Invariant functionals on completely distributive lattices

TL;DR

This paper characterizes invariant functionals on completely distributive lattices, showing they are equivalent to Sugeno integrals with capacities, extending previous results to the infinitary and lattice context.

Contribution

It extends the characterization of invariant functionals to completely distributive lattices and provides normal form representations and axiomatizations for polynomial functionals.

Findings

Invariant functionals coincide with Sugeno integrals with capacities.

Nondecreasing condition is redundant on complete chains.

Canonical normal forms for polynomial functionals are established.

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.