Order-homomorphisms on fuzzes
Suppose that \(L_ 1\) and \(L_ 2\) are fuzzes, i.e. completely distributive lattices with order-reserving involutions. A map \(f: L_ 1\to L_ 2\) is called an order-homomorphism if f is union-preserving and its inverse \(f^{-1}\) is involution-preserving. If the fuzzes \(L_ 1\) and \(L_ 2\) are replaced by fuzzes \(L^ X\) and \(L^ Y\) in the above definition, we get a fuzz function \(f: L^ X\to L^ Y\). In the paper under review, the author studies basic properties of order-homomorphisms. Among others, a condition for a fuzz function to be a function of Zadeh's type (i.e. a fuzz function induced by an ordinary map from X to Y) is obtained (Theorem 2.1). This result is interesting, but the requirement that \(f_ x\) satisfies \((H_ 3)\) may be omitted.
scientific article; zbMATH DE number 3880777