2 Basic Results on Subtraction Algebra

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1 International Mathematical Forum, 2, 2007, no. 59, Vague Ideals of Subtraction Algebra Young Bae Jun Department of Mathematics Education (and RINS) Gyeongsang National University, Chinju , Korea Chul Hwan Park Department of Mathematics University of Ulsan, Ulsan , Korea Abstract The notion of vague ideals in subtraction algebras is introduced, and several properties are investigated. Mathematics Subject Classification: 03G25, 03E72 Keywords: subtraction algebra; vague ideal 1 Introduction B. M. Schein [8] have considered systems of the form (Φ;, \), where Φ is a set of functions closed under the composition of functions (and hence (Φ; ) is a function semigroup) and the set theoretic subtraction \ (and hence (Φ; \) is a subtraction algebra in the sense of [1]). Several authors from time to time have made a number of generalizations of Zadeh s fuzzy set theory [9]. Of these, the notion of vague set theory introduced by Gau and Buehrer [4] is of interest to us. In this paper we introduce a notion of vague ideal in a subtraction algebra, and study some properties of them. 2 Basic Results on Subtraction Algebra By a subtraction algebra we mean an algebra (X; ) with a single binary operation that satisfies the following identities: for any x, y, z X, Corresponding author. H.P.: (C.H.Park)

2 2920 Y. B. Jun and C. H. Park (S1) x (y x) =x; (S2) x (x y) =y (y x); (S3) (x y) z =(x z) y. The last identity permits us to omit parentheses in expressions of the form (x y) z. The subtraction determines an order relation on X: a b a b =0, where 0 = a a is an element that does not depend on the choice of a X. The ordered set (X; ) is a semi-boolean algebra in the sense of [1], that is, it is a meet semilattice with zero 0 in which every interval [0,a]is a Boolean algebra with respect to the induced order. Here a b = a (a b); the complement of an element b [0,a]isa b; and if b, c [0,a], then b c = (b c ) = a ((a b) (a c)) = a ((a b) ((a b) (a c))). In a subtraction algebra, the following are true (see [6]): (a1) (x y) y = x y. (a2) x 0=x and 0 x =0. (a3) (x y) x =0. (a4) x (x y) y. (a5) (x y) (y x) =x y. (a6) x (x (x y)) = x y. (a7) (x y) (z y) x z. (a8) x y if and only if x = y w for some w X. (a9) x y implies x z y z and z y z x for all z X. (a10) x, y z implies x y = x (z y). (a11) (x y) (x z) x (y z). Definition 2.1. [6] A nonempty subset A of a subtraction algebra X is called an ideal of X, denoted by A X, if it satisfies: (b1) a x A for all a A and x X. (b2) for all a, b A, whenever a b exists in X then a b A. Proposition 2.2. [6] Let X be a subtraction algebra and let x, y X. If w X is an upper bound for x and y, then the element x y := w ((w y) x) is a least upper bound for x and y.

3 Vague ideals of subtraction algebras Basic Results on Vague Sets Definition 3.1. [3] A vague set A in the universe of discourse U is characterized by two membership functions given by: 1. A truth membership function and 2. A false membership function t A : U [0, 1] f A : U [0, 1] where t A (u) is a lower bound of the grade of membership of u derived from the evidence for u, and f A (u) is a lower bound on the negation of u derived from the evidence against u, and t A (u)+f A (u) 1. Thus the grade of membership of u in the vague set A is bounded by a subinterval [t A (u), 1 f A (u)] of [0, 1]. This indicates that if the actual grade of membership is μ(u), then The vague set A is written as t A (u) μ(u) 1 f A (u). A = { u, [t A (u),f A (u)] u U}, where the interval [t A (u), 1 f A (u)] is called the vague value of u in A and is denoted by V A (u). Definition 3.2. [3] A vague set A of a set U is called 1. the zero vague set of U if t A (u) =0and f A (u) =1for all u U, 2. the unit vague set of U if t A (u) =1and f A (u) =0for all u U. 3. the α-vague set of U if t A (u) =α and f A (u) =1 α for all u U, where α (0, 1). For α, β [0, 1] we now define (α, β)-cut and α-cut of a vague set. Definition 3.3. [3] Let A be a vague set of a universe X with the truemembership function t A and the false-membership function f A. The (α, β)-cut of the vague set A is a crisp subset A (α,β) of the set X given by A (α,β) = {x X V A (x) [α, β]}.

