Compatible extensions and consistent closures: a fuzzy approach
This work addresses theoretical extensions in fuzzy set theory, providing incremental advancements by adapting existing crisp relation results to the fuzzy domain.
The paper generalizes Duggan's results on compatible extensions from crisp to fuzzy relations, deriving fuzzy versions of key extension theorems like Szpilrajn, Hansson, and Suzumura, and introduces two notions of consistent closure for fuzzy relations.
In this paper $\ast$--compatible extensions of fuzzy relations are studied, generalizing some results obtained by Duggan in case of crisp relations. From this general result are obtained as particular cases fuzzy versions of some important extension theorems for crisp relations (Szpilrajn, Hansson, Suzumura). Two notions of consistent closure of a fuzzy relation are introduced.