Fuzzy posets with fuzzy order applied to fuzzy ordered groups

Abstract

In the framework of lattice valued structures we investigate fuzzy sub-posets of a given poset, in which the underlying set or$\setminus$and the order are fuzzy and the reflexivity is specially defined. We introduce and investigate particular fuzzy sub-posets, e.g, fuzzy up--sets and down--sets, fuzzy convex sub--posets, fuzzy intervals etc. We describe the structure of the lattice of all fuzzy orders contained in a given crisp ordering. Then we apply these in defining a fuzzy ordered subgroup of an ordered group. Main features of fuzzy ordered subgroups are introduced and investigated like fuzzy positive and negative cone and fuzzy convex subgroups.