A STUDY ON INTUITIONISTIC FUZZY IDEALS OF SEMIGROUPS

 

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.

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.

The aim of this project is to study intuitionistic fuzzy ideals of semigroup. The aim is achieved through the following objectives:

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.

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.