Fuzzy Algebra

In this paper we investigate the radical structure of an intuitionistic fuzzy polynomial ideal A x induced by an intuitionistic fuzzy ideal A of a ring and study its properties. Given an intuitionistic fuzzy ideal B of a ring R and a homomorphism f : R → R , we show that if f x : R[x] → R [x] is the induced homomorphism of f , that is, f x (n i=0 a i x i) = n i=0 (f (a i))x i , then f −1 x [(√ B) x ] = (f −1 (B)) x .

In this paper, we define the notion of intuitionistic fuzzy characteristic ideal (IFCI) of a Γ-ring which is analogue of a characteristic ideal in the ordinary ring theory and derive various new results. The correlation between the set of all automorphisms of Γ-ring and the corresponding automorphisms of its operator rings have been innovated. Then a one to one correlation between the set of all intuitionistic fuzzy characteristic ideals of Γ-ring and that of its operator ring has been constituted. This is used to obtain a similar bijection for characteristic ideals.

The category theory deals with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and acting as a unifying notion. In this paper, we study the relationship between the category of groups and the category of intuitionistic fuzzy groups. We prove that the category of groups is a subcategory of category of intuitionistic fuzzy groups and that it is not an Abelian category. We establish a function β : Hom(A, B) → [0, 1] × [0, 1] on the set of all intuitionistic fuzzy homomorphisms between intuitionistic fuzzy groups A and B of groups G and H, respectively. We prove that β is a covariant functor from the category of groups to the category of intuitionistic fuzzy groups. Further, we show that the category of intuitionistic fuzzy groups is a top category by establishing a contravariant functor from the category of intuitionistic fuzzy groups to the lattices of all intuitionistic fuzzy groups.

Let R be a commutative ring with identity and M be an R-module. An intuitionistic L-fuzzy submodule (ILFSM) C of an intuitionistic L-fuzzy module A of R-module M , is called an intuitionistic L-fuzzy essential submodule in A, if C ∩ B = χ {θ} for any non-trivial ILFSM B of A. In this case we say that A is an essential extension of C. Also, if C has no proper essential extension in A, then C is called an intuitionistic L-fuzzy closed submodule in A. Further, for ILFSMs B, C of A, C is called complement of B in A if C is maximal with the property that B ∩ C = χ {θ}. We study these mentioned notations which are generalization of the notions of essential submodule, closed submodule and complement of a submodule in the intuitionistic L-fuzzy module theory. We prove many basic properties of both these concepts.

The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. ABSTRACT It has been established that the fuzzy sets do not hold the Complement laws of Boolean algebra. The aim of this article is to introduce and study a kind of family of fuzzy subsets of a finite set which holds the Complement laws. The complement operation has redefined for this purpose. Further, it has been established that the introduced fuzzy subsets can form Boolean lattice and also Boolean algebra. Keywords fuzzy subset, membership value, the complement operation, the complement laws, Boolean lattice, subsets of a fuzzy subset, fuzzy power set.

The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. Abstract— The aim of this article is to study the behaviour of the fuzzy Boolean algebras formed by the fuzzy subsets of a finite set. The properties of homomorphism, isomorphism and automorphism of the fuzzy Boolean algebras have been investigated. Further, the ideal and filter of the fuzzy Boolean algebras have also been observed with their characteristics.

The concept of intuitionistic fuzzy subset was introduced by K.T.Atanassov and the notion of intuitionistic fuzzy ideals of BG-algebra was developed by A.Zarandi and A.Borumand Saeid in 2005. In this paper, for a set Q, the notion of intuitionistic Q-fuzzy ideals of BG-algebra is introduce and investigate some of their basic properties.

In this paper, we study ordered hyperideals in ordered semihypergroups. Also, we study (m, n)-regular ordered semihypergroups in terms of ordered (m, n)-hyperideals. Furthermore, we obtain some ideal theoretic results in ordered semihypergroups.

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, we propose a ranking algorithm for symmetric octagonal fuzzy numbers, which results in equality only when the numbers coincide. Also the ranking algorithm is verified for some reasonable properties of ordering of fuzzy numbers

In this paper we introduce the notion of intuitionistic fuzzy polynomial ideal A x of a polynomial ring R[x] induced by an intuitionistic fuzzy ideal A of a ring R, and obtain an isomorphism theorem of a ring of intuitionistic fuzzy cosets of A x. It is shown that an intuitionistic fuzzy ideal A of a ring is an intuitionistic fuzzy prime if and only if A x is an intuitionistic fuzzy prime ideal of R[x]. Moreover, we show that if A x is an intuitionistic fuzzy maximal ideal of R[x], then A is an intuitionistic fuzzy maximal ideal of R but converse is not true.

