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

In this article, we study the inconsistency of systems of $\min -\to$ fuzzy relational equations. We give analytical formulas for computing the Chebyshev distances $\mathrm{\nabla }={\inf }_{d\in \mathcal{D}}\mathrm{\parallel }\beta -d\mathrm{\parallel }$ associated to systems of $\min -\to$ fuzzy relational equations of the form $\mathrm{\Gamma }{\mathrm{\square }}_{\to }^{\min }x=\beta$, where $\to$ is a residual implicator among the Gödel implication ${\to }_G$, the Goguen implication ${\to }_{GG}$ or Lukasiewicz’s implication ${\to }_L$ and $\mathcal{D}$ is the set of second members of consistent systems defined with the same matrix $\mathrm{\Gamma }$. The main preliminary result that allows us to obtain these formulas is that the Chebyshev distance $\mathrm{\nabla }$ 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 $\min -{\to }_G$ system, the Chebyshev distance $\mathrm{\nabla }$ may be an infimum, while it is always a minimum for $\min -{\to }_{GG}$ and $\min -{\to }_L$ systems.