In this article, we study the inconsistency of a system of $\max-T$ fuzzy
relational equations of the form $A \Box_{T}^{\max} x = b$, where $T$ is a
t-norm among $\min$, the product or Lukasiewicz's t-norm. For an inconsistent
$\max-T$ system, we directly construct a canonical maximal consistent subsystem
(w.r.t the inclusion order). The main tool used to obtain it is the analytical
formula which compute the Chebyshev distance $\Delta = \inf_{c \in \mathcal{C}}
\Vert b - c \Vert$ associated to the inconsistent $\max-T$ system, where
$\mathcal{C}$ is the set of second members of consistent systems defined with
the same matrix $A$. Based on the same analytical formula, we give, for an
inconsistent $\max-\min$ system, an efficient method to obtain all its
consistent subsystems, and we show how to iteratively get all its maximal
consistent subsystems.

研究了形式为 A Box_T^max x = b 的 max-T 模糊关系方程系统的不一致性，并提出了直接构造规范最大一致子系统的方法，其中使用了计算与不一致 max-T 系统相关的 Chebyshev 距离的分析公式。基于相同的分析公式，提供了一种有效的方法来获取不一致 max-min 系统的所有一致子系统，并展示了如何迭代地获取所有最大一致子系统。

最大一致 Max-T 模糊关系方程的子系统

Maximal Consistent Subsystems of Max-T Fuzzy Relational Equations

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.

研究模糊关系方程组的不一致性，提出计算 Chebyshev 距离的分析公式，阐述了 Chebyshev 距离作为解向量不等式的下界，在 G"odel，Goguen 和 Lukasiewicz 的蕴涵关系中的特殊情况下，Chebyshev 距离可能是下确界或最小值。

处理 $minightarrow$ 模糊关系方程系统的不一致性

Handling the inconsistency of systems of $\min\rightarrow$ fuzzy  relational equations

In this article, we study the inconsistency of a system of $\max$-product
fuzzy relational equations and of a system of $\max$-Lukasiewicz fuzzy
relational equations. For a system of $\max-\min$ fuzzy relational equations $A
\Box_{\min}^{\max} x = b$ and using the $L_\infty$ norm, (Baaj, 2023) showed
that the Chebyshev distance $\Delta = \inf_{c \in \mathcal{C}} \Vert b - c
\Vert$, where $\mathcal{C}$ is the set of second members of consistent systems
defined with the same matrix $A$, can be computed by an explicit analytical
formula according to the components of the matrix $A$ and its second member
$b$. In this article, we give analytical formulas analogous to that of (Baaj,
2023) to compute the Chebyshev distance associated to the second member of a
system of $\max$-product fuzzy relational equations and that associated to the
second member of a system of $\max$-Lukasiewicz fuzzy relational equations.

本文研究了一组 max-product 模糊关系方程组和一组 max-Lukasiewicz 模糊关系方程组的不一致性，并提供了类似于 Baaj（2023）的分析公式，用于计算与其关联的 Chebyshev 距离。

与 Max-product/Lukasiewicz 模糊关系方程二次成员相关的 Chebyshev 距离

Chebyshev distances associated to the second members of systems of Max-product/Lukasiewicz Fuzzy relational equations

In this article, we study the approximate solutions set $\Lambda_b$ of an
inconsistent system of $\max-\min$ fuzzy relational equations $(S): A
\Box_{\min}^{\max}x =b$. Using the $L_\infty$ norm, we compute by an explicit
analytical formula the Chebyshev distance $\Delta~=~\inf_{c \in \mathcal{C}}
\Vert b -c \Vert$, where $\mathcal{C}$ is the set of second members of the
consistent systems defined with the same matrix $A$. We study the set
$\mathcal{C}_b$ of Chebyshev approximations of the second member $b$ i.e.,
vectors $c \in \mathcal{C}$ such that $\Vert b -c \Vert = \Delta$, which is
associated to the approximate solutions set $\Lambda_b$ in the following sense:
an element of the set $\Lambda_b$ is a solution vector $x^\ast$ of a system $A
\Box_{\min}^{\max}x =c$ where $c \in \mathcal{C}_b$. As main results, we
describe both the structure of the set $\Lambda_b$ and that of the set
$\mathcal{C}_b$. We then introduce a paradigm for $\max-\min$ learning weight
matrices that relates input and output data from training data. The learning
error is expressed in terms of the $L_\infty$ norm. We compute by an explicit
formula the minimal value of the learning error according to the training data.
We give a method to construct weight matrices whose learning error is minimal,
that we call approximate weight matrices.
Finally, as an application of our results, we show how to learn approximately
the rule parameters of a possibilistic rule-based system according to multiple
training data.

研究了模糊关系方程的近似解集和学习权重矩阵，提出了一种近似权重矩阵的方法，并应用于可能性规则系统的参数学习。