Résumé: For a distributive lattice L, we consider the problem of interpolating functions f : D → L defined on a finite set D ⊆ Ln , by means of lattice polynomial functions of L. Two instances of this problem have already been solved. In the case when L is a distributive lattice with least and greatest elements 0 and 1, Goodstein proved that a function f : {0, 1}n → L can be interpolated by a lattice polynomial function p : Ln → L if an only if f is monotone; in this case, the interpolating polynomial p was shown to be unique. The interpolation problem was also considered in the more general set- ting where L is a distributive lattice, not necessarily bounded, and where D ⊆ Ln is allowed to range over cuboids D = {a1 , b1 } × · · · × {an , bn } with ai , bi ∈ L and ai < bi . In this case, the class of such partial functions that can be interpolated by lattice polynomial functions was completely described. In this paper, we extend these results by completely characterizing the class of lattice functions that can be interpolated by polynomial functions on arbitrary finite subsets D ⊆ Ln . As in the latter setting, interpolating polynomials are not necessarily unique. We provide explicit descriptions of all possible lattice polynomial functions that interpolate these lattice functions, when such an interpolation is available.

Résumé: Integrals are currently used in multiple criteria analysis for synthesizing into a global evaluation the advantages possessed by a po- tential choice. As such, integrals are operators that increase with the criteria evaluations. However, an item may be also evaluated in terms of its defects. Then the more and the greater the defects, the smaller the evaluation should be. An operator that can provide a synthesis of the defects of an item in this sense is called a desintegral. Desintegrals are maximal when no defects at all are present, while integrals are maximal when all advantages are sufficiently present. So, the greater the value of an integral, or a desintegral, the better the corresponding item since advantages are greater, or defects are smaller respectively. Desintegrals implicitly refer to a negative scale, since an order-reversing mapping of the scale used for evaluating each criterion transforms the degree to which the value is advantageous into a degree to which it is disadvanta- geous, and conversely. In this paper, we provide an organised description of counterparts to Sugeno integrals that synthesize positive or negative evaluations in the qualitative framework of a totally ordered residuated lattice equipped with an involutive negation. We exploit three kinds of criteria weighting schemes that are allowed by this algebraic structure.

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Résumé: CPTS-integrals (Cumulative Prospect Theory Sugeno inte- gral) are agregation functions that return a global rating when the eval- uations are given on a bipolar scale. This paper presents the problem of the elicitation of CPTS-integrals agreeing with a data set composed of pairs concerning some criteria and a global evaluation. Moreover the set of fuzzy measures which are solutions of this inverse problem is identi- fied. When the elicitation of one CPTS-integral is not possible, a set of family of CPTS-integrals is proposed.