\(L\)-fuzzy topological groups (Q1193650)

The concept of \(L\)-fuzzy topological groups is introduced as follows: Let \(X\) be a group and \(J\) be an \(L\)-fuzzy topology on \(X\). The pair \((X,J)\) is said to be an \(L\)-fuzzy topological group, if and only if the following conditions are satisfied: (a) The mapping \(g: (x,y)\to xy\) of the product \(L\)-fuzzy topological space \((X,J)\times(X,J)\) into \((X,J)\) is a continuous order-homomorphism. (b) The mapping \(h: x\to x^{-1}\) of \((X,J)\) into itself is a continuous order-homomorphism. Some basic properties are discussed. It is proved that, if \(X\) is a group with \(e\) then, under some conditions, there exists an \(L\)-fuzzy topology \(J\) on \(X\) such that \((X,J)\) is an \(L\)-fuzzy topological group.