Some members of the class of (quasi-)copulas with given diagonal from the Markov kernel perspective

Calculating Markov kernels of copulas allows not only for a precise description of the way Bertino- and diagonal copulas distribute mass, but also enables a simply proof of the fact that, for certain diagonals, both may degenerate to proper generalized shuffles of the minimum copula. After extending the kernel approach to the case of the maximum quasi-copula Aδ with given diagonal δ, a conjecture on singularity of Aδ by Nelsen et al. (2008 Nelsen, R. B., Quesada-Molina, J. J., Rodríguez-Lallena, J. A., Úbeda-Flores, M. (2008). On the construction of copulas and quasi-copulas with given diagonal sections. Insurance Math. Econ. 42: 473–483.[Crossref], [Web of Science ®] , [Google Scholar]) is established and an alternative simple and short proof of the result by Úbeda-Flores (2008 Úbeda-Flores, M. (2008). On the best-possible upper bound on sets of copulas with given diagonal sections. Soft Comput. 12: 1019–1025.[Crossref], [Web of Science ®] , [Google Scholar]) characterizing diagonals for which Aδ is a copula is given.