Adam Šeliga

Decomposition integrals on finite spaces have a natural linear programming representation, but their evaluation may require exponentially many variables indexed by subsets of the underlying space. Restricting the admissible sets or collections produces computationally smaller lower approximations and raises the problem of quantifying the resulting loss. We develop an approximation theory for this reduction. For a retained collection and a removed set, the central quantity is the retained collection integral evaluated at the indicator of the removed set; its positive shortfall from the monotone measure value defines the reconstruction defect. We aggregate these local defects through a residual packing problem and obtain simultaneous deletion bounds. For a single removed set, the estimate is sharp and equals the worst-case error on the unit cube. We also determine the smallest uniform multiplicative approximation factor, derive a posteriori certificates from dual reduced profits, and characterize pointwise exactness. The results extend to comparisons of arbitrary collections and to decomposition systems, including deletion of whole member collections. For cardinality-based monotone measures, reconstruction values and approximation factors admit closed forms. These results provide rigorous certificates for sparse evaluation while preserving the underlying non-additive structure.

This paper introduces a novel framework for understanding intuitionistic values and fuzzy sets through the automorphism of the unit interval [0, 1]. By generalizing the concepts of classical intuitionistic sets using strong negations generated by these automorphisms, we define the φ -intuitionistic values and explore their arithmetic, establishing a comprehensive set of operations based mainly on additive generators of t-(co)norms and their properties. This approach provides greater flexibility to the modelling with these sets.

Comprehensive families of copulas including the three basic copulas (at least as limit cases) are useful tools to model countermonotonicity, independence, and comonotonicity of pairs of random variables on the same probability space. In this contribution, we study how the transition from a (basic) copula to a copula modeling a different dependence behavior can be realized by means of ordinal sums based on one of the three basic copulas, perturbing one of the three basic copulas (considering some appropriate parameterized transformations) and truncating the results using the Fréchet-Hoeffding bounds. We provide results and examples showing the flexibility and the restrictions for obtaining new copulas or comprehensive families and illustrate the development of their dependence parameters.