Fuzzy functions and their applications
There is an extensive literature devoted to \(L\)-valued similarity relations, i.e., to maps \(E: X\times X\to L\) (where \(X\) is a set and \((L,*)\) is a commutative \(cl\)-monoid with the unit being the top element of the complete lattice \(L\) in which arbitrary sups distribute over \(*\)) satisfying the following conditions: (1) \(E(x,x)= 1\), (2) \(E(x,y)= E(y,x)\), and (3) \(E(x,y)* E(y,z)\leq E(x,z)\) for all \(x,y,z\in X\) [see, e.g., \textit{U. Höhle}, Fuzzy Sets Syst. 27, 31-44 (1988; Zbl 0666.18002)]. The paper under review continues the study of these maps. Typical result (a part of Theorem 3.8): Let \(E_i\) be an \(L\)-valued similarity relation on \(X_i\) such that \(E_i(x,y)= 1\Rightarrow x=y\) \((i= 1,2)\). Let \(\rho: X_1\times X_2\to L\) be such that \(\rho(x_1,x_2)* \rho(y_1,y_2)* E_1(x_1,y_1)\leq E_2(x_2, y_2)\) for all \(x_i,y_i\in X_i\) \((i= 1,2)\), and such that for each \(x\in X_1\) there exists a \(y\in X_2\) with \(\rho(x,y)= 1\). Then there exists a unique map \(f: X_1\to X_2\) such that \(E_1\leq E_2\circ(f\times f)\), \(\rho(x,f(x))= 1\), and \(\rho(x,y)\leq E_2(f(x), y)\) for all \(x\in X_1\) and \(y\in X_2\). Some applications to fuzzy control are given.
scientific article; zbMATH DE number 1563474
scientific article