In this article, we study the inconsistency of systems of $\min-\rightarrow$ fuzzy relational equations. We give analytical formulas for computing the Chebyshev distances $\nabla = \inf_{d \in \mathcal{D}} \Vert \beta - d \Vert$ associated to systems of $\min-\rightarrow$ fuzzy relational equations of the form $\Gamma \Box_{\rightarrow}^{\min} x = \beta$, where $\rightarrow$ is a residual implicator among the G\"odel implication $\rightarrow_G$, the Goguen implication $\rightarrow_{GG}$ or Lukasiewicz's implication $\rightarrow_L$ and $\mathcal{D}$ is the set of second members of consistent systems defined with the same matrix $\Gamma$. The main preliminary result that allows us to obtain these formulas is that the Chebyshev distance $\nabla$ is the lower bound of the solutions of a vector inequality, whatever the residual implicator used. Finally, we show that, in the case of the $\min-\rightarrow_{G}$ system, the Chebyshev distance $\nabla$ may be an infimum, while it is always a minimum for $\min-\rightarrow_{GG}$ and $\min-\rightarrow_{L}$ systems.

翻译：本文研究 $\min-\rightarrow$ 模糊关系方程组系统的不一致性问题。我们给出了计算与形如 $\Gamma \Box_{\rightarrow}^{\min} x = \beta$ 的 $\min-\rightarrow$ 模糊关系方程组相关联的切比雪夫距离 $\nabla = \inf_{d \in \mathcal{D}} \Vert \beta - d \Vert$ 的解析公式，其中 $\rightarrow$ 是哥德尔蕴含 $\rightarrow_G$、戈盖恩蕴含 $\rightarrow_{GG}$ 或卢卡斯维奇蕴含 $\rightarrow_L$ 等剩余蕴含算子之一，$\mathcal{D}$ 是由同一矩阵 $\Gamma$ 定义的一致方程组的第二成员集合。推导这些公式的关键初步结果是：无论使用何种剩余蕴含算子，切比雪夫距离 $\nabla$ 均为某个向量不等式解的下界。最后，我们证明：在 $\min-\rightarrow_{G}$ 系统中，切比雪夫距离 $\nabla$ 可能仅为下确界；而对于 $\min-\rightarrow_{GG}$ 和 $\min-\rightarrow_{L}$ 系统，它始终是最小值。