JID:FSS AID:7884 /FLA [m3SC+; v1.331; Prn:17/06/2020; 15:01] P.1(1-18) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss On a new partial order on bivariate distributions and on constrained bounds of their copulas ✩ Matjaž Omladiˇ c a,∗ , Nik Stopar b,a a Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia b University of Ljubljana, Faculty of electrical engineering, Ljubljana, Slovenia Received 1 April 2020; received in revised form 7 June 2020; accepted 8 June 2020 Abstract In this paper we study the maximal possible difference N of values of a quasi-copula at two different points of the unit square. This study enables us to give upper and lower bounds, called constrained bounds, for quasi-copulas with fixed value at a given point in the unit square, thus extending an earlier result from copulas to quasi-copulas. It turns out that the two bounds are actually copulas. Difference N is also the main tool in exhibiting two new characterizations of quasi-copulas, a major result of this paper, which sheds new light on the subject of copulas as well. Significant applications of our results are also given in the imprecise probability theory, one of the more important non-standard approaches to probability. After a full-scale bivariate Sklar’s theorem has been proven under this approach, we want to establish the tightness of its background before moving to the more general multivariate scene. We present an extension of the partial order on quasi-distributions used in the said theorem, i.e., pointwise order with fixed margins, using again the difference N as a main tool. A careful study of the interplay between the order on quasi-distributions and the order on corresponding quasi-copulas that represent them is also given. Due to a recent result that the quasi-copulas obtained via Sklar’s theorem in the imprecise setting are exactly the same as the ones in the standard setting, it is not surprising that results on quasi-copulas can shed some light both on open questions in the standard probability theory and in the imprecise probability theory at the same time.  2020 Elsevier B.V. All rights reserved. 1. Introduction Our motivation for the development of results presented in this paper is twofold. While some of our investigation was inspired by recent results of the same authors in non-standard probabilities [14,15], the motivation in the standard probability approach goes further back. Theorem 3.2.3 in R. Nelsen’s book [12] can be seen as a prototype of a constrained bound result, as we call it, preceded by a remark on p. 70: “When we possess information about the values of a copula at points in the interior of I 2 (the unit square), the Fréchet-Hoeffding bounds can often be narrowed.” Our Theorem 10 may be itself seen as an outcome of an investigation initiated there. A series of papers lightens the path ✩ The authors acknowledge financial support from the Slovenian Research Agency (research core funding No. P1-0222). * Corresponding author. E-mail addresses: Matjaz@Omladic.net (M. Omladiˇ c), Nik.Stopar@fe.uni-lj.si (N. Stopar). https://doi.org/10.1016/j.fss.2020.06.006 0165-0114/ 2020 Elsevier B.V. All rights reserved.