Capacities on a finite lattice

TL;DR

This paper explores capacities on finite lattices, providing a combinatorial approach, analyzing Fréchet bounds, and examining probabilistic interpretations and inequalities related to completely monotone capacities.

Contribution

It introduces a combinatorial framework for capacities on finite lattices and investigates bounds, interpretations, and inequalities beyond classical monotonicity.

Findings

Development of a combinatorial approach for capacities on finite lattices

Analysis of Fréchet bounds under marginal conditions

Probabilistic interpretation of difference operators and stochastic inequalities

Abstract

In his influential work Choquet systematically studied capacities on Boolean algebras in a topological space, and gave a probabilistic interpretation for completely monotone (and completely alternating) capacities. Beyond complete monotonicity we can view a capacity as a marginal condition for probability distribution over the distributive lattice of dual order ideals. In this paper we discuss a combinatorial approach when capacities are defined over a finite lattice, and investigate Fr\'{e}chet bounds given the marginal condition, probabilistic interpretation of difference operators, and stochastic inequalities with completely monotone capacities.