Some constructions of the join of fuzzy subgroups and certain lattices of fuzzy subgroups with sup property

Abstract

In this paper, some new lattices of fuzzy substructures are constructed. For a given fuzzy set μ in a group G, a fuzzy subgroup S ( μ ) generated by μ is defined which helps to establish that the set L s of all fuzzy subgroups with sup property constitutes a lattice. Consequently, many other sublattices of the lattice L of all fuzzy subgroups of G like L s t , L s n , L s n t , etc. are also obtained. The notion of infimum is used to construct a fuzzy subgroup i ( μ ) generated by a given fuzzy set μ , in contrast to the usual practice of using supremum. In the process a new fuzzy subgroup i ( μ ) ∗ is defined which we shall call a shadow fuzzy subgroup of μ . It is established that if μ has inf property, then i ( μ ) ∗ also has this property.