Abstract
There are many numbers of studies that discuss algebraic structures on fuzzy truth values. But, their main interest is in convex fuzzy truth values. This paper concerns with non-convex fuzzy truth values. A multi-interval truth value is defied as a collection of interval truth values. Therefore, it is not convex in general. This paper focuses on the simplest multi-interval truth values, ternary multi-interval truth values. A ternary multi-interval truth value is a non-empty subset of {0, 1, 2}. Thus, multi-interval truth value {0, 2} is not convex. Min, Max, and involution are defined over multi-interval truth values by Zadeh's extension principle. Then, it is known that a set of multi-interval truth values with the three operations is a de Morgan bi-lattices. This means that non-convex fuzzy truth values do not form a lattice any more. This paper describes functions over ternary multi-interval truth values which are expressed by logic formulas. Then, the paper clarifies that the functions are determined by the four multi-interval truth values {0}, {1}, {2}, and {0, 2}.
