Nilpotency and Theory ofL-Subgroups of anL-Group

Maximal and Frattini L-Subgroups of an L-Group

“…Here we also mention that the dual of completely distributive law is valid in a completely distributive lattice whereas the infinitely meet and join distributive laws are independent from each other. Next, we recall the following from [1,2,5,8,13,17]:…”

“…Remark 1. In order to study the level subsets of maximal L-subgroups of an L-group, we recall the notion of jointly supstar L-subsets from [5]. It is worthwile to mention here that this notion is a generalization of the noion of sup-property and lends itself easily for applications Proposition 3.3.…”

“…However, in the studies of fuzzy groups such an effort is lacking. Recently, a systematic study of L-subgroups (lattice valued fuzzy subgroups) of an L-group has been carried out in a series of papers [3,4,5,6,7] wherein a number of concepts of classical group theory have been extended to L-setting specially keeping in view their compatibility. The present paper is an endeavour to develop and study the maximal L-subgroup of an L-group along with its application to the notion of Frattini subgroups.…”

Subnormality and Theory of L-subgroups

“…In this paper, we present a systematic and successful development of L-group theory which has been made in papers [2][3][4][5][6][7]10]. The important concepts of nilpoent L-subgroups, solvable L-subgroups have been introduced and studied and their inter-relationship is established in [3,6]. Moreover, the concept of normalizer of an L-subgroup in an L-group is formulated in [2].…”

“…Let L be an upper well ordered chain and η, θ ∈ L µ . Then, η and θ are jointly supstar if and only if η and θ possess sup-property.Below, we recall the definition of descending central chain of an L-subgroup η of µ from[3]:Take γ 0 (η) = η, γ 1 (η) = [γ 0 (η), η].And in general, for each i, we defineγ i (η) = [γ i−1 (η), η]. Proposition 9.…”