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 the question of whether a given extended necessity measure on X is induced, through a given implication operator I on I, from some (unknown) normalized fuzzy subset nu of X in the following manner: The measure of mu epsilon I-X is the degree of containment of nu in mu, according to the implication I. We provide a simple complete answer to this question in the case I is in one of the following three infinite sets of implication operators: the S-type implications, the residuated implications, and the continuous implications that satisfy the exchange principle.