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An Approach to Bipolar Vague Group and Its Properties 1 Introduction 2 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 5 Issue 1, January 2018 ISSN (Online) 2348 – 7968 | Impact Factor (2016) – 5.264 www.ijiset.com An Approach to Bipolar Vague Group and its Properties S. Cicily Flora1,∗ I. Arockiarani1 and Ganeshsree Selvachandran2 1Department of Mathematics, Nirmala College for Women, Coimbatore-18, Tamilnadu, India. 2Department of Actuarial Science and Applied Statistic, Faculty of Business and Information Science, UCSI University, Jalan Menara Gading, 56000 Cheras, Kuala Lumpur, MALAYSIA Abstract the standard unit interval of [0; 1] to the interval of [−1; 1]. Previous research on the development The study of fuzzy algebraic theory and of bipolar-valued fuzzy algebraic theory include the development of fuzzy algebraic struc- the study of bipolar-valued fuzzy subgroup of a tures is an important area of study. The group [27], bipolar fuzzy subalgebras and closed study of vague algebraic structures and its ideals of BCH-algebra by Jun et. al. [28], bipo- properties has been extensively studied in lar fuzzy subalgebras and bipolar fuzzy ideals in literature. However, there are no studies BCK/BCI algebra [29]. that deal with vague algebraic theory in a In this paper, we introduce group theory for bipolar setting. As such, this paper aims bipolar vague sets [30,31]through the introduction to initiate the study of the group theory of bipolar vague groups and normal subgroups. for bipolar vague sets. The notion of bipo- lar vague groups, and bipolar vague normal The properties and structural characteristics of subgroups are introduced, and the structural these structures are also studied and verified. characteristics and properties of these struc- tures are studied. Keywords: Bipolar vague set, Vague group, 2 Preliminaries Normal vague subgroup, Conjugate bipolar vague group. Definition 2.1. [2] Let X be a nonempty set. A fuzzy set A over X is defined as A = fhx : µA(x)i : x 2 Xg, where µA : X ! [0; 1] is the membership 1 Introduction function of the fuzzy set A. The study of fuzzy algebraic theory began with Definition 2.2. [26] Let X be a universal set, the introduction of the notion of a fuzzy subgroup and A be a set over X that is defined by a posi- tive membership function µ+ and a negative mem- of a group by Rosenfeld [1]. Since its inception, A bership function µ−, where µ+ : X ! [0; 1] and the study of fuzzy algebraic theory has been ac- A A µ− : X ! [−1; 0]. Then A is called a bipolar- tively studied. However, the single-valued mem- A bership structure of the fuzzy set model [2] makes valued fuzzy set over X, and can be written in the form A = fhx; µ+(x); µ−(x)i : x 2 Xg. it incapable to capture the hesitancy faced by the A A users, and also makes it incapable of expressing Definition 2.3. [27] Let G be a group and A be a the evidence for and against an element effectively. bipolar-valued fuzzy subsets of G. Then A is called This and other problems that are inherent in fuzzy a bipolar-valued subgroup of G(abbr. BV F SG) if sets has led to the expansion of algebraic theory in the following conditions are satisfied. other fuzzy based and soft set [3] based settings. The fuzzy algebraic framework has since been ex- (i) A+(xy) ≥ minfA+(x);A+(y)g tended to other settings which include soft set set- + −1 + ting [4{8], fuzzy soft setting [9{13], intuitionistic (ii) A (x ) ≥ A (x) fuzzy setting [14{16], vague setting [17{21], and a (iii) A−(xy) ≤ maxfA−(x);A−(y)g vague soft set setting [22{24]. Here we are con- cerned with further developing the fuzzy algebraic (iv) A−(x−1) ≤ A−(x). theory in a vague set [25] setting using the prop- Definition 2.4. [25] Let X be a space of points erties of bipolar-valued fuzzy sets [26]. Bipolar- (objects) with element of X denoted by x. A vague valued fuzzy sets [26] is an extension of fuzzy sets set V in X is characterized by a truth-membership whose membership degree range is enlarged from function tV : X ! [0; 1] and a false-membership ∗ Corresponding author: [email protected] function