Self-dual operators and a general framework for weighted nilpotent operators

The main purpose of this paper is to consider generated nilpotent operators in an integrative frame and to examine the nilpotent aggregative operator. As a starting point, instead of associativity, we focus on the necessary and sufficient condition of the self-dual property. A parametric form of the generated operator oν is given by using a shifting transformation of the generator function. The parameter has an important semantical meaning as a threshold of expectancy (decision level). Nilpotent conjunctive, disjunctive, aggregative and negation operators can be obtained by changing the parameter value. The properties (De Morgan property, commutativity, self-duality, fulfillment of the boundary conditions, bisymmetry) of the weighted general operator are examined and the formula of the commutative self-dual generated operator, the so-called weighted aggregative operator is given. It is proved that the two-variable operator with weights w1=w2=1 ∀i is conjunctive for low input values, disjunctive for high ones, and averaging otherwise; i.e. a high input can compensate for a lower one.