(, q)-fuzzy Ideals of BG-Algebra

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1 International Journal of Algebra, Vol. 5, 2011, no. 15, (, q)-fuzzy Ideals of BG-Algebra D. K. Basnet Department of Mathematics, Assam University, Silchar Assam , India dkbasnet@rediffmail.com L. B. Singh Department of Mathematics, Assam University, Silchar Assam , India lbsingh87@rediffmail.com Abstract. We defined (, q)-fuzzy ideal of a BG-algebra and investigated some of its properties. Keywords: BG-algebra, fuzzy ideal, (, q)-fuzzy ideal. 1. Introduction In [2] C. B. Kim, H. S. Kim introduced the notion of BG- algebras which is a generalization of B-algebras [3]. S. K. Bhakat & P. Das [5, 6] used the relation of belongs to and quasi-coincident between fuzzy point and fuzzy set, to introduce the concepts of (, q)-fuzzy subgroup and (, q)-fuzzy subring. Many authors [1, 7] discussed fuzzy and intuitionistic fuzzy subalgebras and ideals of BG-algebra. Antony and Lilly [4] investigated some interesting properties of (α ; β)-fuzzy Lie algebras over an (α ; β)-fuzzy field. In this paper we defined (, q)-fuzzy ideals of BG-algebra and got some interesting results pertaining it. 2. Preliminaries Definition 2.1. A BG-algebra is a non-empty set X with a constant 0 and a binary operation satisfying the following axioms: (i) x x = 0 (ii) x 0 = x and (iii) (x y) (0 y) = x, for all x, y X For brevity we also call X a BG-algebra. A non-empty set S of a BG-algebra X is called a Subalgebra of X if x y S, for all x, y S. Definition 2.2. A nonempty subset I of a BG-algebra X is called an ideal of X if (i) 0 I (ii) x y I and y I imply x I, for all x, y X.

2 704 D. K. Basnet and L. B. Singh Example 2.3. Let X = {0, 1, 2, 3} be a set with the following table: Then (X, *, 0) is a BG-algebra. Definition 2.4. A fuzzy set μ in a BG-algebra X is called a fuzzy ideal of X if ( i) μ(0) μ(x) (ii) μ(x) min{ μ (x y), μ (y)}, for all x, y X. Example 2.5. Consider X = {0, 1, 2, 3} with defined by the table Then (X, *, 0) is a BG-algebra. Define μ: X [0, 1] by μ(0) = μ(1) = 0.5, μ(2) = μ(3) = 0.4. Then μ is a fuzzy ideal of X. 3. (, q)-fuzzy ideals of BG-algebra. Definition 3.1. A fuzzy set μ of the form μ(y) = t if y = x, t (0, 1] = 0 if y x is called a fuzzy point with support x and value t and it is denoted by x t. Definition 3.2. A fuzzy point x t is said to belong to (respectively be quasi coincident with) a fuzzy set μ, written as x t μ (respectively x t q μ) if μ(x) t (respectively μ(x) + t >1). If x t μ or x t q μ then x t q μ. q means q does not hold. Definition 3.3. A fuzzy subset μ of a BG-algebra X is said to be a (, q)-fuzzy ideal of X if (x y) t, y s μ x m(t, s) q μ i.e. μ (x) m(t, s) or μ(x) + m(t, s) >1, for all x, y X, where m(t, s) = min {t, s}.

