Vague Groups Research Articles

Vague Additive Groups

We introduce the concept of vague additive groups (VAG) by linking the vague set and group theory. Also we define the concept of vague normal additive group (VNAG) and vague ring (VR). The properties of VAG are investigated.

Translates of Vague Groups

As an abstraction of the geometric notation of translation, we introduce two operators T α+ and T α- called the vague translation operators. First we define the operators on vague sets and derive some of the properties .Then we investigate their action on vague groups, we prove that all translates of vague subgroups are vague normal sub groups [VNG], we observe that vague abelianness is not translation invariant. Subject classification:- 2000 AMS subject classification : 03G25 ,04A72 ,08A72 Keywords: α-increasing vague translation operator, α-decreasing vague translation operator ,vague translation groups ,translation of vague normal sub groups , vague conjugate property .

On Normalistic Vague Soft Groups and Normalistic Vague Soft Group Homomorphism

We further develop the theory of vague soft groups by establishing the concept of normalistic vague soft groups and normalistic vague soft group homomorphism as a continuation to the notion of vague soft groups and vague soft homomorphism. The properties and structural characteristics of these concepts as well as the structures that are preserved under the normalistic vague soft group homomorphism are studied and discussed.

Permutable indistinguishability operators, perfect vague groups and fuzzy subgroups

Permutability between T-indistinguishability operators is a very interesting property that is related to the compatibility of the operators with algebraic structures. It will be shown that the sup −T product E∘F of two T-indistinguishability operators is also a T-indistinguishability operator if and only if E and F are permutable T-indistinguishability operators (i.e., E∘F=F∘E). This property will be related to the study of fuzzy subgroups, fuzzy normal subgroups and vague groups. The aggregation of fuzzy subgroups will also be analyzed.

"Q-Vague Groups and Vague Normal Sub Groups With Respect to (T, S) Norms"

this Paper, Q-Vague sets and Q-Vague normal subgroups are studied. The study of Vague groups initiated by Ranjit Biswas (2006) is continued and Q-Vague homologous groups characterized as normal groups which admit a particular type of Q-Vague groups with respect to (mini, max) norms.

FILTERS OF MTL-ALGEBRAS BASED ON VAGUE SET THEORY

Abstract. In this paper, we introduce the concept of a vague lter ofMTL-algebra, and then some related properties are investigated. 1. IntroductionZadeh[13] introduced the concept of fuzzy as a new mathematical toolfor dealing with uncertainties, several researches were conducted on the gener-alization of the notion of fuzzy sets. The idea of \vague set was rst publishedby Gau and Buehrer [3], as a generalization of the notion of fuzzy set. Estevaand Godo[2] introduced a new algebra, called an MTL-algebra, and studiedseveral basic properties. MTL-algebras are algebraic structures for monoidalt-norm based logic (MTL), a many-valued propositional calculus that formal-izes the structure of the real interval [0;1];induced by a left-continuous t-norm.They also introduced the notion of lters in MTL-algebras. Zhang [12] stud-ied further properties of lters in MTL-algebras. Using the vague set, Biswas[1] studied vague groups. Jun and Park [6, 8] studied vague ideals and vaguedeductive systems in subtraction algebras. In this paper, we introduce thenotion of vague lters in MTL-algebras, and then some related properties areinvestigated.2. PreliminariesIn this section, we collect some de nition and results that have been used inthe sequel.De nition 1. ([4]) An algebra (L;;^;_; ;!;0;1) with four binaryoperation and two constant is a residuated lattice if it satis es:(R1) (L;;^;_;0;1) is a lattice with the least element 0 and the largestelement 1,(R2) is a commutative semigroup with the unit element 1,

애매 bi-군과 퍼지 bi-함수의 성질에 관한 연구

M. Demirci[Vague groups, J. math. Anal. Appl. vol.230, pp. 142-156, 1999] studied the vague group operation on a crisp set as a fuzzy function and estabished the vague group structure on a crisp set. In this paper we consider bi-groups which are studied by A.A.A. Agboola and L.S. Akinola. And we also will define vague bi-groups and fuzzy bi-functions and we investigate some basic operations on the vague bi-group and fuzzy bi-functions.

SIMILARITY RELATIONS, VAGUE GROUPS, AND FUZZY SUBGROUPS

We define vague groups in terms of similarity relations rather than fuzzy equalities. This yields a bijection between the set of all right-invariant similarity relations on a group and the set of all fuzzy subgroups of the group. Under this bijection, right-invariant and left-invariant similarity relations correspond to normal fuzzy subgroups. We show how this bijection allows for the transfer of results between vague groups and fuzzy subgroups. In particular, certain numerical invariants that characterize fuzzy subgroups of an Abelian group can be used to characterize vague groups.

VAGUE GROUPS AND Ω-VAGUE GROUPS

Given a group G, we show how one can define a vague group structure on G via a chain of subgroups of G. We discuss how a group homomorphism f from a vague group X onto a group Y induces a vague group structure on Y with f satisfying the vague homomorphism property. The notion of Ω-vague groups is introduced, where Ω is a fuzzy subset. The direct product G1 × G2 of two vague groups and the internal vague direct product of subgroups of a vague group is introduced.

