Fuzzy Algebra

In this work, we attempt to investigate the connection between various types of ideals (for examples (m;n)-ideal, bi-ideal, interior-ideal, quasi-ideal, prime-ideal and maximal-ideal) of an or- dered semigroup (S; � ; � ) and the corresponding, hyperideals of its EL-hyperstructure (S; � ) (if exists). Moreover, we construct the class of EL--semihypergroup, associated to a partially-ordered - semigroup.

In this paper, we have enhanced the notion of the direct product of two intuitionistic fuzzy sets to the notion of the generalised direct product of two doubt intuitionistic fuzzy subalgebras and two doubt intuitionistic fuzzy H-ideals of two BCK/BCI-algebras using max-min operations. And we study some interesting properties of such direct product of doubt intuitionistic fuzzy H-ideals of BCK/BCI-algebras. Using level subsets of BCK/BCI-algebras, some characterization theorems are also given.

This study aims to examine intuitionistic fuzzy congruences
and intuitionistic fuzzy submodules on an R-module (near-ring module).
The relationship between intuitionistic fuzzy congruences and intuitionistic
fuzzy submodules of an R-module is also obtained. Furthermore, the
intuitionistic fuzzy quotient R-module of an R-module over an intuitionistic
fuzzy submodule is defined. The correspondence between intuitionistic
fuzzy congruences on an R-module and intuitionistic fuzzy congruences on
the intuitionistic fuzzy quotient R-module of an R-module over an intuitionistic
fuzzy submodule of an R-module is also obtained.

In this paper, we introduce and explore some novel concepts within the frame work of intuitionistic fuzzy module theory. First, we define the notion of an intuitionistic fuzzy chained module as a generalisation of chained modules, establishing it foundational properties. We then characterise intuitionistic fuzzy chained modules, in terms of its level-cut submodules. In addition to this, we describe an intuitionistic fuzzy chained module in terms of its intuitionistic fuzzy cyclic submodules. Finally, we demonstrate that under specific conditions, the intuitionistic fuzzy multiplication module is an intuitionistic fuzzy chained module.

The importance of semigroups and their fuzzy subsystems is evident from their applications and significant role in several applied dis-ciplines 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 (V qk) 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 (V qk)-intuitionistic fuzzy interior ideals. Moreover semigroups are characterized by the properties of these notions.

α-fuzzy cosets of a fuzzy submodule and α-fuzzy quotient modules are defined and discussed. A homomorphism from a given module onto the set of all α-fuzzy quotient module is established. The dimension of α-fuzzy quotient module is determined.

In the literature, there are two different approaches to define entropy of Atanassov Intuitionistic Fuzzy Sets (AIFS, for short). The first approach, given by Szmidt and Kacprzyk, measures how far is an AIFS from its closest crisp set, while the second approach, given by Burrillo and Bustince, measures how far is an AIFS from its closest fuzzy set. On the other hand, divergence measures are functions that measure how different two AIFSs are. This paper studies how to define both types of entropies using measures of local divergence. In this context we provide the necessary and sufficient conditions for defining entropy measures under both frameworks using divergence measures for AIFS. We also show that the usual examples of entropy measures can be obtained as a particular case of our more general framework. Also, we investigate the connection between knowledge measures and divergence measures. Finally, we apply our results in a multi-attribute decision making problem to obtain the weights of the experts.

In this paper, We define some types of Irregular Intuitionistic Fuzzy Graphs (IIFG) and some properties of Irregular intuitionistic fuzzy graphs are studied. Some results on Highly Irregular intuitionistic fuzzy graphs and its complement are established. Also we discussed the isomorphic properties of µ-busy , γ-busy nodes and µ-free, γ-free nodes in Highly Irregular Intuitionistc Fuzzy Graphs.

