A STUDY ON INTUITIONISTIC FUZZY IDEALS OF SEMIGROUPS
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ABSTRACT
In this project, Crispness, vagueness and fussiness were discussed, as well as fuzzy set theory. Some basic definitions on fuzzy set which includes membership functions were also discussed. Also fuzzy set operations which includes: Union of fuzzy sets, intersection of fuzzy sets, compliment, product, equality, and so on were explained also. Similarly, some definitions on group, semigroup were also discussed which includes subsemigroup, regular semigroup, left and right ideals, bi-ideals, two sided ideals.
TABLE OF CONTENTS
DECLARATION.. i
CERTIFICATION.. ii
DEDICATION.. iii
ACKNOWLEDGEMENTS. iv
ABSTRACT. v
CHAPTER ONE. 1
INTRODUCTION.. 1
1.1 Uncertainty. 1
1.2 AIM AND OBJECTIVES. 2
1.3 MOTIVATION.. 2
1.4 ORGANISATION OF THE PROJECT. 2
CHAPTER TWO.. 4
LITERATURE REVIEW.. 4
Crispness, Vagueness, Fuzziness. 4
3.1 Definition of Set 10
3.2 Definition of Binary Operation. 10
3.3 Definition of Groupoid. 10
3.4 Definition of Group. 10
3.5 Definition of Semigroup. 11
3.6 Definition of Subsemigroup. 12
3.7 Definition of Left Ideal and Right Ideal 13
3.8 Definition of Homomorphism.. 13
Monogenic Semigroups. 13
3.9 Definition of Monogenic Subsemigroup. 14
3.10 Definition of Periodic. 14
3.11 Definition of Idempotent Element 14
3.12 Definition of Membership Function. 15
3.13 The magnitude of fuzzy set 15
3.14 Definition of Fuzzy cardinality. 16
3.15 Definition of Subset of fuzzy set 16
3.16 Definition of Complement of fuzzy set 16
3.17 Definition of Union of fuzzy sets. 17
3.18 Definition of Intersection of fuzzy sets. 18
CHAPTER FOUR. 19
RESULT AND DISCUSSION.. 19
CHAPTER FIVE. 28
SUMMARY, CONCLUTION AND RECOMMENDATION.. 28
5.1 Summary. 28
5.2 Conclusion. 28
5.3 Recommendation. 28
REFERENCES. 29
CHAPTER ONE
INTRODUCTION
The important concept of fuzzy set has been introduced by (Zadeh, 1965). Since then many papers on Fuzzy sets appeared showing its importance in different fields and mathematics.(Rosenfeld, 1971) introduced the concept of fuzzy subgroups. (Kuroki, 1981) defined the fuzzy left and right ideals in a semigroup, the idea of fuzzy bi-ideals in a semigroup has been introduced. (Kehayopulu and Tsingelis, 2002) introduced the notation of fuzzy bi-ideals in po-semigroups (ordered semigroup). In this project different properties of fuzzy ideals and its extension in an interior ideal in semigroup have been investigated and in this connection different types of regulations in a semigroup have also been considered. Fuzzy concept is an appropriate methodology for investigating a number of problems characterized by unreliable data, imprecise measures, ambiguous language and unclean decision rules over nearly the past three decades, fuzzy concept has been advanced as a formal means of handling the implicit imprecision in a wide range of problems, e.g. in industrial control, military operations, economics, engineering, medicine, reliability, pattern recognition and classification.
1.1 Uncertainty
Fuzzy concepts can generate uncertainty because they are imprecise (especially if they are refer to a process in motion or a process of transformation where something is “in the process of turning into something else” In that case, they do not provide a clear orientation for action or decision-making (what does really mean or imply?”); reducing fuzziness, perhaps by applying fuzzy logic, would generate more certainty. However, this is not necessarily always so. A concept, even though it is not fuzzy at all, and even though it is very exact, could equally well failto enquire the meaning of something adequately. This is a concept that can be very precise and exact but not- or insufficiently- applicable or relevant to the situation to which it refers. In this sense, a definition can be “very precise”, but “miss the point” altogether.
A fuzzy concept may indeed provide more security, because it provides a meaning for something when an exact concept is unavailable – which is better than not being able to denote it at all. A concept such as God, although not easily definable, for instant can provide security to the believer.
1.2 AIM AND OBJECTIVES
The aim of this project is to study intuitionistic fuzzy ideals of semigroup. The aim is achieved through the following objectives:
- To investigate intuitionistic fuzzy left (right) ideal of a semigroup.
- To investigate intuitionistic fuzzy bi-ideal of a semigroup.
1.3 MOTIVATION
The fact that the concept takes into account all phenomena is the physical universe have a degree of inherent uncertainty gingered me to work on it, the study of intuitionistic fuzzy ideals of semigroup has a lot of application as listed above, the solution and many important problems in science and engineering e.g. in decision- making, in particular it enables me to give a precise mathematical description of what are the concept. The above mention application, interest me to undertake the study of intuitionistic fuzzy ideals of semigroup.
1.4 ORGANISATION OF THE PROJECT
The project consists of four other chapters apart from this introductory chapter. The outline of the remaining chapters are as follows;
In Chapter 2, we present a survey of the necessary and relevant literature for fuzzy set, and fuzzy ideals of semigroup in particular. In chapter 3, we will discuss the fundamentals of fuzzy set and semigroup. In Chapter 4, Result and discussion. In Chapter 5, we give a summary and conclusion of the results obtained in this research.
All notations issued in this work are either standard or explained in the text.
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