A note on specialization L-preorder of L-topological spaces, L-fuzzifying topological spaces, and L-fuzzy topological spaces

It is shown that, for any spatial frame L (i.e., L is a complete lattice generated by the set of all prime elements), both the specialization L-preorder of L-topological spaces introduced by Lai and Zhang [Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems 157 (2006) 1865–1885] and that of L-fuzzifying topological spaces introduced by Fang and Qiu [Fuzzy orders and fuzzifying topologies, Internat. J. Approx. Reasoning 48 (2008) 98–109] are special cases of that of L-fuzzy topological spaces introduced by Fang [I-fuzzy Alexandrov topologies and specialization orders, Fuzzy Sets and Systems 158 (2007) 2359–2374].