On the completion of fuzzy metric spaces

This paper discusses the completion of a fuzzy metric space (in the sense of Kaleva and Seikkala) that is modeled by a pair of two-place functions, L and R. Previous work on this question is based on L=min, R=max. Under the condition that L and R are a generic class of two-place functions, we characterize the completable fuzzy metric space and show that the corresponding completion is uniquely determined up to isometry. Furthermore, we point out that with more conditions on R, including on R=max, the completion of each fuzzy metric space exists and is unique up to isometry. This improves the corresponding result of Kaleva [The completion of fuzzy metric spaces, J. Math. Anal. Appl. 109 (1985) 194–198].