The Symmetric Sugeno Integral

We propose an extension of the Sugeno integral for negative numbers, in the spirit of the symmetric extension of Choquet integral, also called \Sipos\ integral. Our framework is purely ordinal, since the Sugeno integral has its interest when the underlying structure is ordinal. We begin by defining negative numbers on a linearly ordered set, and we endow this new structure with a suitable algebra, very close to the ring of real numbers. In a second step, we introduce the M"obius transform on this new structure. Lastly, we define the symmetric Sugeno integral, and show its similarity with the symmetric Choquet integral.

💡 Research Summary

The paper introduces a symmetric extension of the Sugeno integral that can handle negative values, mirroring the way the Choquet integral is symmetrically extended in the so‑called Sipos integral. The authors begin by constructing a notion of “negative numbers” on any linearly ordered set (L). For each element (a\in L) a formal opposite (-a) is introduced, satisfying (-(-a)=a) and the order reversal property (-a\le -b\iff b\le a). This yields an enlarged ordered universe (\tilde L) that contains positive elements, their negatives, and a distinguished zero element.

On (\tilde L) they define two binary operations that play the role of addition and multiplication but are based on the underlying order rather than arithmetic. The “addition” (\oplus) is essentially a max (or min) operator chosen to be compatible with the sign of the operands, while the “multiplication” (\odot) is an order‑preserving combination that also reduces to a max/min depending on the region of the lattice. Both operations are associative, commutative, and distribute over each other, giving (\tilde L) a ring‑like algebraic structure: there exists a neutral element (0) for (\oplus), a unit (1) for (\odot), and each element possesses an additive inverse (its formal negative) and, when appropriate, a multiplicative quasi‑inverse. This algebraic framework is deliberately close to the real numbers, yet it remains purely ordinal, preserving the essence of the Sugeno setting.

With this algebra in place, the authors turn to the Möbius transform, a classic tool for representing capacities (monotone set functions) as linear combinations of elementary set indicators. Because the Sugeno integral is non‑linear and ordinal, the usual linear Möbius transform does not apply. The paper therefore defines an “ordered Möbius transform” that uses the (\oplus) and (\ominus) (the symmetric subtraction derived from (\oplus)) operations together with set inclusion to decompose a capacity (\mu) into basic components (m). The transform and its inverse are shown to satisfy the usual inclusion–exclusion type identities, but with max/min in place of addition and subtraction. This ordered Möbius machinery is essential for expressing the new integral in a compact form and for proving its main properties.

The central contribution is the definition of the symmetric Sugeno integral. For a function (f:\tilde L\to\tilde L) the authors split it into its positive part (f^{+}) and negative part (f^{-}) (the latter being (-) of the negative values of (f)). They then apply the classical Sugeno integral separately to each part:

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