A Study of Fuzzy Subgroups and Its Level Subgroups
Chapter One
Aim and Objectives of the Dissertation
The aim of this research work is to study an algebraic structure called -fuzzy subgroup and its level -subgroup. The objectives are to:
- Review some of the fundamental works done in M-fuzzy subgroup theory;
- Provide some new or alternative methods of proving some existing theorems in M- fuzzy subgroup theory;
- Provide proofs of some theorems on M-fuzzy subgroup theory which, to the very best of our knowledge, do not exist in the literature;
- Obtain independent proof of several theorems on Level M- subgroups of M-fuzzy subgroups.
CHAPTER TWO
LITERATURE REVIEW
Introduction
The study of fuzzy groups was started firstly by Rosenfeld (1971). Rosenfeld used the min operation to define his fuzzy groups and showed that many results in group theory can be extended in an elementary manner to develop the theory of fuzzy group.
Das (1981) obtain a similar characterization of all fuzzy subgroups of finite cyclic groups and study what are called “level subgroups” of a fuzzy subgroup in the first part of the paper. These level subgroups in turn play an important role in the above characterization.
Jacobson (1951) introduced the concept of M-group, M-subgroup.
Gu et al (1994) studied the theory of fuzzy groups and developed the concept of M-fuzzy groups.
Kundu (1998) presented a counter example to recent result on truly closed level subgroup of a fuzzy group and proves a correct form of that result.
Mordeson et al (2005) presented a book on fuzzy group theory, where they define the notion of a fuzzy subgroup and examine its properties; they also introduced some operations on a fuzzy subset of a group G in terms of the group operation.
Solairaju and Nagarajan (2010) discussed some structure properties of M-fuzzy groups.
Muthuraj et al (2010) introduced the concept of M-homomorphism and M-anti homomorphism of an M-fuzzy subgroups.
Sundararajan and Muthuraj (2011) introduced the concept of an anti M-fuzzy subgroup of an M-group and lower level subset of an anti M-fuzzy subgroup and discussed some of its properties.
Mourad and Massa‟deh (2012) studied the theory of fuzzy subgroups and discussed some concepts such as fuzzy subgroups with operator, normal fuzzy subgroups with operator, homomorphism with operator, etc, while some elementary properties were discussed, such as intersection operation, the image and inverse image of fuzzy subgroups with operator.
Subramanian et al (2012) gave an independent proof of several theorems on M- fuzzy groups. They discussed M- fuzzy groups and investigated some of their structures on the concept of M- fuzzy group family.
CHAPTER THREE
FUNDAMENTALS OF FUZZY SUBGROUP
Introduction
Let G denotes an arbitrary group with a multiplicative binary operation and identity e. In order to define the notion of a fuzzy subgroup and to examine its properties, some operations on a fuzzy subset of a group G in terms of the group operation are introduced. Also, the fundamental definitions that will be used in the sequel are sited.
Concept of Fuzzy Subgroup Definition 3.2.1: (Rosenfeld, 1971)
Let be a group. A fuzzy subset of a group is called a fuzzy subgroup of the group
CHAPTER FOUR
FUZZY SUBGROUP AND ITS LEVEL – SUBGROUP
Introduction
In this section, we discuss about M-fuzzy group and investigate some of their structures on the concept of M-fuzzy subgroup and its level M-subgroups. We provide some independent proof of several theorems on M-fuzzy subgroup and its level M-subgroups.
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS
This chapter gives the summary and conclusion of the whole study and supplies recommendations for further research on the concept of M-fuzzy subgroup.
Summary
This research studied the concept of -fuzzy subgroup and its level -subgroups. After comprehensive review of some literatures on fuzzy group and -fuzzy group, new propositions were established.
In chapter one, a general introduction of the dissertation, which includes the early history of fuzzy set, algebraic structures of fuzzy sets, aims and objective of the study and organization of the dissertation were presented.
Chapter two presented literature review of the development of algebraic structures of fuzzy groups and -fuzzy group.
Chapter three presents definition of fuzzy subgroup as definition by Rosenfeld (1971). The notions of normal fuzzy subgroups and order of fuzzy subgroups were also introduced. Algebraic operations and properties of fuzzy subgroups, normal fuzzy subgroups and order of fuzzy subgroups were also presented. This chapter forms a stepping stone to the study of fuzzy subgroup
In chapter four, the concept of -fuzzy subgroup and its level -subgroups is proposed. Algebraic operations and properties of -fuzzy subgroup and it level -subgroup were also presented.
Conclusion
Fuzzy subgroups can be considered as a good development in the field of pure mathematics and engineering mathematics. The aim of this work is to study the fundamentals structure of fuzzy subgroup and extend this study to -fuzzy subgroups and its level -subgroups.
Recommendations
We wish this topic “ -fuzzy subgroup” could be extended and applied to other areas where fuzzy subgroups have been applied especially in Normal fuzzy subgroup, Order of fuzzy subgroup and many other fields of mathematics and engineering.
REFERENCES
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- Zadeh, L. A. (1965). Fuzzy Sets. Inform. Control, 8, 338-353. Rosenfeld, A. (1971). Fuzzy groups,. J. Math. Anal. Appl., 35, 512-517.
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- Gu, W.X., Li, S.Y. and Chen, D.G. (1994). Fuzzy groups with operators, fuzzy sets and system, 66, 363-371.
- Kundu, S. (1998). The correct form of a recent result on level-subgroups of a fuzzy group. Fuzzy Sets and Systems, 97, 2, 261-263.
- Mordeson, J. N., Bhutani, K. R. and Rosenfeld, A. (2005). Fuzzy group Theory, Studies in fuzziness and soft computing, 182
- Sundararajan, P., Palaniappan, N. and Muthuraj, R. (2009). Anti M-fuzzy subgroup and anti M-fuzzy sub-bigroup of an M-group. Antratica J. Math, 6 (1), 33-37.
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