Representation and construction of self-dual aggregation operators

Two different characterizations of self-dual aggregation operators are available in the literature: one based on C(x,y)=x/(x+1-y)C(x,y)=x/(x+1-y) and one based on the arithmetic mean. Both approaches construct a self-dual aggregation operator by combining an aggregation operator with its dual. In this paper, we fit these approaches into a more general framework and characterize N-invariant aggregation operators, with N an involutive negator. Various binary aggregation operators, fulfilling some kind of symmetry w.r.t. N and with a sufficiently large range, can be used to combine an aggregation operator and its dual into an N-invariant aggregation operator. Moreover, using aggregation operators to construct N-invariant aggregation operators seems rather restrictive. It suffices to consider n-ary operators fulfilling some weaker conditions. Special attention is drawn to the equivalence classes that arise as several of these n-ary operators can yield the same N-invariant aggregation operator.