Fuzzy groups and collections of subgroups

Abstract

The aim of the paper is to investigate fuzzy subgroups of a group from a general point of view, using collections of ordinary subgroups. Fuzzy subgroups are taken to be special mappings from a group to a partially ordered set, which can also be a lattice, and particularly the unit interval [0,1]. As it is known, every fuzzy subgroup uniquely determines a poset of level subgroups of the same group. Properties of that poset are investigated. On the other hand, it is proved that every collection of subgroups can be used for the construction of a suitable fuzzy subgroup of a group. For that purpose, particular fuzzy completions of collections of subsets are introduced. It turns out that fuzzy subgroup is an intrinsic notion, expressible in terms of collections of subgroups under set-theoretic relations and operations.