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Title: Dedekind-MacNeille completion of multivariate copulas via ALGEN method Authors:  Matjaz Omladic - Institute of Mathematics, Physics, and Mechanics (Slovenia) [presenting]
Abstract: The problem of the Dedekind-MacNeille completion of the class of $d$copulas originated in 2005 and is well-known to the experts in the field. Unlike in the bivariate case, where the solution is just the class of $2$-quasi-copulas, in the case that $d>2$, this class is simply too big. The presented solution to the problem identifies an appropriate concrete subclass together with concrete meet and join operations so that the requirements for the desired completion are satisfied. The construction is made so that the induced order coincides with the starting pointwise order on $d$-quasi-copulas. However, this causes the two operations to be adjusted accordingly. The solution is based on a method called {Al}gebraic Obstacles in the {Ge}ometry of {N}egative Volumes (ALGEN for short). This technique has been developed by the same authors to solve some questions in imprecise probability. Since the presented solution is based on a small subclass of $d$-quasi-copulas, a natural question arises whether this class is too big as the completion of $d$-copulas with respect to order suprema and infima. An answer to this question is also given.