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.