1.1. History and Development Summary of the Thesis

Transcription

1 CHPTER 1 INTRODUCTION 1.1. History and Development 1.2. Summary of the Thesis

2 1.1. History and Development The crisp set is defined in such a way as to dichotomize the elements in some given universe of discourse into two groups: members, which are certainly belong in the set and non-members, those that certainly do not belong in the set. sharp unambiguous distinction exists between the members and non-members of the set. However, many classification concepts we commonly employ and express in natural language describe sets that do not exhibit this characteristic. Examples are the set of tall people, expensive cars, highly contagious diseases, numbers much greater than one, modest profits, sunny days etc. We perceive these sets as having imprecise boundaries that facilitate gradual transitions from membership to non-membership and vice versa.there was no proper frame work to describe study and formulate such problems in classical mathematics until the emergence of fuzzy set theoretical approach. L.. Zadeh [76] in 1965 introduced the notion of fuzzy set to describe vagueness mathematically in its very abstractness and tried to solve such problems by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller membership grade. These membership grades are very often represented by real numbers ranging in the closed interval between 0 and 1. The grade of membership of an element in a given fuzzy set can be viewed as a degree of our certainty or degree of truth of the vague notion. For instance, a fuzzy set representing our concept of sunny day might assign a degree of membership of 1 to a cloud cover of 0%,.8 to a cloud cover of 20%,.4 to a cloud cover of 30%, and 0 to a cloud cover of 80% and above. These grades signify the degree to which each percentage of cloud cover approximates our subjective

3 Chapter 1 concept of sunny, and the set itself models the semantic flexibility inherent in such a common linguistic term. Because full membership and full non-membership in the fuzzy set can still be indicated by the values of 1 and 0, respectively, we can consider the concept of a crisp set to be a restricted case of the fuzzy set with the membership function as a characteristic function. But in a fuzzy set, the transition from membership to non-membership is gradual rather than abrupt. Thus fuzzy set theory can be considered as a generalization of classical set theory. Because of this generalization, the theory of fuzzy sets has a wider scope of applicability than classical set theory in solving various problems. Extensive applications of the fuzzy set theory have been found in various fields such as Mathematics, Computer Science, rtificial Intelligences, Medical Sciences,Economics, Statistics, Neural networks etc. fuzzy set in a universe X is defined by membership function that maps X to the interval [0, 1] and therefore implies a linear, i.e. total ordering of the elements of X, one could argue that this makes them inadequate to deal with incomparable information. possible solution, however, was already implicit in Zadeh s seminal paper [76] in a footnote, he mentioned that in a more general setting, the range of the membership function can be taken to be a suitable partially ordered set P. In 1967, Goguen [32] formally introduced the notion of an L- fuzzy set with a membership function taking values in a lattice L. nother generalizations of fuzzy sets called interval valued fuzzy sets, apparently first studied by Sambue [61] who called them φ -flou functions, serve to capture a feature of uncertainty with respect to the assignment of membership degrees by intervals in [0, 1] because assigning an exact number to an expert s opinion is too restrictive, and that the assignment of an interval of values is more realistic. Finally intuitionistic fuzzy sets (IFS) were introduced in 1983 by K. T. tnassov [12] as generalization of fuzzy sets. IFS theory basically defies the 2 PPLICTION OF FUZZY SETS IN LTTICE THEORY

4 Introduction claim that from the fact that an element x belongs to a given degree say µ ( x ) to a fuzzy set, naturally follows that x should not belong to to the extent 1- µ ( x ), an assertion implicit in the concept of a fuzzy set. On the contrary, IFSs assign to each element of the universe both a degree of membership µ ( x ) and one of non - membership ν ( x ) such that µ ( x) + ν ( x) 1, thus relaxing the enforced duality ν ( x ) = 1- µ ( x) from fuzzy set theory. For each intuitionistic fuzzy set in X, we will call π (x) = 1 µ ( x) ν ( x) a hesitation margin or intuitionistic fuzzy index of x and it expresses lack of knowledge of whether x belongs to or not. It is obvious that 0 π (x) 1, for each x X. The application of intuitionistic fuzzy sets instead of fuzzy sets means the introduction of another degree of freedom into a set description. Such a generalization of fuzzy sets gives us an additional possibility to represent imperfect knowledge what leads to describing many real problems in a more adequate way. Basically intuitionistic fuzzy sets based models may be adequate in situations when we face human testimonies, opinions, etc. involving answers of the type yes, no, does not apply. Imagine, for instance, a voting procedure in which delegates have to express their feelings with respect to a number of proposals. It is obvious that while one can be in favour or in disfavour of a proposal to a certain extent, one can also abstain from the vote, an attitude inspired by, e.g., a lack of background or interest, or simply because no obvious arguments for or against the cause at stake have been raised. In such a situation, using only a [0, 1] valued degree expressing support for the proposal is arguably too committing, and we should be duly hesitant to classify him as a supporter or an opponent of the proposal. IFSs circumvent this problem by allowing one to address the positive and the negative side of an imprecise concept separately, and by not insisting that these assessments be exactly complementary. Intuitionistic fuzzy sets have been found to be very useful in diversely applied areas of science and technology. In fact there are situations where IFS PPLICTION OF FUZZY SETS IN LTTICE THEORY 3

