**8. Conclusion**

It has been said earlier that *L*-group theory developed in [10, 25–34] is a very rich generalization of classical group theory. For if we replace the lattice *L* by the closed unit interval *L*, then we retrieve fuzzy group theory. Moreover, if we replace *L* by the two elements set {0,1}, we obtain results of classical group theory as simple corollaries of corresponding results of *L*-group theory.

In view of this development we suggest the researchers, working in the areas of other fuzzy algebraic structures to shift their studies to lattice valued fuzzy sets (*L*-subsets). Also, in order to obtain consistency, the parent structure of classical algebra should be replaced by the corresponding fuzzy algebraic structures.

For those who are involved in active research in these areas, we propose here few research problems: The formation of quotient structure in fuzzy algebraic structures has been a problem child since its very inception. Its proper formulation is still awaited. We invite the researchers to construct a quotient of *L*-group *μ* by a normal *L*-subgroup in *μ* in the sense of Wu [22]. The second problem which is likely to be

*Development of* L*-Group Theory DOI: http://dx.doi.org/10.5772/intechopen.110387*

tackled more easily is related to nilpotent *L*-subgroups of an *L*-group which is discussed in the present work, that is, the investigation of nilpotency by upper central series. The upper central series is not yet formulated in the theory of *L*-subgroups.

Finally, to mention the further richness of this generalization, we emphasize that here we study the group theoretic properties of posets of subgroups of a group or in particular chains of subgroups of a group rather than properties of a single subgroup. This way, *L*-group theory provides us a new language and a new tool for the study of the classical group theory. The classical group theory has been founded on abstract sets and therefore the language used for its development is formal set theory. On the other hand, *L*-group theory expresses itself through the language of functions. The functions which are lattice valued. Therefore the approach adopted in the studies of L-group theory can be looked upon as a modernization of the approach of classical group theory.
