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On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations

机译：关于模糊关系不等式和方程的弱线性系统的最大解

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In this paper we study systems of fuzzy relation inequalities and equations of the form U o V_i≤V_i o V (i ∈ I), where U is an unknown and V_i (i ∈I) are given fuzzy relations, the dual systems V_i o U≤U o V_i (i ∈ I), their conjunctions, the systems of the form U o V_i = V_i o U (i ∈ I), and certain special types of these systems. We call them weakly linear systems.rnFor each weakly linear system, with a complete residuated lattice as the underlying structure of truth values, we prove the existence of the greatest solution, and we provide an algorithm for computing the greatest solution, which works whenever the underlying complete residuated lattice is locally finite. Otherwise, we determine some sufficient conditions under which the algorithm works. The algorithm is iterative, and each its single step can be viewed as solving of a particular linear system.rnWeakly linear systems emerged from the fuzzy automata theory, but we show that they also have important applications in other fields, e.g. in the concurrency theory and social network analysis.
机译：本文研究模糊关系不等式的系统和形式为U oV_i≤V_io V（i∈I）的方程，其中U是未知数，而V_i（i∈I）具有模糊关系，对偶系统V_i o U≤Uo V_i（i∈I），它们的合取，形式为U o V_i = V_i o U（i∈​​I）的系统以及这些系统的某些特殊类型。我们将它们称为弱线性系统。对于每个具有完整剩余格作为真值基础结构的弱线性系统，我们证明了最大解的存在，并且我们提供了一种计算最大解的算法，该算法可以在底层完整残差格是局部有限的。否则，我们将确定算法可在哪些条件下工作。该算法是迭代的，每个步骤都可以看作是解决特定线性系统的问题。模糊自动机理论产生了微弱的线性系统，但我们证明了它们在其他领域也有重要的应用。在并发理论和社交网络分析中。

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