Preorder relations in fuzzy compactifications (Q1262563)

Let L be a completely distributive lattice with an order reversing involution. We consider the compactifications for L-fuzzy topological spaces. We introduce a preorder for the collection of compactifications of an L-fuzzy topological space and give a counterexample to show that the largest compactification is not unique under the preorder, namely the preorder need not satisfy anti-symmetric law. The construction of this counterexample involves many special and deeper properties of fuzzy product spaces. After introducing some separation axioms, we prove that the preorder is just a partial order and therefore ensures the uniqueness of the largest compactification under the assumption of separation properties.