 It has been accepted that the set of all fuzzy subsets of a finite set cannot satisfy all the properties of Boolean algebra and the main cause is that for a fuzzy subset A and its complement A , neither AA   is the empty fuzzy subset, nor AA   is the universal fuzzy subset; where  and  represents the operations fuzzy union and fuzzy intersection respectively. Accordingly, the set of all the fuzzy subsets of a finite set have been accepted not to form a Boolean algebra. The fuzzy subsets of a finite set form a Boolean algebra if the proposed definition of the complement operation is considered. Since, this Boolean algebra is formed by the fuzzy subsets of a set we called this as a special fuzzy Boolean algebra. Throughout the article different aspects of the introduced special fuzzy Boolean algebra have been discussed.

In this paper, we define the concept of an intuitionistic Q-fuzzy normal HX group and some related properties are investigated. We also define the level subsets of an intuitionistic Q-fuzzy normal HX group and discussed some of its properties. A few equivalent propositions of fuzzy HX group constituting an intuitionistic Q-fuzzy normal HX group are given. Keywords: Intuitionistic fuzzy set, intuitionistic Q-fuzzy set, intuitionistic fuzzy subgroup , intuitionistic Q-fuzzy subgroup, intuitionistic Q-fuzzy HX subgroup and level sub HX groups, intuitionistic normal Q-fuzzy HX subgroup.

In current paper we have defined (α, β)-fuzzy KU-ideals of KU-algebras and then some useful characterizations have provided. Finally we have introduced the concept of (α, β)-fuzzy KU-relation and then by the use of this notion we have proved some more interesting results.

In this paper, we introduce the notion of -complement of intuitionistic fuzzy graph structure = (A, B1, B2,..., Bk) where  is a permutation on {B1, B2,...,Bk} and obtain some results. We also define some elementary definitions like self complementary, totally self complementary, strong self complementary intuitionistic fuzzy graph structure and study their properties.

The concept of intuitionistic fuzzy G-modules and its properties including representation, reducibility and injectivity are already defined by the author et al. In this paper, we extend this idea to define semi-simplicity of intuitionistic fuzzy G-modules. The existence of a semi-simple intuitionistic fuzzy G-module for every finite dimensional G-module is proved and the relationships of semi-simplicity with other properties of intuitionistic fuzzy G-modules are also discussed.

In this paper, we study ordered hyperideals in ordered semihypergroups. Also, we study (m, n)-regular ordered semihypergroups in terms of ordered (m, n)-hyperideals. Furthermore, we obtain some ideal theoretic results in ordered semihypergroups.

The main aim of this paper is to investigate the relationship between the characteristic of a ring and the order of an element in the intuitionistic fuzzy environment. We also characterise the term divisor of zero in a ring and study their connection with the ideal of the ring in the intuitionistic fuzzy setting. Some related results are also analyse.

In the present manuscript, we introduce and study the notion of primary submodules as well as P-primary submodules of a module in the intuitionistic L-fuzzy environment. Apart from investigating basic properties of these submodules, we explore some foundational results analogous to corresponding submodules. A suitable characterization of intuitionistic L-fuzzy primary (P-primary) submodules in terms of primary (P-primary) submodules are presented.

In this paper some basic definitions of soft set
theory and fuzzy set theory are given. Then the concept of
fuzzy isomorphism theorems of idealistic soft gamma rings are
introduced. The first, second and third fuzzy gamma
isomorphism theorems of idealistic soft gamma rings are
established and their proofs are given, respectively. Moreover,
several basic conclusions are given.

— The set of all mappings from a finite set X into a closed interval   0,1 is the set of fuzzy sets denoted by F. This set F is closed under the binary operation absolute difference,  of fuzzy set F satisfies the axioms, closure, commutativity, identity and inverse law under the binary operation . The associative law is not satisfied by F. In this article, we wish to introduce the subset B of F with binary operation absolute difference  and fuzzy intersection  , as a special fuzzy Boolean ring briefly denoted by SFBR. Keywords— Special fuzzy Boolean ring (SFBR), absolute difference, subs SFBR, Isomorphic SFBR, Divisor of empty fuzzy set.

