The Mobius transform on symmetric ordered structures and its application to capacities on finite sets

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 detail computing rules, which we endow with a partial order. This permits to find solutions to the inversion formula underlying the Mobius 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 Mobius transform of the capacity. Properties and examples of Mobius transform of sup-preserving and inf-preserving capacities are given. (C) 2004 Elsevier B.V. All rights reserved.