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Abstract: In this paper we are interested in functionals defined on completelydistributive lattices and which are invariant under mappings preserving{arbitrary} joins and meets. We prove that the class of nondecreasing invariantfunctionals coincides with the class of Sugeno integrals associated with$\{0,1\}$-valued capacities, the so-called term functionals, thus extendingprevious results both to the infinitary case as well as to the realm ofcompletely distributive lattices. Furthermore, we show that, in the case offunctionals over complete chains, the nondecreasing condition is redundant.Characterizations of the class of Sugeno integrals, as well as its superclasscomprising all polynomial functionals, are provided by showing that theaxiomatizations given in terms of homogeneity of their restriction tofinitary functionals still hold over completely distributive lattices. We alsopresent canonical normal form representations of polynomial functionals oncompletely distributive lattices, which appear as the natural extensions totheir finitary counterparts, and as a by-product we obtain an axiomatization ofcomplete distributivity in the case of bounded lattices.

Autor: Marta Cardin, Miguel Couceiro


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