The lattices of fuzzy subgroups and fuzzy normal subgroups

Distributivity in lattices of fuzzy subgroups☆

“…This is closely connected to that of fuzzy congruence, studied in [2,4,9,11], or [13]. The set FLðGÞ of all fuzzy subgroups of a given group G forms a complete lattice under the usual ordering of fuzzy set inclusion (see [1]). Several remarkable sublattices of FLðGÞ have been investigated in [1][2][3].…”

“…The set FLðGÞ of all fuzzy subgroups of a given group G forms a complete lattice under the usual ordering of fuzzy set inclusion (see [1]). Several remarkable sublattices of FLðGÞ have been investigated in [1][2][3]. From these, we mention the lattice FNðGÞ of all fuzzy normal subgroups which is modular, as show [3], or [6].…”

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

“…Ajmal and Thomas initiated such studies in the realm of fuzzy group theory and did extensive work in this field [1][2][3][4][5][6][7][8][9]. It was shown in [2] that L , the set of all fuzzy subgroups of a group, constitutes a lattice under the ordering of fuzzy set inclusion.…”

“…Ajmal and Thomas initiated such studies in the realm of fuzzy group theory and did extensive work in this field [1][2][3][4][5][6][7][8][9]. It was shown in [2] that L , the set of all fuzzy subgroups of a group, constitutes a lattice under the ordering of fuzzy set inclusion. It was also demonstrated in [2] that L t , the set of all fuzzy subgroups of a group G with the same tip 't' forms a complete sublattice of the lattice L. Various other sublattices of the lattice L have been constructed by Ajmal and Thomas in [2,5,8].…”

When do L-fuzzy ideals of a ring generate a distributive lattice?

“…Liu [3] introduced the fuzzy subring, etc. Standing upon these achievements, many researchers explored the lattice theoretical properties of these structures, such as, modularity of the lattice of the fuzzy normal subgroups was established in a systematic and step wise manner in [4][5][6][7][8][9][10]. By supposing the value "t " at the additive identity of a given ring, the fact that the set of fuzzy ideals with sup property forms a sublattice of the lattice of fuzzy ideals was proved by Ajimal and Thomas [11].…”