Distributivity in lattices of fuzzy subgroups☆

A new equivalence relation to classify the fuzzy subgroups of finite groups

“…The fuzzy (normal) subgroups of G can be classified up to some natural equivalence relations on F(G). One of them (used in [16][17][18][19][20][21][22][23][24][25], as well as in [27]) is defined by…”

“…The number h(G) has been also computed in several particular situations for dihedral groups and symmetric groups in [18] and [20], respectively. A complete study of finding the number h (G) for all above classes of finite groups can be found in [22].…”

“…In Theorem 3 of [18] we have proved that the number h(D 2p m ) of all distinct fuzzy subgroups relative to ∼ of the dihedral group D 2p m (where p is prime and m ∈ N * ) is…”

“…It can be made with respect to some natural equivalence relations on the fuzzy subgroup lattices, as the Murali's equivalence relation used in [7][8][9][10][11], ∼ used in [16][17][18][19][20][21][22][23][24][25] and [27], or ≈ used in this paper. Obviously, other such relations can be introduced and investigated.…”

“…The explicit formula obtained for the first class leads in [16] to an important combinatorial result: a precise expression of the well-known central Delannoy numbers in an arbitrary dimension. Next, the study has been extended to some remarkable classes of nonabelian groups: dihedral groups, symmetric groups, finite p-groups having a cyclic maximal subgroup and hamiltonian groups (see [18,20,21] and [25], respectively). The same problems were also investigated for the fuzzy normal subgroups of a finite group (see [22]).…”