3 Fuzzy ideals of BG-algebra 705 Theorem 3.4. A fuzzy subset μ of a BG algebra X is a fuzzy ideal if and only if μ is a (, )-fuzzy ideal. Proof. Let μ be a fuzzy ideal of X. Let x, y X such that (x y) t, y s μ, where t, s (0, 1]. Then μ(x y) t and μ(y) s. Now μ(x) min{μ (x y), μ (y)} min{t, s}= m(t, s) x m(t, s) μ. Conversely let μ be a (, )-fuzzy ideal. Let x, y X and t = μ (x y) and s = μ (y). Then μ(x y) t and μ(y) s, i.e., (x y) t, y s μ. By the given condition, x m(t, s) μ μ(x) m(t, s) = min{μ (x y), μ (y)}. Hence μ is a fuzzy ideal. Theorem 3.5. If μ is (q, q)-fuzzy ideal of a BG algebra X then it is also a (, )-fuzzy ideal of X. Proof. Let μ be a (q, q)-fuzzy ideal of a BG-algebra X. Let x, y X be such that (x y) t, y s μ. Then μ(x y) t and μ(y) s. If δ be an arbitrarily small positive number then we get μ(x y) + δ > t and μ(y) +δ > s μ(x y) + (1 + δ t) > 1 and μ(y) + (1 + δ s ) > 1 (x y) (1 + δ t) q μ and y (1 + δ s ) q μ. Since μ is (q, q)-fuzzy ideal of X so we get x m(1 + δ t, 1 + δ s) q μ μ(x) + m(1 + δ t, 1 + δ s) > 1 μ(x) + 1+ δ M(t, s) > 1, where M(t, s) = max{t, s}. μ(x) > M (t, s) δ. Since δ is arbitrary so μ(x) M (t, s) m(t, s). This shows that x m(t, s) μ and hence μ is a (, )-fuzzy ideal of X. Remark 3.6. However the converse of the above theorem is not true for example consider the BG algebra X = {0, 1, 2, 3} with defined by the table and let us define μ: X [0, 1] by μ(0) = μ(1) = 0.45, μ(2) = μ(3) = Then μ is a (, )-fuzzy ideal but it is not (q, q)-fuzzy ideal of X. Because if x = 2, y = 1, t = 0.7, s = 0.6, then (x y) t q μ and y s q μ but μ(x) + m(t, s) = μ(2) + m(0,7, 0.6) = = 0.97 < 1. Theorem 3.7. A fuzzy subset μ of a BG algebra X is a (, q)-fuzzy ideal of X if and only if μ(x) m(μ(x y), μ(y), 0.5) for all x, y X.

4 706 D. K. Basnet and L. B. Singh Proof. First let μ be a (, q)-fuzzy ideal of X. Case I. Let m(μ(x y), μ(y)) < 0.5, for all x, y X. Then m(μ(x y), μ(y), 0.5) = m(μ(x y), μ(y)) If possible let μ(x) < m(μ(x y), μ(y)). Choose a real number t such that μ(x) < t <m(μ(x y), μ(y)). Then (x y) t and y t μ. But μ(x) < t i.e., x t μ and μ(x) + t < t + t = 2t < 2 m(μ(x y), μ(y)) < 1. This contradicts the fact that μ is a (, q)-fuzzy ideal of X. Hence μ(x) m(μ(x y), μ(y)) = m(μ(x y), μ(y), 0.5). Case II. Let m(μ(x y), μ(y)) 0.5, for all x, y X. Then m(μ(x y), μ(y), 0.5) = 0.5. If possible let μ(x) < m(μ(x y), μ(y), 0.5) = 0.5. Then μ(x y) 0.5 and μ(y) 0.5, so (x y) 0.5, y 0.5 μ. But μ(x) < 0.5 i.e., x 0.5 μ and μ(x) < = 1, which is again a contradiction. Hence we must have μ(x) 0.5 = m(μ(x y), μ(y), 0.5). Conversely let μ(x) m(μ(x y), μ(y), 0.5). Let x, y X such that (x y) t, y s μ. Then μ(x y) t and μ(y) s. So m(μ(x y), μ(y)) m(t, s). By the given condition μ(x) m(μ(x y), μ(y), 0.5) m(t, s, 0.5). If m(t, s) 0.5, then m(t, s, 0.5) = m(t, s) and so μ(x) m(t, s) i.e., x m(t,s) μ. Again if m(t, s) > 0.5, then μ(x) 0.5 and so μ(x) + m(t, s) > = 1 i.e., x m(t, s) q μ. Hence x m(t, s) q μ and consequently μ is a (, q)-fuzzy ideal of X. Remark 3.8. A (, )-fuzzy ideal is always a (, q)-fuzzy ideal but not conversely. For example consider the BG algebra X = {0, 1, 2, 3} with defined by the table and let us define μ: X [0, 1] by μ(0) = μ(1) = μ(2) = 0.8, μ(3) = 0.6. Then μ is a (, q)-fuzzy ideal by theorem 3.7. But it is not a (, )-fuzzy ideal because = (3 1) 0.8, μ but μ. Theorem 3.9. If a fuzzy subset μ of a BG algebra X is a (, q)-fuzzy ideal of X and μ(x) < 0.5, for all x X then μ is also a (, )-fuzzy ideal of X. Proof. Let μ be a (, q)-fuzzy ideal of X and μ(x) < 0.5, for all x X. Let (x y) t μ and y s μ. Then t μ(x y) <0.5 and s μ(y) <0.5 and so m(t, s) < 0.5. Also μ(x) < 0.5. Thus μ(x) + m(t, s) < 1. But μ is a (, q)-fuzzy ideal of X, so we must have (x) m(t, s) μ. This shows that μ is a (, )-fuzzy ideal of X.

