We address the (generalized) extension problem for possibility measure
s: given a map defined on a family of(fuzzy) sets, is it possible to e
xtend 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 so
lution (of type inf-J, with J a border implicator). When the family of
sets considered is a semi-partition, another important solution (of t
ype sup-T, with T a t-norm) can be identified. In the treatment of the
generalized possibilistic extension problem, we show that a fuzzifica
tion of the greatest solution also plays a central role. On the other
hand, an immediate fuzzification of the sup-T type solution is investi
gated. General necessary and sufficient conditions for this fuzzificat
ion to be a solution are established. This fuzzification is then furth
er discussed in the case of a T-semi-partition or a T-partition. Final
ly, we investigate possible criteria for extendability, inspired by Wa
ng's classical criterion of P-consistency. (C) 1998 Elsevier Science I
nc. All rights reserved.