Redefined Fuzzy K-algebras

World Applied Sciences Journal 7(7): 805-811, 2009 ISSN 1818-4952 c °IDOSI Publications, 2009

Redefined Fuzzy K-algebras M. Akram1 , K. H. Dar 2 , B. L. Meng3 and Y. L. Liu4 1

PUCIT, University of the Punjab, Old Campus, P. O. Box 54000, Lahore, Pakistan Department of Mathematics, G. C. University Lahore, P.O. Box 54000, Pakistan 3 Department of Basic Courses, Xian University of Science and Technology, P. R. China 4 Department of Mathematics, Wuyi University, Wuyishan, Fujian 354300, P. R. China 2

Abstract: We first introduce a new sort of fuzzy K-subalgebra called, an (≺, ≺ ∨Υ)∗ - fuzzy K-subalgebra of K-algebras using the notion of an anti fuzzy point and its besideness to and non-quasi-coincidence with a fuzzy set, and investigate some of its interesting properties. Then we introduce the concept of an anti fuzzy K-subalgebra with thresholds. Finally, we discuss generalization of an (≺, ≺ ∨Υ)∗ - fuzzy K-subalgebra. Key words: K-algebra, Besidesness, non-quasi-coincidence, (≺, ≺ ∨Υ)∗ - fuzzy subalgebra, (≺, ≺ ∨Υm )∗ fuzzy K-subalgebras

INTRODUCTION The notion of a K-algebra (G, ·, ¯, e) was first introduced by Dar and Akram [1] in 2003 and published in 2005. A K-algebra is an algebra built on a group (G, ·, e) by adjoining an induced binary operation ¯ on G which is attached to an abstract K-algebra (G, ·, ¯, e). This system is, in general non-commutative and nonassociative with a right identity e, if (G, ·, e) is noncommutative. For a given group G, the K-algebra is proper if G is not an elementary abelian 2-group. Thus, a K-algebra is abelian and non-abelian purely depends on the base group G. Dar and Akram further renamed a K-algebra on a group G as a K(G)-algebra [2] due to its structural basis G. The K(G)-algebras have been characterized by their left and right mappings in [2] when group is abelian. In 1965, Zadeh [3] introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. Since then it has become a vigorous area of research in different domains such as engineering, medical science, social science, physics, statistics, graph theory, artificial intelligence, signal processing, multiagent systems, pattern recognition, robotics, computer networks, expert systems, decision making, automata theory. On the other hand, Murali [4] proposed a definition of a fuzzy point belonging to a fuzzy subset under a natural equivalence on a fuzzy set. The idea of quasi-coincidence of a fuzzy point with a fuzzy set, which is mentioned in [5] played a vital role to generate some different types of fuzzy subgroups. A new type of fuzzy subgroups, (∈, ∈ ∨q)-fuzzy subgroups, was introduced in earlier paper Bhakat and Das [6] by using the combined notions of belongingness and quasi-coincidence of fuzzy point and fuzzy set. In fact, (∈, ∈ ∨q)-fuzzy subgroup

is an important and useful generalization of Rosenfeld’s fuzzy subgroup. Following this idea, Akram introduced the notion of (∈, ∈ ∨q)-fuzzy ideals of a K-algebra in [7]. In this paper, we introduce a new sort of fuzzy Ksubalgebra called, an (≺, ≺ ∨Υ)∗ - fuzzy K-subalgebra of K-algebras using the notion of an anti fuzzy point and its besideness to and non-quasi-coincidence with a fuzzy set, and investigate some of its interesting properties. We then introduce the concept of an anti fuzzy K-subalgebra with thresholds. In particular, we discuss generalization of an (≺, ≺ ∨Υ)∗ - fuzzy K-subalgebra. Some recent results obtained by Akram [8] and Saeid-Jun [9] are extended. The definitions and terminologies that we used in this paper are standard. For other notations, terminologies and applications, the readers are referred to [10-16]. PRELIMINARIES In this section we review some elementary facts that are necessary for this paper. Definition 1 [1] Let (G, ·, e) be a group in which each non-identity element is not of order 2. Then a K- algebra is a structure K = (G, ·, ¯, e) on a group G in which induced binary operation ¯ : G × G → G is defined by ¯(x, y) = x ¯ y = x . y −1 and satisfies the following axioms: (K1) (x ¯ y) ¯ (x ¯ z) = (x ¯ ((e ¯ z) ¯ (e ¯ y))) ¯ x, (K2) x ¯ (x ¯ y) = (x ¯ (e ¯ y)) ¯ x, (K3) (x ¯ x) = e, (K4) (x ¯ e) = x, (K5) (e ¯ x) = x−1 for all x, y, z ∈ G.