5 Fuzzy ideals of BG-algebra Homomorphism of BG-algebra and fuzzy ideals. Definition 4.1. Let X and X / be two BG algebras. Then a mapping f : X X / is said to be a homomorphism if f(x y) = f(x) f(y) for all x, y X. Theorem 4.2. Let X and X / be two BG algebras and f : X X / be a homomorphism. If μ / be a (, q)-fuzzy ideal of X / then f -1 (μ / ) is also a (, q)-fuzzy ideal of X. Proof. Recall that f -1 (μ / ) is defined as (f -1 (μ / ))(x) = μ / (f(x)) for all x X Let μ / be a (, q)-fuzzy ideal of X /. Let x, y X such that (x y) t, y s f -1 (μ / ). Then (f -1 (μ / ))(x y) t and (f -1 (μ / ))(y) s μ / (f(x y)) t and μ / (f(y)) s (f(x y)) t, f(y) s μ / (f(x) s f(y)) t, f(y) s μ /. But μ / is (, q)-fuzzy ideal of X /, so we have (f(x)) m(t, s) μ / or μ / (f(x)) + m(t, s) > 1 μ / (f(x)) m(t, s) or μ / (f(x)) + m(t, s) > 1 ((f -1 (μ / )(x) m(t, s) or (f -1 (μ / ))(x) + m(t, s) > 1 x m(s, t) f -1 (μ / ) or x m(t, s) q f -1 (μ / ). Hence f -1 (μ / ) is a (, q)-fuzzy ideal of X. Theorem 4.3. Let X and X / be two BG algebras and f : X X / be an onto homomorphism. If μ / be a fuzzy subset of X / such that f -1 (μ / ) is a (, q)-fuzzy ideal of X then μ / is also a (, q)-fuzzy ideal of X /. Proof. Let u, v X / such that (u v) t, v s μ /, where 0 t, s 1. Then μ / (u v) t and μ / (v) s. Since f is onto so there exists x, y X such that f(x) = u and f(y) = v. Also f is a homomorphism so f(x y) = f(x) f(y) = u v. So we get μ / (f(x y)) t and μ / (f(y)) s (f -1 (μ / ))(x y) t and (f -1 (μ / ))(y) s. But f -1 (μ / ) is a (, q)- fuzzy ideal of X, so we get (x) m(t,s) q f -1 (μ / ) (f -1 (μ / ))(x) m(t, s) or (f -1 (μ / ))(x) + m(t, s) > 1 μ / (f(x)) m(t, s) or μ / (f(x)) + m(t, s) > 1 μ / (u) m(t, s) or μ / (u) + m(t, s) > 1 u m(t,s) q μ /. Hence μ / is a (, q)-fuzzy ideal of X /. References [1] A. Zarandi and A. Borumand Saeid, Intuitionistic Fuzzy Ideals of BGalgebras, World Academy of Science, Engineering and technology 5 (2005),

6 708 D. K. Basnet and L. B. Singh [2] C. B. Kim, H. S. Kim, On BG-algebras, (submitted). [3] J. Neggers and H. S. Kim, On B-algebras, Math. Vensik 54 (2002), [4] P. L. Antony and P. L. Lilly, (α ; β)-fuzzy Lie algebras over an (α ; β)-fuzzy field, Journal of Generalized Lie Theory and Applications, Vol. 4 (2010), Article ID G090701, 8 pages, doi: /jglta/g [5] S. K. Bhakat and P. Das, Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems, 81(1996), [6] S. K. Bhakat and P. Das, (, q)-fuzzy subgroup, Fuzzy Sets and Systems, 80 (1996), [7] S. S. Ahn and D. Lee, Fuzzy subalgebras of BG algebras, Commun. Korean Math. Soc, 19 ( 2004 ), No2, Received: October, 2010