The M\"obius transform on symmetric ordered structures and its application to capacities on finite sets

TL;DR

This paper introduces a symmetric algebraic structure on ordered sets, explores its non-associative properties, and applies the developed M"obius transform framework to capacities in decision theory, providing new computational methods and examples.

Contribution

It develops a symmetric, near-ring algebraic structure on ordered sets, analyzes non-associativity, and applies the M"obius transform to capacities in decision theory.

Findings

Established a symmetric algebraic structure with non-associativity.

Developed computing rules with a partial order for the M"obius transform.

Provided properties and examples of M"obius transforms of capacities.

Abstract

Considering a linearly ordered set, we introduce its symmetric version, and endow it with two operations extending supremum and infimum, so as to obtain an algebraic structure close to a commutative ring. We show that imposing symmetry necessarily entails non associativity, hence computing rules are defined in order to deal with non associativity. We study in details computing rules, which we endow with a partial order. This permits to find solutions to the inversion formula underlying the M\"obius transform. Then we apply these results to the case of capacities, a notion from decision theory which corresponds, in the language of ordered sets, to order preserving mappings, preserving also top and bottom. In this case, the solution of the inversion formula is called the M\"obius transform of the capacity. Properties and examples of M\"obius transform of sup-preserving and inf-preserving capacities are given.