The main goal of this paper is to study the extent of freedom one has in constructing quasi-copulas vs. copulas. Specifically, it exhibits three construction methods for quasi-copulas based on recent developments: a representation of multivariate quasi-copulas by means of infima and suprema of copulas, an extension of a classical result on shuffles of min to the setting of quasi-copulas, and a construction method for quasi-copulas obeying a given signed mass pattern on a patch.

We answer a 15-year-old open question about the exact upper bound for bivariate copulas with a given diagonal section by giving an explicit formula for this bound. As an application, we determine the maximal asymmetry of bivariate copulas with a given diagonal section and construct a copula that attains it. We derive a formula for the maximal asymmetry that is simple enough to be used by practitioners.

We prove that every quasi-copula can be written as a uniformly converging infinite sum of multiples of copulas. Furthermore, we characterize those quasi-copulas which can be written as a finite sum of multiples of copulas, i.e., that are a linear combination of two copulas. This generalizes a recent result of Fernández-Sánchez, Quesada-Molina, and Úbeda-Flores who considered linear combinations of discrete copulas.

It is well known that every bivariate copula induces a positive measure on the Borel σ-algebra on [0,1]2, but there exist bivariate quasi-copulas that do not induce a signed measure on the same σ-algebra. In this paper we show that a signed measure induced by a bivariate quasi-copula can always be expressed as an infinite combination of measures induced by copulas. With this we are able to give the first characterization of measure-inducing quasi-copulas in the bivariate setting.

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value 1 at (1,1,…,1). We characterize the existence of a k-increasing n-variate function C fulfilling A⩽C⩽B for standardized n-variate functions A,B and discuss methods for constructing such functions. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when A respectively B coincides with the pointwise infimum respectively supremum of the set of all k-increasing n-variate functions C fulfilling A⩽C⩽B.

In this paper we present a surprisingly general extension of the main result of a paper that appeared in this journal: I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48--66. The main tools we develop in order to do so are: (1) a theory on quasi-distributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) $p$-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) $p$-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladi\v{c} and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) $p$-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual $\sigma$-additive probability space or on an additive space can be obtained as a copula of its margins meaning that its possible extraordinariness depends solely on its margins. This might indicate that copulas are a stronger probability concept than probability itself.

A notion of real number graph labellings captures the dependence of the span of an optimal channel assignment on the separations that are required between frequencies assigned to close transmitters. We determine the spans of such optimal labellings for a subfamily of Kneser graphs formed by the complements of the line graphs of complete graphs. This subfamily contains (among others) the Petersen graph.