In this article, we introduce the notion of injectivity and quasi injectivity of an intuitionistic fuzzy G-module. We also establish a condition on finite dimensional G-modules for which an intuitionistic fuzzy G-module is injective with respect to another intuitionistic fuzzy G-module. We discuss the injectivity of an intuitionistic fuzzy G-module with respect to the quotient intuitionistic fuzzy G-module of another G-module. Some important properties of injectivity of intuitionistic fuzzy G-modules with regards to direct sum of intuitionistic fuzzy G-modules are also studied. A relationship between injectivity and quasi injectivity of intuitionistic fuzzy G-modules is also obtained.

In this paper, we will introduce the notion of intuitionistic fuzzy G-module by defining the group action of G on the intuitionistic fuzzy set of Z-module M on which G-acts. We obtain the conditions when an intuitionistic fuzzy submodule be an intuitionstic fuzzy G-submodule. Some new notion like intuitionistic fuzzy prime submodule, intuitionistic fuzzy prime G-submodule, intuitionistic fuzzy semi prime submodule, intuitionistic fuzzy G-semi prime submodule, G-invariant intuitionistic fuzzy submodule and G-invariant intuitionistic fuzzy prime submodule of M will be introduced and studied. The homomorphic image and pre-image of intuitionistic fuzzy G-submodules, G-invariant intuitionistic fuzzy submodules and G-invariant intuitionistic fuzzy prime submodules of M will be established.

Result concerning various ideals in near rings (as defined by Yakabe, Chelvam and
Ganesan, Kim and Zhan and Xueling as well as some generalization thereof) that have been
established over the past three decays, will be discussed. These include the fuzzy ideals, fuzzy
quasi ideals and fuzzy bi-ideals in near rings. In this talk, we extend these notion to intuitionistic
fuzzy ideals, intuitionistic fuzzy quasi-ideals, intuitionistic fuzzy bi-ideals in near-rings. A relationship between various types of ideals has been established.

In this paper, we introduce the concept of intuitionistic fuzzy small submodule with respect to an arbitrary intuitionistic fuzzy submodule of an R-module M. We derive the condition when an intuitionistic fuzzy submodule to be a small submodule with respect to another intuitionistic fuzzy submodule with the crisp small submodule of the R-module M. It is also shown that the sum of two intuitionistic fuzzy small submodules with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy submodule with respect to the same fixed intuitionistic fuzzy submodule. This result can be extended to an arbitrary sum of intuitionistic fuzzy submodules. Further, we prove that the homomorphic image of an intuitionistic fuzzy small submodule with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy small submodule with respect the homomorphic image of the fixed intuitionistic fuzzy submodule.

In this paper, we introduce the concept of bipolar fuzzy soft ordered Γ -hyperideals in Γ -hypersemigroups. We define
bipolar fuzzy soft ordered hyperideals, bi-ideals and interior ideals of ordered Γ -hypersemigroups and discuss some properties

In this paper, we introduce the notion of direct sum of intuitionistic fuzzy submodules of G-module M and discuss some related properties. If B and C are intuitionistic fuzzy submodules of G-module M and G-module N respectively, then we extend the definition of intuitionistic fuzzy submodules B and C to formulate intuitionistic fuzzy submodules of the direct sum M ⊕ N. It is proved that the support of the direct sum of intuitionistic fuzzy submodules B and C is equal to the direct sum of supports of intuitionistic fuzzy submodules B and C. Also we analyze the direct sum of arbitrary family of intuitionistic fuzzy submodules and obtained some results. We also introduce the notion of decomposability and indecomposility of intuitionistic fuzzy G-modules and prove some results. 2010 AMS Classification: 03F55, 16D10

In this talk we introduce the notion of intuitionistic fuzzy
polynomial ideal Ax of a polynomial ring R[x] induced by an
intuitionistic fuzzy ideal A of a ring R. Then many properties of Ax
will be discussed. We shall also establish an isomorphism theorem
of a ring of intuitionistic fuzzy cosets of Ax . It will be shown that an
intuitionistic fuzzy ideal A of a ring R is an intuitionistic fuzzy prime
if and only if Ax is an intuitionistic fuzzy prime ideal of R[x].
However, if Ax is an intuitionistic fuzzy maximal ideal of R[x], then
A is an intuitionistic fuzzy maximal ideal of R, but converse is not
true. We will also investigate the nil radical structure of Ax . The
homomorphic image and inverse image of an intuitionistic fuzzy
polynomial ideal Ax and nil radical of Ax when A is an intuitionistic
fuzzy prime ideal of a ring R will also be discussed.