5 Chapter 1 theory is more appropriate to deal with. To name only a few areas of applications, IFSs have been applied to logic programming, medical diagnosis, decision making problems and microelectronic fault analysis. mong various branches of pure and applied mathematics modern algebra was one of the first few subjects where the notion of intuitionistic fuzzy set was applied. In [20] R. Biswas defined intuitionistic fuzzy subgroups and their properties. It is demonstrated that the concepts of group theory can be used to form the building blocks of new theory of intuitionistic fuzzy subgroups. lso in the same paper the characterizations of intuitionistic fuzzy subgroups in terms of level subsets are provided. From 2001 to 2006 K. H. Kim et.al [45,46,47] applied the notion of intuitionistic fuzzy sets to semigroups and studied the notions of interior ideals, bi-ideals, Q- fuzzy semi prime ideals etc. B. Banerjee and D. K. Basent in [16] introduced intuitionistic fuzzy subrings and Ideals. Sum and product of intuitionistic fuzzy ideals are defined and prove that they are again intuitionistic fuzzy ideals. They also studied the properties of intuitionistic fuzzy ideals under ring homomorphism. fter this work D. K. Basent gave a detailed description of level subsets of an intuitionistic fuzzy sets in [17]. In the light of these developments, we discuss in this thesis, intuitionistic fuzzy sublattices and ideals in a lattice. Their characterization in terms of level subsets are provided and their homomorphic images under various conditions are studied. In 2005 K. Hur et.al [37, 40 ] studied in detail the notion of intuitionistic fuzzy ideals of a ring and established their characterization in terms of level subsets. Moreover they studied the lattice structure of intuitionistic fuzzy ideals of a ring and their modularity. W. Dudek [31] introduced and studied intuitionistic fuzzy h-ideals of hemirings and provided their characterization in terms of upper and lower level subsets. In this work, we introduced various operations between intuitionistic fuzzy ideals of a lattice and studied their properties under these operations. The notions of ideal of an intuitionistic fuzzy sublattice and residual of ideals are introduced and studied. 4 PPLICTION OF FUZZY SETS IN LTTICE THEORY

6 Introduction Relations are a suitable tool for describing correspondences between objects. Crisp relations like,, = etc. have served well in developing mathematical theories. The use of fuzzy relations originated from the observation that real-life objects can be related to each other to a certain degree. However, in real-life situations, a person may assume that a certain object is in relation R with another object B to a certain degree, but it is possible that he is not so sure about it. In other words, there may be a hesitation or uncertainty about the degree that is assigned to the relationship between and B. Fuzzy relations are able to model vagueness, however they cannot model uncertainty completely. Intuitionistic fuzzy relations defined by tnassov [13] give us a way to incorporate uncertainty in an additional degree. Intuitionistic fuzzy relations are intuitionistic fuzzy sets in a cartesian product of universe. P. Burillo and H. Bustince [ 23, 24] have introduced the intuitionistic fuzzy relations and their properties. One of the main concepts in relational calculus is the composition of two relations. They also studied the composition of intuitionistic fuzzy relations, and have analyzed the effect of tanassov s operators on the properties of the intuitionistic fuzzy relations, and established the conditions with which the reflexivity, symmetry and transitivity properties of these relations are maintained. In [29] G. Deschrijver and E.E. Kerre presented an intuitionistic fuzzy version of the triangular compositions and the variants of these compositions. Some properties like containment, convertibility, monotonic, interaction with union and intersection etc. of these compositions are also investigated. K. Hur et.al [38, 39] introduced the notion of intuitionistic fuzzy congruences on a semigroup and lattice and studied their lattice structure. lso they studied the relation between intuitionistic fuzzy ideals and intuitionistic fuzzy congruences and established an isomorphism between them. In this thesis, we thoroughly discussed the concepts of intuitionistic fuzzy equivalence and congruence relations by generalizing their usual definition. We established an isomorphism between lattice of intuitionistic fuzzy congruences and ideals with PPLICTION OF FUZZY SETS IN LTTICE THEORY 5

