Completeness of Hutton [0,1]-quasi-uniform spaces

This paper deals with completeness of Hutton [0,1]-quasi-uniform spaces. Recently, the first two authors, [J. Gutiérrez García, M.A. de Prada Vicente, Hutton [0,1]-quasi-uniformities induced by fuzzy (quasi-)metric spaces, Fuzzy Sets and Systems 157 (2006), 755–766], have constructed a Hutton [0,1]-quasi-uniformity induced by a fuzzy metric space (in the sense of George and Veeramani). In this paper, we define completeness of Hutton [0,1]-quasi-uniform spaces as convergence of any stratified tight Cauchy [0,1]-filter. Our main result states the equivalence between completeness of any fuzzy metric space (X,M,*) and completeness of the induced Hutton [0,1]-quasi-uniformity UM. Also it is proved that the Hutton [0,1]-quasi-uniform space (X,UM) has, in this context, a kind of completion that is unique up to uniform isomorphism. The obtained results come from an appropriate definition of Cauchy L-filter (where L stands for a complete lattice, with additional properties).