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

• URL: http://arxiv.org/abs/2308.12385v1
• Date: Tue, 22 Aug 2023 16:12:26 GMT
• Title: Handling the inconsistency of systems of $\min\rightarrow$ fuzzy relational equations
• Authors: Isma\"il Baaj
• Abstract summary: We give analytical formulas for computing the Chebyshev distances $nabla = inf_d in mathcalD Vert beta. We show that, in the case of the $min-rightarrow_G$ system, the Chebyshev distance $nabla$ may be an infimum.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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