Fuzzy concepts in non associative and non commutative algebraic structures

Abstract

ian semigroups have been generalized in a class of non associative and non commutative algebraic structures  means to left almost semigroup. A new pseudo associative law: , has been established in [23] by introducing the braces on the left side of ternary commutative law: . This is called a left invertive law. A groupoid is said to be a left almost semigroup abbreviated as LA-semigroup if it satisfies the left invertive law . An LAsemigroup is a non associative and non commutative algebraic structure. It is a midway algebraic structure between a groupoid and an abelian semigroup with several interesting features. A number of important papers have been appeared since 1972, when the concept was introduced. For further information about this structure, one can refer 3, 42 and 43. In 19 resp. 8 a groupoid is called medial resp. paramedial if . In 3 an LA-semigroup is medial, but in general LA-semigroup needs not to be paramedial. Every LA-semigroup with left identity is paramedial by Protic et al 50 and also satisfies An LA-semigroup cannot contain the right identity, because an LA-semigroup with right identity becomes an abelian semigroup with identity means twosided identity. Many authors, for example Murdock 41, Henry 14 and Kazim et al 2 have generalized the concept of a group and have investigated the structural properties of these generalizations. The notion of left almost group is one among them, which though is a non associative and non commutative algebraic structure has interesting resemblance with an abelian group. Kamran 21 extended the notion of LA-semigroup to the left almost group abbreviated as LA-group. An LA-semigroup G is said to be a left almost group, if there exists left identity such that for all and for every there exists an element such that . From a commutative group we can always obtain an LA-group by defining for all . There are various concept of universal algebra, generalizing an associative ring as near-ring and semiring etc. A near-ring satisfies all axioms of an associative ring expect the commutativity of addition and one of the two distributive laws. The second type of such algebra is called semiring or halfring arise from a ring such that is supposed to be a semigroup rather than a commutative group. An LA-ring is another generalization of an associative ring, such that is assumed to be an LA-group rather than a commutative group and is assumed to be an LA-semigroup rather than a semigroup. Shah et al 56 initiated the concept of left almost ring of finitely nonzero functions, which is a generalization of commutative semigroup ring and investigated the some motivating results. By a left

almost ring it is a generalization of a ring, we mean a non-empty set R with at least two elements such that is an LA-group, is an LA-semigroup, both left and right distributive laws hold. For example, from a commutative ring , we can always obtain an LA-ring by defining for all is same as in the ring. Although the structure is non associative and non commutative, nevertheless, it possesses many interesting properties which we usually find in associative and commutative algebraic structures. Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical computer sciences, control engineering, information sciences, coding theory, topological spaces and the like. These provide sufficient motivation for researchers to review various concepts and results from the realm of abstract algebra in broader framework of fuzzy setting.

A fuzzy subset  of a given set or a fuzzy set in , is described as an arbitrary function , where , is the usual closed interval of real numbers. The fundamental concept of fuzzy set, was first initiated by Zadeh in his seminal paper 62 of 1965, was applied to generalize the some basic concepts of algebra. It soon invoked a natural question concerning a possible connection between fuzzy sets and algebraic structures like set, group, semigroup, ring, semiring, near-ring, measure theory, groupoids, real analysis, topology and differential equations and so forth. In 1971, Rosenfeld 54 applied this concept to the theory of group and studied the fuzzy subgroups of a group. The concept of fuzzy ideals in semigroups was first developed by Kuroki and studied fuzzy ideals, fuzzy bi-, generalized bi-, quasi-, semiprime, semiprime quasi- ideals of semigroups see 27, 28, 29, 30, 31, 32. Dib et al in 11 introduced the definition of fuzzy groupoid and fuzzy semigroup. They studied fuzzy ideals and fuzzy bi-ideals of fuzzy semigroups. The monograph deals with the application of fuzzy approach to the concepts of automata and formal languages by Mordeson and Malik 36. A systematic exposition of fuzzy semigroups by Mordeson et al appeared in 37 where one can find the theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy finite state machines and fuzzy languages. Liu 34, 35 initiated the concept of fuzzy subrings and fuzzy ideals of a ring and also discussed the operations on fuzzy ideals. Ren 53 studied the fuzzy ideals of a ring and defined the fuzzy quotient rings. Subsequently mukherjee et al 39, 40, Swamy et al 58 and Yue 61 fuzzified certain standard concepts and results on rings and ideals. Gupta et al 13 defined the intrinsic product of fuzzy subsets of a ring. Kuroki 33 investigated the notion fuzzy left, right, bi-, quasi- ideals of a ring and also characterized regular, intra regular, both regular and intra-regular rings in terms of such ideals. Sherwood 57 introduced the concept of product of fuzzy subgroups. Further study on this concept explored by Osman 46, 47 and Ray 52. Mukherjee et al 38 gave the idea of fuzzy normal subgroups. Moreover study on fuzzy normal subgroups established by Kumar et al 6. After the introduction of fuzzy set by zadeh, there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy set introduced by Atanassov 1, 2 is one among

