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

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