Type-2 lattice-valued preorders: A common framework of lattice-valued preorders and various kinds of metrics

The aim of this paper is to introduce a concept of type-2 L-preorders for L being a complete residuated lattice. It can be considered as a common framework of L-preorders and hemimetrics, and also contains various kinds of fuzzy metrics, including Morsi fuzzy metrics, KM-fuzzy metrics and modular metrics, as natural examples. It is shown that the category of L-preordered sets can be reflectively and coreflectively embedded in that of type-2 L-preordered sets. A type-2 L-preorder can be supplied as different models for further study.