4 2922 Y. B. Jun and C. H. Park Clearly A (0,0) = X. The (α, β)-cuts are also called vague-cuts of the vague set A. Definition 3.4. [3] The α-cut of the vague set A is a crisp subset A α of the set X given by A α = A (α,α). Note that A 0 = X, and if α β then A β A α and A (α,β) = A α. Equivalently, we can define the α-cut as A α = {x X t A (x) α}. For our discussion, we shall use the following notations, which are given in [3], on interval arithmetic. Notation Let I[0, 1] denote the family of all closed subintervals of [0, 1]. If I 1 =[a 1,b 1 ] and I 2 =[a 2,b 2 ] be two elements of I[0, 1], we call I 1 I 2 if a 1 a 2 and b 1 b 2. Similarly we understand the relations I 1 I 2 and I 1 = I 2. Clearly the relation I 1 I 2 does not necessarily imply that I 1 I 2 and conversely. We define the term imax to mean the maximum of two intervals as imax(i 1,I 2 ) = [max(a 1,a 2 ), max(b 1,b 2 )]. Similarly we define imin. The concept of imax and imin could be extended to define isup and iinf of infinite number of elements of I[0, 1]. It is obvious that L = {I[0, 1], isup, iinf, } is a lattice with universal bounds [0, 0] and [1, 1] (see [3]). 4 Vague ideals In what follows let X be a subtraction algebra unless otherwise specified. Definition 4.1. A vague set A of X is called a vague ideal of X if the following conditions are true: (c1) ( x, y X) (V A (x y) V A (x)), (c2) ( x, y X) ( x y V A (x y) min{v A (x),v A (y)}), that is, 1. t A (x y) t A (x), 1 f A (x y) 1 f A (x), 2. t A (x y) min{t A (x),t A (y)}, 1 f A (x y) min{1 f A (x), 1 f A (y)} whenever there exists x y. (4.1)

5 Vague ideals of subtraction algebras 2923 Example 4.2. Consider a subtraction algebra X = {0, 1, 2} with the following Cayley table: Clearly, the vague set A = { 0, [0.7, 0.2], 1, [0.2, 0.4], 2, [0.5, 0.3] } is a vague ideal of X. Proposition 4.3. Zero vague set, unit vague set and α-vague set of X are trivial vague ideals of X. Proposition 4.4. If a vague set A of X satisfies ( x, a, b X)(V A (x ((x a) b)) min{v A (a),v A (b)}), (4.2) then A is a vague ideal of X. Proof. Let A be a vague set of X that satisfies (4.2). Then t A (x y) = t A ((x y) (((x y) x) x)) min{t A (x),t A (x)} = t A (x), 1 f A (x y) = 1 f A ((x y) (((x y) x) x)) min{1 f A (x), 1 f A (x)} =1 f A (x). Now suppose x y exists for x, y X. Putting w := x y, we have x y = w ((w x) y) by Proposition 2.2. It follows from (4.2) that and t A (x y) =t A (w ((w x) y)) min{t A (x),t A (y)}, 1 f A (x y) =1 f A (w ((w x) y)) min{1 f A (x), 1 f A (y)}. Hence A is a vague ideal of X. Proposition 4.5. For every vague ideal A of X, we have the following inequality: ( x X)(V A (0) V A (x)). (4.3) Proof. If we take y := x in (c1), then t A (0) = t A (x x) t A (x) and Hence (4.3) is valid. 1 f A (0) = 1 f A (x x) 1 f A (x).

6 2924 Y. B. Jun and C. H. Park Proposition 4.6. Let A be a vague set of X such that (k1) ( x X) (V A (0) V A (x)), (k2) ( x, y, z X) (V A (x z) min{v A ((x y) z),v A (y)}). Then we have the following implication: ( a, x X)(x a V A (x) V A (a)). (4.4) Proof. Let a, x X be such that x a. Then t A (x) = t A (x 0) min{t A ((x a) 0),t A (a)} = min{t A (0),t A (a)} = t A (a), 1 f A (x) = 1 f A (x 0) min{1 f A ((x a) 0), 1 f A (a)} = min{1 f A (0), 1 f A (a)} =1 f A (a). Hence V A (x) V A (a). The following proposition is straightforward. Proposition 4.7. A necessary and sufficient condition for a vague set A = (x, t A,f A ) of X to be a vague ideal of X is that t A and 1 f A are fuzzy ideals of X. Theorem 4.8. Let A be a vague ideal of X. Then for α [0, 1], the α-cut A α is a crisp ideal of X. Proof. Let x X and a A α. Then t A (a) α, and so t A (a x) t A (a) α. Thus a x A α. Let a, b A α and assume that there exists a b. Then t A (a) α and t A (b) α, which imply from (c2) that t A (a b) min{t A (a),t A (b)} α so that a b A α. Therefore A α is a crisp ideal of X. Theorem 4.9. Let A be a vague ideal of X. Then for any α, β [0, 1], the vague-cut A (α,β) is a crisp ideal of X. Proof. Let x X and a A (α,β). Then t A (a) α and 1 f A (a) β. Thus t A (a x) t A (a) α and 1 f A (a x) 1 f A (a) β. Therefore a x A (α,β). Now let a, b A (α,β) and assume that there exists a b. Then t A (a b) min{t A (a),t A (b)} α, 1 f A (a b) min{1 f A (a), 1 f A (b)} β, which shows that a b A (α,β). This completes the proof.

