ON THE RELATIONSHIP OF EXTENDED NECESSITY MEASURES TO IMPLICATION OPERATORS ON THE UNIT INTERVAL

By an extended necessity measure on a set X we mean a function from I- X to the unit interval I such that the measure of an arbitrary meet of fuzzy subsets of X equals the infimum of their measures. We address t he question of whether a given extended necessity measure on X is indu ced, through a given implication operator I on I, from some (unknown) normalized fuzzy subset nu of X in the following manner: The measure o f mu epsilon I-X is the degree of containment of nu in mu, according t o the implication I. We provide a simple complete answer to this quest ion in the case I is in one of the following three infinite sets of im plication operators: the S-type implications, the residuated implicati ons, and the continuous implications that satisfy the exchange princip le.