It is known that for every s∈]1,2[ there is a copula whose support is a self-similar fractal set with Hausdorff —and box-counting— dimension s. In this paper we provide similar results for (proper) quasi-copulas, in both the bivariate and multivariate cases.

In this article, we answer positively an open question posed in [G. Mayor, J. Suñer, and J. Torrens, “Copula-like operations on finite settings,”IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp. 468–477, Aug. 2005] concerning the extensions of discrete copulas to shuffles of Min. Moreover, we also use the extension of discrete copulas to sparse copulas, associated with discrete copulas, of idempotent copulas, in particular, the copula$M$, in order to answer the open question in an alternative way.

In this paper we provide an alternative proof to that given in Ref. 1 to the fact that the class of multivariate copulas with a given diagonal section δ — denoted by 𝒞δ — is a singleton if, and only if, the unique element of the class is the copula of comonotonicity among random variables. This will be used to prove that, for any diagonal δ different from the diagonal of the comotonic copula, 𝒞δ is strictly included in the class of all multivariate quasi-copulas whose diagonal section is δ.

In this note we provide a new proof of Sklar’s theorem by using Zorn’s lemma.

A framework for constructing copulas that can be regarded as a patchwork-like assembly of arbitrary copulas, with nonoverlapping rectangles as patches, is studied in [B. De Baets and H. De Meyer, “Orthogonal grid constructions of copulas,” IEEE Trans. Fuzzy Syst., vol. 15, no. 6, pp. 1053-1062, Dec. 2007], where the authors provide sufficient conditions to derive a family of construction methods. Our aim in this paper is to provide a characterization of such constructions.

Let [Formula: see text] and [Formula: see text] be, respectively, the classes of all copulas and quasi-copulas whose diagonal section is δ. We determine under which conditions on δ we have: (a) both [Formula: see text] and [Formula: see text] consist of a singleton; (b) [Formula: see text]. Moreover, a simple construction of copulas with a given convex diagonal section is introduced.