7 Vague ideals of subtraction algebras 2925 The ideals like A (α,β) are also called vague-cut ideals of X. Clearly we have the following result. Proposition Let A be a vague ideal of X. Two vague-cut ideals A (α,β) and A (ω,γ) with [α, β] < [ω, γ] are equal if and only if there is no x X such that [α, β] V A (x) [ω, γ]. Theorem Let X be finite and let A be a vague ideal of X. Consider the set V (A) given by V (A) :={V A (x) x X}. Then A i are the only vague-cut ideals of X, where i V (A). Proof. Consider [a 1,a 2 ] I[0, 1] where [a 1,a 2 ] / V (A). If [α, β] < [a 1,a 2 ] < [ω, γ] where [α, β], [ω, γ] V (A), then A (α,β) = A (a1,a 2 ) = A (ω,γ). If [a 1,a 2 ] < [a 1,a 3 ] where [a 1,a 3 ]=imin{x x V (A)}, then A (a1,a 3 ) = X = A (a1,a 2 ). Hence for any [a 1,a 2 ] I[0, 1], the vague-cut ideal A (a1,a 2 ) is one of A i for i V (A). This completes the proof. Theorem Any ideal J of X is a vague-cut ideal of some vague ideal of X. Proof. Consider the vague set A of X given by { [t, t], if x J, V A (x) = [0, 0], if x/ J, (4.5) where t (0, 1). It can be proved that V A (x y) V A (x) for all x, y X, and V A (x y) min{v A (x),v A (y)} whenever there exists x y for all x, y X. Thus A is a vague ideal of X. Clearly J = A (t,t). Theorem Let A be a vague ideal of X. Then the set is a crisp ideal of X. K := {x X V A (x) =V A (0)}

8 2926 Y. B. Jun and C. H. Park Proof. Let a K and x X. Then V A (a) =V A (0), and so V A (a x) V A (a) =V A (0) by (c1). It follows from (4.3) that V A (a x) =V A (0) so that a x K. Let a, b K and assume that there exists a b. By means of (c2), we know that V A (a b) min{v A (a),v A (b)} = V A (0). Thus V A (a b) =V A (0) by (4.3), and so a b K. Therefore K is a crisp ideal of X. Before finishing our discussion, we pose the following question: Question Does any vague ideal of a subtraction algebra satisfy the condition (4.2)? References [1] J. C. Abbott, Sets, Lattices and Boolean Algebras, Allyn and Bacon, Boston [2] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., Vol. 25, second edition 1984; third edition, 1967, Providence. [3] R. Biswas, Vague groups, Internat. J. Comput. Cognition 4 (2006), no. 2, [4] W. L. Gau and D. J. Buehrer, Vague sets, IEEE Transactions on Systems, Man and Cybernetics 23 (1993), [5] Y. B. Jun and H. S. Kim, On ideals in subtraction algebras, Sci. Math. Jpn. 65 (2007), no. 1, , :e2006, [6] Y. B. Jun, H. S. Kim and E. H. Roh, Ideal theory of subtraction algebras, Sci. Math. Jpn. 61 (2005), no. 3, , :e-2004, [7] Y. B. Jun and M. Kondo, On transfer principle of fuzzy BCK/BCIalgerbas, Sci. Math. Jpn. 59 (2004), no. 1, 35-40, :e9, [8] B. M. Schein, Difference Semigroups, Comm. in Algebra 20 (1992), [9] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), [10] B. Zelinka, Subtraction Semigroups, Math. Bohemica, 120 (1995), Received: May 24, 2007