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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:
 3e+72 Theory of fuzzy sets, etc.
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