fV : X ! [0; 1]. The value tV (x) is a 1 70 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 5 Issue 1, January 2018 ISSN (Online) 2348 – 7968 | Impact Factor (2016) – 5.264 www.ijiset.com lower bound on the grade of membership of x de- Definition 2.8. [30] Let X be a universe of dis- rived from the evidence for x and fV (x) is a lower course, and A be an object over X. Then A is bound on the negation of x derived from the evi- called a bipolar vague set which is of the form: + + − − dence against x. The values tV (x) and fV (x) both A = fhx; [tA(x); 1 − fA (x)]; [−1 − fA (x); tA(x)]i : + + associate a real number in the interval [0; 1] with x 2 Xg; where [tA; 1 − fA ]: X ! [0; 1] and − − each point in X, where 0 ≤ tV (x) + fV (x) ≤ 1. [−1−fA ; tA]: X ! [−1; 0] are mappings such that + + − − This approach bounds the grade of membership of tA +fA ≤ 1 and −1 ≤ tA +fA . The positive mem- + + x to a closed subinterval [tV (x); 1−fV (x)] of [0; 1]. bership degree [tA(x); 1 − fA (x)] denotes the inter- val of satisfaction of an element x to the property Next, we present some basic results pertaining corresponding to a bipolar-valued fuzzy set A, and to the concepts and operations of vague sets. Let − − the negative membership degree [−1−fA (x); tA(x)] A and B be two vague sets over the universe U, denotes the interval of satisfaction of x to some where A and B are as defined below: implicit counter property of A. A = fhu; [tA(x); 1 − fA(x)]i : u 2 Ug; + For the sake of simplicity, the notation vA = [t+; 1 − f +] and v− = [−1 − f −; t−] will be used B = fhu; [t (x); 1 − f (x)]i : u 2 Ug: A A A A A B B to denote a bipolar vague set. Definition 2.5. [25] The vague value and unit vague set of a vague set A are as defined below: 3 Bipolar Vague Groups (i) The interval [tA(x); 1 − fA(x)] is called the vague value of x in A, and is denoted by Definition 3.1. Let (X; ∗) be a group and A be VA(x) . a bipolar vague set over X. Then A is called a bipolar vague group of X if it satisfies the following (ii) A vague set A of U is called a unit vague set conditions: if tA(x) = 1 and fA(x) = 0, for all x 2 U: + + + + (i) tA(xy) ≥ min(tA(x); tA(y)) and 1−fA (xy) ≥ (iii) A vague set A of U is called a null vague set + + min(1 − fA (x); 1 − fA (y)); if tA(x) = 0 and fA(x) = 1, for all x 2 U: + −1 + + −1 (ii) tA(x ) ≥ tA(x) and 1 − fA (x ) ≥ 1 − Definition 2.6. [25] The subset, complement, f +(x); union and intersection of vague sets are as defined A − − − below: (iii) tA(xy) ≤ max(tA(x); tA(y)) and −1 − − − − fA (xy) ≤ max(−1 − fA (x); −1 − fA (y)); (i) If for all x 2 U, tA(x) ≤ tB(x) and 1 − f (x) ≤ 1 − f (x), then A is called a vague − −1 − − −1 A B (iv) tA(x ) ≤ tA(x) and −1 − fA (x ) ≤ −1 − subset of B, denoted as A ⊆ B. − fA (x): (ii) The complement of A, denoted as Ac is de- + − Definition 3.2. Let A = (X; VA ;VA ) be a bipolar fined as: + vague group over X and H = fx 2 G=VA (x) = + − − c VA (e) and VA (x) = VA (e)g, then O(A), order of A = f(x; [fA(x); 1 − tA(x)]) : x 2 Xg: A is defined as O(A) = O(H). + − (iii) The union of A and B, denoted as A [ Definition 3.3. Let A = (X; VA ;VA ) and B = + − B, is a vague set C, defined as C = (X; VB ;VB ) be two bipolar vague groups over f(x; [max(tA(x); tB(x)); max(1 − fA(x); 1 − group X. Then A and B are said to be conjugate fB(x))]) : x 2 Xg: bipolar vague groups in X if for some g 2 G, (iv) The intersection of A and B, denoted as + + −1 VA (x) = VB (g xg) A \ B, is a vague set D, defined as D = f(x; [min(tA(x); tB(x)); min(1 − fA(x); 1 − and − − −1 fB(x))]) : x 2 Xg: VA (x) = VB (g xg) Definition 2.7. [17] Let (X; ∗) be a group and A for every x 2 X. be a vague set over X. Then A is called a vague 2 group over X if the following conditions are satis- Example 3.4. Let G = f1; !; ! g where ! is the fied: cubic root of unity with the binary operation de- fined as below: (i) tA(xy) ≥ min(tA(x); tA(y)) and 1−fA(xy) ≥ 2 min(1 − fA(x); 1 − fA(y)); * 1 ! ! 1 1 ! !2 −1 −1 (ii) tA(x ) ≥ tA(x) and 1 − fA(x ) ≥ 1 − ! ! !2 1 fA(x): !2 !2 1 ! 2 71 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol.
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