In this paper we solve in the negative the problem proposed in this journal (I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48-66) whether an order interval defined by an imprecise copula contains a copula. Namely, if C is a nonempty set of copulas, then C = inf{C} C∈C and C = sup{C} C∈C are quasi-copulas and the pair (C, C) is an imprecise copula according to the definition introduced in the cited paper, following the ideas of p-boxes. We show that there is an imprecise copula (A, B) in this sense such that there is no copula C whatsoever satisfying A C B. So, it is questionable whether the proposed definition of the imprecise copula is in accordance with the intentions of the initiators. Our methods may be of independent interest: We upgrade the ideas of Dibala et al. (Defects and transformations of quasi-copulas, Kybernetika, 52 (2016), 848-865) where possibly negative volumes of quasi-copulas as defects from being copulas were studied.1991 Mathematics Subject Classification. AMS.
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 quasidistributions 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. Omladič 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 σ-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.1991 Mathematics Subject Classification. AMS.
We define a compressed zero-divisor graph Θ(K) of a finite commutative unital ring K, where the compression is performed by means of the associatedness relation. We prove that this is the best possible compression which induces a functor Θ, and that this functor preserves categorial products (in both directions). We use the structure of Θ(K) to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.
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