7 Chapter 1 the help of strong level subsets. lso we studied the concept of intuitionistic fuzzy quotient lattice and defined the quotient of an intuitionistic fuzzy lattice relative to ordinary congruence relation and proved the intuitionistic fuzzy version of fundamental homomorphism theorem. In 1975 Zadeh [77] made an extension of the concept of fuzzy set by an interval-valued fuzzy set. In [52] P. P. Ming and L. Y. Ming introduced the concept of quasi-coincidence of a fuzzy point with a fuzzy subset. Based on quasi-coincidence,s. K. Bhakat and P. Das [19] introduced a new type fuzzy subring of ring called an (, q) -fuzzy subring. Recently D. S. Lee and C. H. Park [50] introduced the notion of interval valued (, q) - fuzzy subring and ideal, and investigated their properties. X. Ma, J. Zhan and Y. B. Jun [41] introduced and study interval valued (, q) -fuzzy ideals of pseudo-mv algebra. Using the belongs to relation and quasi-coincidence with relation between intuitionistic fuzzy points and intuitionistic fuzzy sets, Y. B. Jun [43] introduced the concept of ( φ, ψ )- intuitionistic fuzzy subgroup. Later H. Hedayati [35] introduced interval valued intuitionistic fuzzy substructures in semirings with respect to t- norm T and s- norm S and their characterization based on level subsets. Motivated by these works, in this thesis we initiate a study of interval valued (, q) -intuitionistic fuzzy sublattices and ideals of a lattice. Rough set theory has been proposed by Pawlak [56] is an another tool for dealing with the vagueness and granularity in information systems. It is a method to conceptualize, organize and analyze various types of data in data mining. Equivalence relation is a key notion in Pawlak s rough set model. The equivalence classes are employed to construct the lower and upper approximations. Replacing equivalence relations by arbitary relations, or replacing equivalence classes by coverings or neighborhood systems a variety of generalizations of Pawlak s rough set model have been made. Theories of rough sets and fuzzy sets are related but distinct and complementary theories. Integration of these two theories has been made in recent years. The fuzzy generalization of 6 PPLICTION OF FUZZY SETS IN LTTICE THEORY

8 Introduction rough sets is typical example. Dubois and Prade [30] proposed the notions of rough fuzzy sets and fuzzy rough sets Rough set theory has found practical application in many areas such as knowledge discovery, machine learning, data analysis etc. lso there is connection between rough sets and algebraic systems. Some authors for example Biswas and Nanda [21] introduced the notion of rough subgroups. Kuroki [48] introduced rough ideals in a semigroup, Kuroki and Wang [49] studied properties of lower and upper approximation with respect to normal subgroups. B Davaaz [28] introduced rough subring and ideals with respect to an ideal of a ring. In this work, we studied the rough set approximation of a set with respect to an ideal of a lattice and introduced rough sublattice (ideals and prime ideals). lso we introduced the concepts of rough fuzzy sublattice, rough fuzzy ideal and prime ideal in a lattice and their properties are investigated Summary of the Thesis The whole thesis is divided into seven chapters and each chapter is further subdivided into a number of sections. In chapter 1, a brief history of the theory and development of fuzzy sets, intuitionistic fuzzy sets and rough sets are provided. This chapter also presents a summary of the research work carried out in the thesis. Chapter 2 emphasizes on basic definitions, results and properties of lattices, fuzzy sets and intuitionistic fuzzy sets, which serve as a prerequisite for the research work done in the thesis. Chapter 3 starts with the concepts intuitionistic fuzzy sublattice (IFL) and intuitionistic fuzzy ideal (IFI). Firstly we discussed some properties of these concepts. s in the case of fuzzy setting, in intuitionistic fuzzy setting also the notion of level subset is found to be an important tool for establishing various lattice theoretic properties. Here we have given characterizations of intuitionistic fuzzy sublattices and intuitionistic fuzzy ideals in terms of its level subsets. In the sequel, intuitionistic fuzzy convex sublattice is defined and its level subset characterization is provided. lso properties of intuitionistic fuzzy ideals under PPLICTION OF FUZZY SETS IN LTTICE THEORY 7

