Contributions to L-Fuzzy Soft Nearrings

Abstract

To solve complicated problems in economics, engineering and environment sciences, the classical methods cannot be successfully used due to various uncertainties for those problems. There are theories viz, theory of probability, theory of fuzzy sets , theory of intuitionistic fuzzy sets , theory of vague sets, theory of interval mathematics and theory of rough sets which can be considered as mathematical tools for dealing with uncertainties but all these theories have their inherent difficulties. Molodtsov [10] initiated the concept of soft set as mathematical tool for dealing with uncertainties which is free from above difficulties. Maji et al. [16] defined some operations on soft sets. Ali et al.introduced several new operations of soft sets. The theory has also seen a wide-ranging applications in the mean of algebraic structures such as groups, semirings, rings, BCK/BCI-algebras , nearrings and soft substructures and union soft substructures. Abstract: The fundamental concept of fuzzy set was introduced by Zadeh [27] in 1965. Rosenfeld inspired the fuzzification of algebraic structures and introduced the notions of fuzzy subgroups. Das [7] characterized fuzzy subgroups by their level subgroups. W. Liu [8] studied fuzzy ideals of rings.Abou-Zaid introduced the notion of a fuzzy subnearring and studied fuzzy ideals of a nearring.The concept of fuzzy subnearring and fuzzy ideal was discussed further by many researchers. Davvaz for a complete lattice L, introduced interval-valued L-fuzzy ideal (prime ideal) of a nearring which is an extended notion of a fuzzy ideal (prime ideal) of a nearring. This dissertation is devoted to the discussion of algebraic structures of L-fuzzy soft sets and basic concepts of lattices and L-fuzzy sets. This dissertation consists of three chapters. Chapter one consists of some basic definitions and examples of Nearrings and basic concept of soft sets, fuzzy sets and L-fuzzy soft sets. In Chapter two, We initiated the study of L-fuzzy soft ideals along with L-fuzzy soft nearrings. In Chapter three, We introduced L-fuzzy soft prime and semiprime ideals . Moreover, We have done the characterization of nearrings by the properties of their L-fuzzy soft ideals. We have characterized those nearrings for which each L-fuzzy soft ideal is Prime and also those nearrings for which each L-fuzzy soft ideal is idempotent