In this article, we try to obtain a more general form than (∈, ∈ ∨ q)-intuitionistic fuzzy bi-ideals in ordered semigroups. The notion of an (∈, ∈ ∨ q k)-intuitionistic fuzzy bi-ideal is introduced, and several properties are investigated. Characterizations of an (∈, ∈ ∨ q k)-intuitionistic fuzzy bi-ideal are established. A condition for an (∈, ∈ ∨ q k)-intuitionistic fuzzy bi-ideal to be an intuitionistic fuzzy bi-ideal is provided. It is shown that every (∈, ∈)-intuitionistic fuzzy bi-ideal is an (∈, ∈ ∨ q)-intuitionistic fuzzy bi-ideal, and every (∈, ∈ ∨ q)-intuitionistic fuzzy bi-ideal is an (∈, ∈ ∨ q k)-intuitionistic fuzzy bi-ideal but the converse is not true. The important achievement of the study with an (∈, ∈ ∨ q k)-intuitionistic fuzzy bi-ideal is that the notion of an (∈, ∈ ∨ q)-intuitionistic fuzzy bi-ideal is a special case of an (∈, ∈ ∨ q k)-intuitionistic fuzzy bi-ideal, and, thus, several results in the paper (Jun et al. in Bi-ideals of ordered semigroups based on the intuitionistic fuzzy points (submitted)) are the corollaries of our results obtained in this paper.

In this article, we deal with ordered generalized $\Gamma$-hyperideals in ordered $\Gamma$-semihypergroups. In particular, we study $(m, n)$-regular ordered $\Gamma$-semihypergroups in terms of ordered $(m, n)$-$\Gamma$-hyperideals. Moreover, we obtain some ideal theoretic results in ordered $\Gamma$-semihypergroups.

In this paper, we define the algebraic structures of anti Q-fuzzy subgroup and some related properties are investigated. The purpose of this study is to implement the fuzzy set theory and group theory in anti-Q fuzzy subgroups. Characterizations of lower level subsets of an anti-Q fuzzy subgroup of a group are given.

A fuzzy subset A defined on a set X is represented as     X x x x A A   where , ) ( , . It is not always possible for membership functions of type ] 1 , 0 [ :  X A  to associate with each point x in a set X a real number in the closed unit interval ( ] 1 , 0 [ ) without the loss of some useful information. The importance of the ideas of “belongs to” () and “quasi coincident with” ( q ) relations between a fuzzy point and fuzzy set is evident from the research conducted during the past two decades. Ordered  -semigroup (generalization of ordered semigroups) play an important role in the broad study of ordered semigroups. In this paper we provide an extension of fuzzy generalized bi  -ideals and introduce ) , ( k q    -fuzzy generalized bi  -ideals of ordered  -semigroup. The purpose of this paper is to link this new concept with ordinary generalized bi  -ideals by using level subset and characteristic function.

The advancement in the fascinating area of fuzzy set theory has become area of much interest, generalization of the existing fuzzy subsystems of other algebraic structures is very important to tackle more current real life problems. In this paper, we give more generalized form of regular ordered gamma semigroups in terms of (E,EVq_k)-fuzzy gamma ideals. Particularly, we characterized left regular, right regular, simple and completely regular ordered gamma semigroups in terms of this new notion. Some necessary and sufficient conditions for ordered gamma semigroup to be completely regular are provided in this paper.

A fuzzy subset A defined on a set X is represented as A = {(x, A (x), where x ∈ X}. It is not always possible for membership functions of type λA : X → [0,1] to associate with each point x in a set X a real number in the closed unit interval [0,1] without the loss of some useful information. The importance of the ideas of “belongs to” (∈) and “quasi coincident with” (q) relations between a fuzzy point and fuzzy set is evident from the research conducted during the past two decades. Ordered Γ-semigroup (generalization of ordered semigroups) play an important role in the broad study of ordered semigroups. In this paper, we provide an extension of fuzzy generalized bi Γ-ideals and introduce (∈,∈∨qk)-fuzzy generalized bi Γ-ideals of ordered Γ-semigroup. The purpose of this paper is to link this new concept with ordinary generalized bi Γ-ideals by using level subset and characteristic function.

