Representation of fuzzy subalgebras by crisp subalgebras (Q1873684)

For any nonempty set \(X\), let \(X_L\) denote the set of all fuzzy points of \(X\). A subset \(Z\) of \(X_L\) is called closed if for any \(x\in X\) and any \(M\subseteq L\), the fuzzy point \(F_x^{\sup M}\in Z\Leftrightarrow F^a_x\in Z\) for all \(a\in M\). The authors prove that the complete lattice of fuzzy subsets of \(X\) is isomorphic to the complete lattice of closed subsets of \(X_L\). Hence for any algebra \(X\) of type \(\tau_1\), \(X_L\) is an algebra of the same type and the lattice of fuzzy subalgebras of \(X\) is isomorphic to the lattice of closed subalgebras of \(X_L\).