In this paper, we study ordered hyperideals in ordered semihypergroups. Also, we study (m, n)-regular ordered semihypergroups in terms of ordered (m, n)-hyperideals. Furthermore, we obtain some ideal theoretic results in ordered semihypergroups.

In this talk, we study the concept of residual quotient of IF subsets of ring
and module and develop many properties out of these. Using the concept
of residual quotient, we investigate some important characterization of
annihilator of IF submodules, IF prime(primary) submodules and IF
prime(primary) decomposition.

In this paper, we define the concept of an intuitionistic Q-fuzzy HX group and define a new algebraic structure of intuitionistic Q-fuzzy HX group and some related properties are investigated. We also define the level subsets of an intuitionistic Q-fuzzy HX group and discussed some of its properties.

In this paper, we introduce the notion of intuitionistic fuzzy superfluous (or small) submodule of a module and study some of their properties. We establish the condition of an intu-itionistic fuzzy submodule to be an intuitionistic fuzzy superfluous submodule. A relationship between superfluous submodule and the intuitionistic fuzzy superfluous submodule is derived. We also study the nature of intuitionistic fuzzy superfluous submodules under intuitionistic fuzzy direct sum. A relation regarding intuitionistic fuzzy superfluous submodule and intuitionistic fuzzy quotient module is established. It is shown that the well-known relation between the Jacobson radical and the superfluous submodules does not hold in case of intuitionistic fuzzy superfluous submodules.

The importance of semigroups and their fuzzy subsystems is evident from their applications and significant role in several applied disciplines like computer sciences, control engineering, error-correcting codes and fuzzy automata theory. In this paper, we give generalizations of in-tuitionistic fuzzy interior ideals of semigroups and introduced the notions of intuitionistic fuzzy interior ideals of type (∈, ∈ ∨ q k) and (∈, ∈) of semi-groups. The important mile stone of the present paper is to link ordinary intuitionistic fuzzy interior ideals, (∈, ∈)-intuitionistic fuzzy interior ideals and (∈, ∈ ∨ q k)-intuitionistic fuzzy interior ideals. Moreover semigroups are characterized by the properties of these notions.

The concept of intuitionistic fuzzy subset was introduced by K.T.Atanassov and the notion of intuitionistic fuzzy ideals of BG-algebra was developed by A.Zarandi and A.Borumand Saeid in 2005. In this paper, for a set Q, the notion of intuitionistic Q-fuzzy ideals of BG-algebra is introduce and investigate some of their basic properties

In this paper, we redefine the definition of a fuzzy HX group and define a new algebraic structure of Q-fuzzy HX group and anti Q-fuzzy HX subgroup and some related properties are ivestigated. We establish the relation between Q-fuzzy HX group and anti Q-fuzzy HX group of a HX group . The purpose of this study is to implement the fuzzy set theory and group theory in anti Q-fuzzy HX subgroups.

In this paper, we study intuitionistic fuzzy prime submodules with the help of residual quotient subsets of ring and modules. Also a complete characterisation of an intuitionistic fuzzy prime submodule is given. A relationship between intuitionistic fuzzy prime submodule and intuitionistic fuzzy maximal submodule is established. Homeomorphic image and pre-image of intuitionistic fuzzy prime submodules are obtained.

This article is in continuation of the first author's previous paper on intuitionistic fuzzy prime submodules, [13]. In this paper, we explore the decomposition of intuitionistic fuzzy submodule as the intersection of finite many intuitionistic fuzzy prime submodules. Many other forms of decomposition like irredundant and normal decomposition are also investigated.

Let M be an R-module, A and B are intuitionistic fuzzy submodules of M with A  B. Then A is called an intuitionistic fuzzy cosmall submodule of B in M if B / A << IF M /A (=  (M) / A *). In this paper an attempt has been to study intuitionistic fuzzy cosmall submodules and investigate various properties of such intuitionistic fuzzy submodules. The notion of an intuitionistic fuzzy hollow module is also introduce and a relationship of this with the intuitionistic fuzzy indecomposable module and the factor module are established.

In this paper, we further study the theory of intuitionistic fuzzy rings and give some new concepts such as intuitionistic fuzzy ring with operators, intuitionistic fuzzy ideal with operators , intuitionistic fuzzy quotient ring with operators, etc. while their some elementary properties are discussed.