9 Chapter 1 lattice homomorphism are studied. We defined f-invariant class of IFIs and established a correspondence between the IFIs of a lattice which are f-invariant and IFIs of its homomorphic image. This chapter is concluded with the result that the homomorphism image of intuitionistic fuzzy prime ideal with supremum and infimum property or f-invariant property is again an intuitionistic fuzzy prime ideal. In chapter 4, some operations on intuitionistic fuzzy sets are introduced and based on that properties of intuitionistic fuzzy ideals under these operations are studied. Different characterizations of intuitionistic fuzzy sublattices and ideals are given in terms of these operations. Using these operations for certain subclass of the class of intuitionistic fuzzy ideals, the join of two intuitionistic fuzzy ideals is constructed. This leads to the formation of various lattices and sublattices of intuitionistic fuzzy ideals. In the next section we defined the intuitionistic fuzzy ideal of an intuitionistic fuzzy lattice and studied their properties under the operations defined in the first section. Moreover the concept of quotient (or residual) of ideals of an intuitionistic fuzzy lattice is introduced and proved that it is again an ideal of the intuitionistic fuzzy lattice. Chapter 5 is devoted to a detailed study of intuitionistic fuzzy equivalence and congruence relations including their lattice structures. Firstly we slightly modify the present definitions of equivalence and congruence relations and referred to it as (t, k) equivalence and congruence relations. We studied certain properties of these relations and the characterization of (t, k) equivalence and congruence in terms of their level subsets and proved that the class of all intuitionistic fuzzy congruence relations forms a distributive lattice. We established the relationship between intuitionistic fuzzy congruence and intuitionistic fuzzy ideals in a distributive lattice and using the notion of upper and lower level subsets we proved that in a generalized Boolean algebra their lattices are isomorphic. In this chapter we also defined an intuitionistic fuzzy partition called (t, k)- partition and a canonical one-to-one correspondence between (t, k)-partitions and (t, k)-equivalences is established. Moreover it is verified that in a lattice, (t, k)- 8 PPLICTION OF FUZZY SETS IN LTTICE THEORY

10 Introduction partitions with respect to a (t, k)-congruence constitute a lattice. This lattice is called the quotient lattice with respect to a given (t, k) - congruence relation. We also studied the concept of quotient of an IFL relative to ordinary congruence relation and proved the intuitionistic fuzzy version of fundamental homomorphism theorem. systematic development of the theory of interval valued intuitionistic fuzzy sublattices is carried out in chapter 6. Firstly we defined interval valued intuitionistic fuzzy sublattices and ideals and their properties in terms of level subsets. lso we gave some properties of interval valued intuitionistic fuzzy ideals under lattice homomorphism. Further based on the concepts of belongingness and quasi-coincidence of an interval valued intuitionistic fuzzy point with an interval valued intuitionistic fuzzy set, a new type of intuitionistic fuzzy sublattice (ideal) is provided and called(, q) intuitionistic fuzzy sublattice (ideal). Properties of these new classes of sublattices and ideals are studied and their characterizations in terms of level subsets are established. Finally, in chapter 7 we discussed about rough sets in lattice theory. In a distributive lattice there exist a one to one correspondence between ideals and congruence relations. Using this notion we defined the rough set approximation of a set with respect to an ideal of a lattice and introduced rough sublattice (ideals and prime ideals). lso some properties of the lower and upper approximations of a set in a lattice are studied. Moreover we introduced the concepts of rough fuzzy sublattices (ideal and prime ideal) in a lattice. Further we discussed the homomorphic images of upper and lower rough ideals (prime ideals). The developments achieved so far allows as to give a rough approximation of a intuitionistic fuzzy set and which introduces rough intuitionistic fuzzy sublattices, ideals etc. detailed study of these concepts including intuitionistic fuzzy rough sets, intuitionistic fuzzy rough sublattices (ideals) are provided. PPLICTION OF FUZZY SETS IN LTTICE THEORY 9