Characterization of weighted maximum and some related operations
Information Sciences—Intelligent Systems: An International Journal
On comparison meaningfulness of aggregation functions
Journal of Mathematical Psychology
Lattice functions and equations
Lattice functions and equations
On order invariant aggregation functionals
Journal of Mathematical Psychology
On order invariant synthesizing functions
Journal of Mathematical Psychology
Aggregation Functions (Encyclopedia of Mathematics and its Applications)
Aggregation Functions (Encyclopedia of Mathematics and its Applications)
Aggregation Functions: A Guide for Practitioners
Aggregation Functions: A Guide for Practitioners
A quantile approach to integration with respect to non-additive measures
MDAI'12 Proceedings of the 9th international conference on Modeling Decisions for Artificial Intelligence
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In this paper we are interested in functionals defined on completely distributive lattices and which are invariant under mappings preserving arbitrary joins and meets. We prove that the class of nondecreasing invariant functionals coincides with the class of Sugeno integrals associated with {0,1}-valued capacities, the so-called term functionals, thus extending previous results both to the infinitary case as well as to the realm of completely distributive lattices. Furthermore, we show that, in the case of functionals over complete chains, the nondecreasing condition is redundant. Characterizations of the class of Sugeno integrals, as well as its superclass comprising all polynomial functionals, are provided by showing that the axiomatizations (given in terms of homogeneity) of their restriction to finitary functionals still hold over completely distributive lattices. We also present canonical normal form representations of polynomial functionals on completely distributive lattices, which appear as the natural extensions to their finitary counterparts, and as a by-product we obtain an axiomatization of complete distributivity in the case of bounded lattices.