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On fuzzy relational equations and the covering problem. (English) Zbl 1231.03047
In this study, the authors consider finite fuzzy relational equations of the form \(X\circ A= B\), where \(A\) and \(B\) are a fuzzy relation and a fuzzy set, respectively, while \(X\) is the fuzzy set to be determined. The symbol \(\circ\) stands for the max-continuous \(u\)-norm composition operator with “\(u\)” being a bivariate function \([0,1]^2\to [0,1]\) with \(u(0,0)=0\), \(u(1,1)= 1\) and strictly increasing on the domain where \(u(x,y)>0\). The greatest solution to these equations is provided. It is shown how to transform such fuzzy relational equations into a format involving a binding matrix so that solving the equation is equivalent to solving the covering problem. An alternative approach is presented as well; its essence is to transform the original equation into one with the max-product composition operator.

MSC:
03E72 Theory of fuzzy sets, etc.
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