This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0-1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion.

Recent years have witnessed a growing volume in Chinese interregional trade, along with the increasing disparities in environmental pressures. This has prompted an increased attention on where the responsibilities for environmental impacts should be placed. In this paper, we quantify the environmental responsibility of SO 2 emissions and biodiversity impacts due to terrestrial acidification at the provincial level for the first time. We examine the environmental responsibility from the perspectives of production, consumption, and income generation by employing a Multi-Regional Input-Output (MRIO) model for 2007, 2010, and 2012. The results indicate that ~40% of SO 2 emissions were driven by the consumption in provinces other than where the emissions discharged. In particular, those developed provinces were net importers of SO 2 emissions and mainly outsourced their emissions to nearby developing provinces. Over the period of analysis, environmental inequality among 30 provinces was larger than GDP inequality. Furthermore, environmental inequality continued to increase while GDP inequality decreased over the time period. The results of a shared income-and consumption-based responsibility approach suggest that the environmental responsibility of SO 2 emissions and biodiversity impacts for developed provinces can reach up to ~4-to 93-fold the environmental pressure occurred within those provinces. This indicates that under these accounting principles the developed northern provinces in China would bear a much larger share of the environmental responsibility. Capsule: We calculate the shared responsibilities for SO 2 emissions in China and find them to differ significantly from the production-based reduction targets set by governments.

Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems, from both of the theoretical and practical viewpoints. The notion of fuzzy relational equations is associated with the concept of "composition of binary relations." In this survey paper, fuzzy relational equations are studied in a general lattice-theoretic framework and classified into two basic categories according to the duality between the involved composite operations. Necessary and sufficient conditions for the solvability of fuzzy relational equations are discussed and solution sets are characterized by means of a root or crown system under some specific assumptions.

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