In this paper, we introduced the notion of intuitionistic fuzzy prime radical of an intuitionistic fuzzy ideal in Γ-rings. We also characterise intuitionistic fuzzy primary ideal of Γ-rings. We also analyse homomorphic behaviour of intuitionistic fuzzy primary ideal and intuitionistic fuzzy prime radical of Γ-rings.

In this paper we consider the fuzzy ideals of N-group G where N is a nearring. We introduce fuzzy ideal θ µ of the quotient N-group G/µ with respect to a fuzzy ideal µ of G. If µ is a fuzzy ideal of G and θ a fuzzy ideal of G/µ such that θ µ ⊆ θ and θ µ (0) = θ(0), then corresponding σ θ : G → [0, 1] is defined and proved that it is a fuzzy ideal of G such that µ ⊆ σ θ and µ(0) = σ θ (0). We also prove some results on homomorphisms and fuzzy ideals of N-groups. The image and preimage of fuzzy ideal µ under N-group homomorphism were studied. Finally it is obtained that if f : G → G 1 is an epimorphism of N-groups, then there is an order preserving bijection between the fuzzy ideals of G 1 and the fuzzy ideals of G that are constant on kerf. Some examples related to these concepts were illustrated.

In this paper, new powerful modification of homotopy analysis technique (NMHAM) was submitted to create an approximate solution of nonhomogeneous nonlinear ordinary and partial differential equations. The NMHAM is a combination of the new technique of homotopy analysis method(NHAM) [4] and the new technique of homotopy analysis method(nHAM) [7].Three illustrative examples are employed to illustrate the accuracy and computational proficiency of this approach. The outcomes uncover that the NMHAM is more accurate than the NHAM and nHAM.

In this paper, the notion of fuzzy quasi ideal is extended to intuitionistic fuzzy quasi ideal with respect to a triangular norm and various results of such intuitionistic fuzzy ideals are studied. It has also been established that if an intuitionistic fuzzy subring with respect to a triangular norm is an intuitionistic fuzzy quasi ideal with respect to that norm, then it is also intuitionistic fuzzy quasi ideal with respect to the annihilation of that triangular norm.

In this paper, the notion of fuzzy quasi ideal is extended to intuitionistic fuzzy quasi ideal with respect to a triangular norm and various results of such intuitionistic fuzzy ideals are studied. It has also been established that if an intuitionistic fuzzy subring with respect to a triangular norm is an intuitionistic fuzzy quasi ideal with respect to that norm, then it is also intuitionistic fuzzy quasi ideal with respect to the annihilation of that triangular norm.

In this paper, we introduced the notion of interval-valued (θ, δ)fuzzy KU-ideals of KU-algebras and some related properties are investigated. We proved that U (η; t) is a KU-ideal if and only if the interval-valued fuzzy subset η is an interval-valued (θ, δ)-fuzzy KU-ideal of a KU-algebra for θ ≺ t δ.

In this paper, we introduce the concept of Anti Q-fuzzy PMS-ideals of PMS-algebras, lower level cuts of a fuzzy set and proved some results. We discussed few results of anti Q-fuzzy PMS-ideals of PMS-algebras in homomorphism and Cartesian product.

In this paper, we introduce normal hyperideal and bi-hyperideal in normal hypersemigroups. We study (normal)hypersemigroup and normal regular hypersemigroup based on bi-hyperideal proving some equivalent conditions. In particular, we prove, among the other results, that if are any two normal hyperideals of a hypersemigroup , then their product and are also normal hyperideals of and. We also prove that the minimal normal hyperideal of a hypersemigroup is a hypergroup.

