A SUFFICIENT AND NECESSARY CONDITION FOR AN OWA BAG MAPPING HAVING THE SELF-IDENTITY

A mapping f : (Un=1In)-I-infinity --> I is called a bag mapping having the self-identity if for every (x(1),...,x(n)) epsilon (Ui=1In)-I-inf inity we have (1) f(x(1),...,x(n)) = f(x(i1),,...,x(in)) for any arran gement (i(1),..., i(n)) of {1,..., n}; (2) f is monotonic; (3) f(x(1), ...,x(n),) f(x(1),...,x(n))) = f(x(1),...,x(n)). Let (omega(i,n) : i = 1,...,n;n = 1,2,...) be a family of non-negative real numbers satisfy ing Sigma(i=1)(n)omega(i,n) = 1 for every n. Then one calls the mappin g f : (Ui=1In)-I-infinity --> I defined as follows an OWA bag mapping: for every (x(1),...,x(n)) epsilon (Ui=1In)-I-infinity, f(x(1),...,x(n )) = Sigma(i=1)(n)omega(i,n)y(i), where yi is the ith largest element in the set {x(1),...,x(n)}. In this paper, we give a sufficient and ne cessary condition for an OWA bag mapping llaving the self-identity. (C ) 1997 Elsevier Science B.V.