n this paper we study the set of copulas for which both a horizontal section and a vertical section have been given. We give a general construction for copulas of this type and we provide the lower and upper copulas with these sections. Symmetric copulas with given horizontal section are also discussed, as well as copulas defined on a grid of the unit square. Several examples are presented.
Based on a recent representation of copulas invariant under univariate conditioning, a new class of copulas linked to a distortion of the identity function is introduced and studied.
Quasi-homogeneity of copulas is introduced and studied. Quasi-homogeneous copulas are characterized by the convexity and strict monotonicity of their diagonal sections. As a by-product, a new construction method for copulas when only their diagonal section is known is given.
Several open problems posed during FSTA 2006 (Liptovský Ján, Slovakia) are presented. These problems concern the classification of strict triangular norms, Lipschitz t-norms, interval semigroups, copulas, semicopulas and quasi-copulas, fuzzy implications, means, fuzzy relations, MV-algebras and effect algebras.
Eighteen open problems posed during FSTA 2010 (Liptovský Ján, Slovakia) are presented. These problems concern copulas, triangular norms and related aggregation functions. Some open problems concerning effect algebras are also included.