In this paper, we study the concept of ordered (m, n)-Γ-hyperideals in an ordered LA-Γ-semihypergroup. We show that if (S, Γ, •,) is a unitary ordered LA-Γ-semihypergroup with zero 0 and satisfies the hypothesis that it contains no non-zero nilpotent (m, n)-Γ-hyperideals and if R(L) is a 0minimal right (left) Γ-hyperideal of S, then either (R•Γ•L] = {0} or (R•Γ• L] is a 0-minimal (m, n)-Γ-hyperideal of S. Also, we prove that if (S, Γ, •,) is a unitary ordered LA-Γ-semihypergroup; A is an (m, n)-Γ-hyperideal of S and B is an (m, n)-Γ-hyperideal of A such that B is idempotent, then B is an (m, n)-Γ-hyperideal of S.

In this paper, we establish the intuitionistic fuzzy version of the Lasker-Noether theorem for a commutative Γ-ring. We show that in a commutative Noetherian Γ-ring, every intuitionistic fuzzy ideal A can be decomposed as the intersection of a finite number of intuitionistic fuzzy irreducible ideals (primary ideals). This decomposition is called an intuitionistic fuzzy primary decomposition. Further, we show that in case of a minimal intuitionistic fuzzy primary decomposition of A, the set of all intuitionistic fuzzy associated prime ideals of A is independent of the particular decomposition. We also discuss some other fundamental results pertaining to this concept.

The concept of an intuitionistic fuzzy set was introduced by Krassimir T. Atanassov in 1986. As an application of this to algebra, in this article we study some properties of intuitionistic fuzzy submodules (IFSM's) of an R-Module and the characteristics of the image & the pre-image of an intuitionistic fuzzy submodule under an R-Module homomorphism. Also we introduce two methods of constructing IFSM's of a quotient module.

The concept of an intuitionistic fuzzy set, which is a generalization of the concept of a fuzzy set, was introduced by Krassimir T. Atanassov in 1986. In this article the algebraic nature of an intuitionistic fuzzy subset of an R-Module is studied. Many properties of an ...

As a generalization of fuzzy sets,the concept of intuitionistic fuzzy sets was introduced by Krassimir T. Atanassov in 1986. In this article, we study about the support and level sets of intuitionistic fuzzy sets in an R-module. Also we study some properties of intuitionistic fuzzy submodules (IFSM's) of an R-module and prove some characterizations for IFSM's.

The Fuzzy set theory has been applied in many fields such as management, engineering and almost in every business enterprise as well as day to day activities. Ordering fuzzy numbers plays an important role in approximate reasoning, optimization, forecasting, decision making, risk analysis, controlling, scheduling and various other usages in our day to day activities. This paper describes a ranking method for ordering fuzzy numbers based on Area, Mode, divergence, Spreads and Weights of generalized (non-normal) hexagonal fuzzy numbers.

In this paper all rings A are commutative with identity and all modules(mods)X are unitary left mods over a ring A .Weintroduce a generalization of essential submodule(s.mods) namely annihilator semi essential submodules(for short; Ann.Si.Ess.S.Mods) .We study some properties and the relation between the submodule and his annihilator and we give some basic properties . Also we introduce the concept of annihilator uniform modules and annihilator semi maximal submodules(Ann.Si.Max.S.Mod) .

We consider the intuitionistic fuzzification of the concept of bi-Γ-ideal in a Γ-LA-semigroup S, and investigate some properties of bi-Γ-ideal. We prove that if B = (µ B , γ B) is an intuitionistic fuzzy bi-Γideal of a Γ-LA-semigroup T, then the pre image f − 1 (B) = (f − 1 (µ B), f − 1 (γ B)) of B under f is an intuitionistic fuzzy bi-Γ-ideal of S. Moreover, every Γ-LA band is an intuitionistic fuzzy right and left Γ-due.

In this paper, we introduced the notion of interval-valued (θ, δ)fuzzy KU-ideals of KU-algebras and some related properties are investigated. We proved that U (η; t) is a KU-ideal if and only if the interval-valued fuzzy subset η is an interval-valued (θ, δ)-fuzzy KU-ideal of a KU-algebra for θ ≺ t δ.

