Invariant functionals on completely distributive lattices

Abstract

Abstract. 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.

An ordinal approach to risk measurement

“…Note that the polynomial functionals obtained solely from projections by taking arbitrary meets and joins, coincide exactly with those Sugeno integrals associated with {0, 1}-valued capacities, i.e., nondecreasing mappings : ( ) → {0, 1} such that ( ) ∈ {0, 1}. Sugeno integrals over completely distributive lattices were axiomatized in [3] in terms of nondecreasing monotonicity and homogeneity.…”

“…In [15], Tunnicliffe presented a characterization of complete distributivity which relied on the notion of a "cone". We shall make use of the following alternative characterization given in [3].…”

“…Remark Even though our underlying universe is implicitly assumed to be bounded, this condition is not really necessary as discussed in [3]. This boundness condition is not required in the literature, however the functionals considered are defined on spaces of measurable functions which are bounded.…”

Aggregation Functionals on Complete Lattices

“…So we study aggregation functionals based on a complete lattices and we consider in particular the class of completely distributive lattices. A general approach to aggregation on bounded posets is considered also in [3], [6] and [12]. The plan of the paper is the following.…”