them. Application of intuitionistic fuzzy concept has been already done by Atanassov and others in algebra, topological space, knowledge engineering, natural language and neural network, etc. An intuitionistic fuzzy set (briefly, IFS) in a non-empty set is an object having the form , where the functions and denote the degree of membership and the degree of nonmembership, respectively and for all 0,1. An intuitionistic fuzzy set in can be identified to be an ordered pair in , where is the set of all functions from to . For the sake of simplicity, we shall use the symbol for the IFS . Biswas 7 applied the concept of intuitionistic fuzzy set to the theory of groups and studied the intuitionistic fuzzy subgroups of a group. Kim et al 4 introduced the notion of intuitionistic fuzzy subquasigroups of quasigroups. Kim and Jun 5 introduced the concept of intuitionistic fuzzy ideals of semigroups. Banerjee et al 3 and Hur et al 18 applied the concept of intuitionistic fuzzy set to the theory of ring. They introduced the notion of intuitionistic fuzzy subrings and intuitionistic fuzzy ideals of a ring. Further many authors studied the intuitionistic fuzzy subrings and intuitionistic fuzzy ideals of a ring by discussing the different properties in different ways see 17. Palaniappan et al 48, 49 explored the notion of homomorphism, antihomomorphism of intuitionistic fuzzy normal subrings and also discussed some properties of intuitionistic fuzzy normal subrings. Moreover intuitionistic fuzzy ring and its homomorphism image have been investigated by Yan 59. Bhakat et al 4, 5, 6 initiated the concept of -fuzzy subgroups. They gave an important generalization of Rosenfeld fuzzy subgroups as -fuzzy subgroups. Jun et al 0, Davvaz 10 and Narayanan et al 45 studied the algebraic structures by utilizing the notion of -fuzzy subsets introduced by Bhakat et al. Yuan et al 60 introduced the concept of fuzzy subgroups with thresholds, which is a generalization of fuzzy subgroups and -fuzzy subgroups. Shabir et al 55 investigated the notion of fuzzy ideals with thresholds of semigroups. This thesis comprises five chapters. First chapter consists three sections. In the first section, we give a brief history of semigroup resp. group and ring theory, containing the basic definitions, some fundamental results and examples. In the second section, we introduce the notion of LA-semigroups resp. LA-groups and LA-rings. We provide the basic definitions, imperative results and examples. In the third section, we give the concept of fuzzy sets and intuitionistic fuzzy sets. In chapter two, we initiate the notion of intuitionistic fuzzy LA-subrings, in fact it is a generalization of intuitionistic fuzzy subrings. This chapter contains three sections. In the first section, we give the concept of intuitionistic fuzzy normal LA-subrings and the intuitionistic characteristic function of a non

empty subset of an LA-ring . We establish the some fundamental properties of intuitionistic fuzzy normal LA-subrings. In the second section, we extend the concept of intuitionistic fuzzy normal LA-subrings to the direct product of intuitionistic fuzzy normal LA-subrings by initiating the direct product of intuitionistic fuzzy sets and the intuitionistic characteristic function of a non-empty subset of an LA-ring . We examine the essential properties of intuitionistic fuzzy normal LA-subrings of an LA-ring . In the third section, further we extend the concept of intuitionistic fuzzy normal LA-subrings to the direct product of finite intuitionistic fuzzy normal LA-subrings by defining the direct product of finite intuitionistic fuzzy sets and the intuitionistic characteristic function of a non-empty subset of an LA-ring . We explore the imperative properties of intuitionistic fuzzy normal LA-subrings of an LA-ring . In chapter three, we introduce the concept of intuitionistic fuzzy left, right, interior, quasi-, bi-, generalized bi- ideals of an LA-ring . We give the idea of regular and intra-regular LA-rings. We establish a brief synopsis by discussing the different properties of the intuitionistic fuzzy left, right, interior, quasi-, bi-, generalized bi- ideals. We also characterize regular, intra-regular, both regular and intra-regular LA-rings in terms of such ideals. In chapter four, we recall the definition of intuitionistic fuzzy left, right, bi-, generalized bi- ideals of an LA-ring from chapter three, which are needed for our discussion. We give the concept of intuitionistic fuzzy -ideal of an LA-ring . We introduce the notion of left regular, right regular, -regular, left weakly regular, right weakly regular LA-rings. We establish a study of regular, left regular, right regular, -regular, left weakly regular, right weakly regular, intra-regular LA-rings in terms of intuitionistic fuzzy left, right, bi-, generalized bi- ideals. We also characterize left weakly regular LA-rings by the properties of intuitionistic fuzzy right, two-sided, bi-, generalize bi- ideals. In chapter five, we extend the concept of intuitionistic fuzzy ideals of an LA-ring with thresholds where such that . We define the notion of intuitionistic fuzzy LA-subrings with thresholds and intuitionistic fuzzy left, right, interior, quasi-, bi-, generalized bi- ideals with thresholds of an LA-ring . We investigate a summary by discussing the different properties of such ideals. We characterize regular, intra-regular, both regular