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 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. Characterizations of lower level subsets of an anti Q-fuzzy HX subgroup of a HX group are given.

In this article, we introduce the notion of fuzzy G-module by defining the group action of G on a fuzzy set of a Z-module M. We establish the cases in which fuzzy submodules also become fuzzy Gsubmodules. Notions of a fuzzy prime submodule, fuzzy prime G-submodule, fuzzy semi prime submodule, fuzzy G-semi prime submodule, G-invariant fuzzy submodule and G-invariant fuzzy prime submodule of M are introduced and their properties are described. The homomorphic image and pre-image of fuzzy G-submodules, G-invariant fuzzy submodules and G-invariant fuzzy prime submodules of M are also established.

In this paper we propose the use of enhanced mid-level information, such as information obtained from the application of supervised or unsupervised learning methodologies on low-level characteristics, in order to improve semantic multimedia analysis. High-level, a priori contextual knowledge about the semantic meaning of objects and their low-level visual descriptions are combined in an integrated approach that handles in a uniform way the gap between semantics and low-level features. Prior work on low-level feature extraction is extended and a region thesaurus containing all mid-level features is constructed using a hierarchical clustering method. A model vector that contains the distances from each mid-level element is formed and a neural network-based detector is trained for each semantic concept. Contextual adaptation improves the quality of the produced results, by utilizing fuzzy algebra, fuzzy sets and relations. The novelty of the presented work is the contextdriven mid-level manipulation of region types, utilizing a domain-independent ontology infrastructure to handle the knowledge. Early experimental results are presented using data derived from the beach domain.

In this paper we propose an algorithm to improve the results of knowledge-assisted image analysis, based on contextual information. In order to achieve this, we utilize fuzzy algebra, fuzzy sets and relations, towards efficient manipulation of image region concepts. We provide a novel context modelling, based on the OWL language, using RDF reification. Initial image analysis results are enhanced by the utilization of domain-independent, semantic knowledge in terms of concepts and relations between them. The novelty of the presented work is the context-driven readjustment of the degrees of confidence of the detected concepts produced by any image analysis technique, utilizing a domainindependent ontology infrastructure to handle the knowledge, as well as multiple application domains.

The article is devoted to classification of close sets of functions in k-valued logic. We build the classification of finite closed sets. The sets contain only constants and unary functions since sets containing even a two-ary function are infinite. Formulas of the number of finite sets and minimal sets exist for all natural k. We find the numbers of closed sets containing the identity function f (x)=x or a constant for all k. We give the number of sets on levels of inclusions at k up to 10. The inclusion diagrams are present at k up to 5, at k=6 we give inclusion diagrams of sets containing only function f (x)=x and constant 0. We find isomorphic sets and use only one of the isomorphic sets to build the diagrams.

In this paper, we defined the terms M-N-fuzzy ring and studied some basic properties. Also defined M-N-fuzzy ideal and its some elementary properties are discussed. Keywords Fuzzy ring , Fuzzy ideal , Fuzzy ring with operator , M-N-fuzzy ring , M-N-fuzzy ideal , M-N-homomorphism

A fuzzy subset A defined on a set X is represented as     X x x x A A   where , ) ( , . It is not always possible for membership functions of type ] 1 , 0 [ :  X A  to associate with each point x in a set X a real number in the closed unit interval ( ] 1 , 0 [ ) without the loss of some useful information. The importance of the ideas of “belongs to” () and “quasi coincident with” ( q ) relations between a fuzzy point and fuzzy set is evident from the research conducted during the past two decades. Ordered  -semigroup (generalization of ordered semigroups) play an important role in the broad study of ordered semigroups. In this paper we provide an extension of fuzzy generalized bi  -ideals and introduce ) , ( k q    -fuzzy generalized bi  -ideals of ordered  -semigroup. The purpose of this paper is to link this new concept with ordinary generalized bi  -ideals by using level subset and characteristic function.