The Construction of Possibility Measures from Samples of T-Semi-Partitions

We address the (generalized) extension problem for possibility measures: given a map defined on a family of (fuzzy) sets, is it possible to extend it to a (generalized) possibility measure? The extension problem for possibility measures is known to be equivalent to a system of sup-T equations, with T a t-norm. A key role is played by the greatest solution (of type inf-I, with I a border implicator). When the family of sets considered is a semi-partition, another important solution (of type sup-T , with T a t-norm) can be identified. In the treatment of the generalized possibilistic extension problem, we show that a fuzzification of the greatest solution also plays a central role. On the other hand, an immediate fuzzification of the sup-T type solution is investigated. General necessary and sufficient conditions for this fuzzification to be a solution are established. This fuzzification is then further discussed in the case of a T -semi-partition or a T -partition. Finally, we investigate possible criteria for extendability, inspired by Wang’s classical criterion of P-consistency.