset will be called discrete interval truth values) and operations on the truth values. The operations discussed in this paper will be called regular because they are one of the extensions of the regularity, which was first introduced by Kleene in his ternary logic. Mukaidono investigated some properties of ternary functions which can be represented by the regular operations. He called such ternary functions “regular ternary logic functions.” Regular ternary logic functions are useful for representing and analyzing ambiguities such as transient states and/or initial states in binary logic circuits that Boolean functions cannot cope with. Furthermore, they are also applied to studies of fail-safe systems for binary logic circuits. In this paper, we will discuss an extension of regular ternary logic functions to functions on the discrete interval truth values. First, we will suggest an extension of the regularity, in the sense of Kleene, into operations on the discrete interval truth values. We will then present some mathematical properties of functions on the discrete interval truth values consisting of regular operations and one application of these functions.</p>