Logic expressions of monotonic multiple-valued functions

This paper presents some fundamental properties of multiple-valued logic functions monotonic with respect to a partial-ordering relation which is introduced in the set of truth values and does not necessarily have the greatest or least element. Two kinds of necessary and sufficient conditions for monotonic p-valued functions are given with the proofs. Their logic formulas using unary operators defined in the partial-ordering relation and a simplification method for those logic formulas are also given. These results include our former results for p-valued functions monotonic with respect to the ambiguity relation which is a partial-ordering relation with the greatest element