Handling the inconsistency of systems of min→ fuzzy relational equations
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In this article, we study the inconsistency of systems of min-→ fuzzy relational equations. We give analytical formulas for computing the Chebyshev distances ∇ = inf_d ∈𝒟‖β - d ‖ associated to systems of min-→ fuzzy relational equations of the form Γ_→^min x = β, where → is a residual implicator among the Gödel implication →_G, the Goguen implication →_GG or Lukasiewicz's implication →_L and 𝒟 is the set of second members of consistent systems defined with the same matrix Γ. The main preliminary result that allows us to obtain these formulas is that the Chebyshev distance ∇ 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-→_G system, the Chebyshev distance ∇ may be an infimum, while it is always a minimum for min-→_GG and min-→_L systems.
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