Algebraic structures of interval truth values in fuzzy logic

Any statement in fuzzy logic takes a value in the unit interval of [0,1] as a truth value, which is called a numerical truth value, apart from only 0 and 1 in two-valued logic. This truth value has been extended into an interval called an interval truth value, where an interval truth value is a closed interval [a,b] in [0,1] such that a and b are numerical truth values and a ⩽ b. In this paper the fundamental properties of the set of interval truth values are shown when three fundamental logic operations AND(·), OR(V) and NOT(~) are defined on the truth values, and the algebraic structures of the set are clarified. Finally, algebraic structures of subsets of interval truth values generated from finite generators are explained with examples