Handling the inconsistency of systems of minrightarrow 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 = infd ∈ D Vert β - d Vert associated to systems of min-rightarrow fuzzy relational equations of the form Gamma Boxrightarrowmin x = β, where rightarrow is a residual implicator among the G"odel implication rightarrowG, the Goguen implication rightarrowGG or Lukasiewicz's implication rightarrowL and 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-rightarrowG system, the Chebyshev distance nabla may be an infimum, while it is always a minimum for min-rightarrowGG and min-rightarrowL systems.