The generalized associative law in vague groups and its applications—II

As a continuation of Part I, vague integral powers of elements in vague groups and their representation properties are introduced in this paper. Thereafter, some rudimentary algebraic properties of vague integral powers of elements, obtained from the generalized vague associative law formulated in Part I, are established.The present paper particularly provides the abstract foundations of integral powers and multiples of real numbers in vague arithmetic. For this reason, special attention is also paid to the calculation of integral powers and multiples of real numbers in vague arithmetic, and some practical applications related to the discrete structure of measurement instruments are also given.

The generalized associative law in vague groups and its applications—I

Products of elements and integral powers of elements in vague groups have a significant concern for the development and the applications of vague algebra. Formulation of properties of these notions basically depends on a many-valued counterpart to the generalized associative law (called the generalized vague associative law in vague groups). For this reason, the present paper and the forthcoming paper [M. Demirci, The generalized associative law in vague groups and its applications—II, Information Sciences, Submitted for publication] are devoted to the formulation of the generalized vague associative law and its applications in vague groups. The generalized vague associative law in vague semigroups, which is the main contribution of this exposition, and some elementary properties of the notion of a product of a finite number of elements in vague semigroups, which cover necessary preparatory results for Part II, are the subjects of this paper. Since the present paper forms an abstract foundation of the product and sum of a finite number of real numbers in vague arithmetic, some practical applications of the notions of product and sum of a finite number of real numbers in vague arithmetic are also given.

Vague groups and generalized vague subgroups on the basis of ([0, 1], ⩽, ∧)

In this paper, various elementary properties of vague groups and some properties of vague binary operations related with their associativity aspects are obtained. Furthermore, the concept of vague isomorphism is defined and some basic properties of this concept are studied. The concept of external direct product of vague groups is established. Later the definition of generalized vague subgroup, which is a generalization of the vague subgroup defined by Demirci, is introduced, and the validity of some classical results in this setting is investigated on the basis of the particular integral commutative, complete quasi-monoidal lattice ([0, 1], ⩽, ∧).

Fuzzy groups, fuzzy functions and fuzzy equivalence relations

To each fuzzy subgroup a fuzzy equivalence relation is associated and it is proved that a fuzzy subgroup is normal if and only if the operation of the group is compatible with its associated fuzzy equivalence relation. In the definition of fuzzy subgroup only the subset is fuzzy whilst the group operation remains crisp. In an opposite direction, another structure can be constructed maintaining crisp the basic set X of the group and fuzzifying the group operation. When in addition it is imposed to the fuzzy operation to be compatible with a given fuzzy equivalence relation, this structure is called a vague group. The results of the paper allow us to associate a vague group to every fuzzy subgroup μ in such a way that it can be interpreted as the fuzzy quotient group X/ μ. Special attention is paid to normal fuzzy groups and vague groups of the additive group ( R,+) of the real numbers. Two examples related to triangular numbers illustrating the results of the paper are provided.

Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part I: fuzzy functions and their applications

Strong fuzzy functions and perfect fuzzy functions based on *-fuzzy equalities were studied, and applied to fuzzy control [Demirci, M. (2000) “Fuzzy functions and their applications”, J. Math. Anal. Appl. 252, 495–517] and vague algebra [Demirci, M. (1999b) “Vague groups”, J. Math. Anal. Appl. 230, 142–156; Demirci, M. (2002a) “Fundamentals of M-vague algebra and M-vague arithmetic operations”, Int. J. Uncertainty Fuzziness Knowledge-Based Systems 10 (1), 25–75]. This paper introduces the general theory of strong (perfect) fuzzy functions on the basis of many-valued equivalence relations, and establishes the fundamental tools of this theory. Another aim of the present paper is to develop the modelling of input/output systems by strong (perfect) fuzzy functions, and is to propose the description of laws of nature by strong (perfect) fuzzy functions. The present paper also forms the rudimentary base of M-vague algebra on the basis of many-valued equivalence relations, presented in the cognate papers [Demirci, M. (2003d) “Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, Part II: Vague algebraic notions”, Int. J. General Systems 32, 157–175; Demirci, M. (2003e) “Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, Part III: Constructions of vague algebraic notions and vague arithmetic operations”, Int. J. General Systems 32, 177–201].

Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part III: constructions of vague algebraic notions and vague arithmetic operations

As a continuation of Demirci [(2003b) “Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part II: vague algebraic notions”, Int. J. Gen. Syst. 32, 157–175], starting from the algebraic concepts in the classical sense, this paper deals with the constructions of vague semigroups, vague monoids, vague groups, vague rings and vague fields on the basis of many-valued equivalence relations. Vague addition operations and vague multiplication operations on the basis of many-valued equivalence relations, which are naturally derived from vague algebraic notions in Demirci [(2003b) “Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part II: vague algebraic notions”, Int. J. Gen. Syst. 32, 157–175], and their constructions from the arithmetic operations in the classical sense are also subjects of this paper.