**7. Discussion and analysis**

The subject matter discussed in this work presents a systematic and compatible theory of *L*-subgroups(fuzzy subgroups). In this work, firstly we replace ordinary fuzzy subsets by lattice valued fuzzy subsets. This puts our work beyond the purview of Tom Head's metathoerem [35]. This is contrary to the situation of fuzzy group theory, where the results which are obtained, become simple instances of application of metatheorem. Secondly, throughout our work, we have replaced the parent structure of an ordinary group by an *L*-subgroup which is called an *L*-group. In order to carry out our studies successfully, we need to use the notion of normality of a fuzzy subgroup in a fuzzy group defined by Wu [22] rather than the notion of normality of a fuzzy subgroup in a group introduced by Liu [21]. Without following this approach, we can not construct various types of series of *L*-subgroups, such as the subinvariant series, normal closure series, subnormal series or derived series: even the results of classical group theory such as a characteristic subgroup of a normal subgroup of a group is a normal subgroup of the given group or the property of transitivity of characteristic subgroup can not be formulated and extended to the *L*-setting. By following Wu's normality, we have successfully extended the above results to the *L*-setting. Moreover, we have proved that the commutator *L*-subgroup of an *L*-group is a characteristic *L*-subgroup. Furthermore, we proved that a commutator *L*-subgroup of a supersolvable *L*-subgroup is nilpotent. Thus an application of Wu's normality establishes a very high degree of compatibility among various notions studied in *L*-group theory.

Another deviation in our work from the work of earlier researchers in fuzzy group theory is the construction of the trivial *L*-fuzzy subgroup. This is a proper fuzzification of the notion of trivial subgroup of an ordinary group and it makes possible a successful study of various types of series, arising while dealing with the notions of nilpotency, solvability, supersolvability, subnormality etc.. While defining

a trivial *L*-subgroup, the notion of infimum of an *L*-set comes into play. So is the case at various other places in our investigations where the infimum of an *L*-subgroup plays a significant role. This is due to the fact that we are carrying out our investigations in an *L*-group rather than an ordinary group and we use Wu's normality in place of Liu's normality. There are several peculiarities of *L*-setting which involves the notion of infimum of *L*-subgroups. For example Theorem 1.60 states that the infimum of the set product of two *L*-subsets lies in between the meet and the join of the infimums of given *L*-subsets. This result has been used in Theorem 1.81 wherein we have obtained a necessary and sufficient condition for the set product of two trivial *L*-subgroups of an *L*-group to be a trivial *L*-subgroup. This result is instrumental in establishing a sufficient condition for the set product of two nilpotent *L*-subgroups to be nilpotent. Moreover, the role of infimum (tail) of *L*-subgroups is displayed while dealing with homomorphic and inverse homomorphic images of commutator and solvable *L*-subgroups. To show that the the concepts of nilpotency, solvability and supersolvability are closed under the formation of subgroups, the notion of infimum is again helpful. The infimums (tails) of all the members of almost all the series discussed in our work are identical with the tails of their respective trivial *L*-subgroups.

Last but not the least, another pleasing feature of our study is the formation of lattices of normal *L*-subgroups and characteristic *L*-subgroups of an *L*-group. In the process, the notion of sup-property has played a very significant role. We obtain a characterization of sup-property by using the notion of image set of the given *L*-subset. This characterization of sup-property forms the basis for our generalization. This gave rise to the notions of supstar family of *L*-subsets and jointly supstar *L*-subsets. The notion of image of an *L*-subset is intimately related with the above mentioned concepts. Theorem 1.36 shows that each member of supstar family possesses sup-property. Further, in Theorem 1.70, a relationship has been established between the image of *i*th members of descending central series of an *L*-subgroup and the image of the given *L*-subgroup. A similar relationship has been obtained in Theorem 1.84 and Theorem 1.120.

These concepts are subsequently used in the development of *L*-group theory. A co-ordinated approach between all these concepts paved a way for a successful development of *L*-group theory.