Corresponding Author: Muhammad Akram, [email protected] 805

World Appl. Sci. J., 7(7): 805-811, 2009

Definition 2 [10] A K-algebra K is called abelian if and only if x ¯ (e ¯ y) = y ¯ (e ¯ x) for all x, y ∈ G.

• xt ≺ γ and xt Υγ will be denoted by xt ≺ ∧Υγ. • xt ≺ γ or xt Υγ will be denoted by xt ≺ ∨Υγ.

If a K-algebra K is abelian, then the axioms (K1) and (K2) can be written as:

• The symbol ≺ ∧Υ means neither ≺ nor Υ hold. (≺, ≺ ∨Υ)∗ − FUZZY K-SUBALGEBRA

(K1) (x ¯ y) ¯ (x ¯ z) = z ¯ y . (K2) x ¯ (x ¯ y) = y.

Definition 7 A fuzzy set γ in G is called an (≺, ≺ ∨Υ)∗ −fuzzy K-subalgebra of K if the following conditions are satisfied:

A nonempty subset H of a K-algebra K is called a subalgebra [1] of the K-algebra K if a ¯ b ∈ H for all a, b ∈ H. Note that every subalgebra of a K-algebra K contains the identity e of the group G. A mapping f : K1 = (G1 , ·, ¯, e1 ) → K2 = (G2 , ·, ¯, e2 ) of Kalgebras is called a homomorphism [10] if f (x ¯ y) = f (x) ¯ f (y) for all x, y ∈ K1 . We note that if f is a homomorphism, then f (e) = e. We now introduce the following notions:

(3) xs ≺ γ ⇒ es ≺ ∨Υγ, (4) xs , yt ≺ γ ⇒ (x ¯ y)max(s,t) ≺ ∨Υγ for all x, y ∈ G, s, t ∈ [0, 1). Example 8 Consider the K-algebra K=(S3 , ·, ¯, e) on the symmetric group S3 ={e, a, b, x, y, z} where e = (1), a = (123), b = (132), x = (12), y = (13), z = (23), and ¯ is given by the following Cayley’s table:

Definition 3 A fuzzy set γ : G → [0, 1] is called an anti fuzzy subalgebra of K if γ(e) ≤ γ(x),

γ(x ¯ y) ≤ max{γ(x), γ(y)}

¯ e x y z a b

hold for all x, y ∈ G. We give the following Lemma without proof. Lemma 4 Let γ be a fuzzy set in K. Then γ is an anti fuzzy subalgebra of K if and only if

is a subalgebra of K for all t ∈ (0, 1]. Definition 5 For a family {µi : i ∈ Λ} of fuzzy sets of K, we define for all x ∈ K: S (1) µi (x) = (sup µi )(x) = sup µi (x), (2)

T i∈Λ

i∈Λ

a b b a z y x z y x e b a e

if t = e, if t = a, if t = b, x if t = y, z.

By routine computations, it is easy to see that γ is an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K.

i∈Λ

µi (x) = (inf µi )(x) = inf µi (x). i∈Λ

y z y z a b e a b e x y z x

Define a fuzzy set γ in K by  0.1,    0.3, γ(t) = 0.4,    0.6,

L(γ; t) = {x ∈ G : γ(x) ≤ t}

i∈Λ

e x e x x e y b z a a z b y

Theorem 9 Let γ be a fuzzy set in a algebra K. Then γ is an (≺, ≺ ∨Υ)∗ −fuzzy subalgebra of K if and only if it satisfies the following

i∈Λ

Definition 6 [8, 9] A fuzzy set γ in a set G of the form ½ t ∈ [0, 1), if y = x γ(y) = 1, y 6= x

γ(x ¯ y) ≤ max(γ(x), γ(y), 0.5), f or all x, y ∈ G.

is called an anti fuzzy point with support x and value t and is denoted by xt . An anti fuzzy point xt is said to “besides to" a fuzzy set γ, written as xt ≺ γ if γ(x) ≤ t. An anti fuzzy point xt is said to be “non-quasicoincident with" a fuzzy set γ, denoted by xt Υγ if γ(x) + t ≤ 1.

Proof: Let x, y ∈ G. We consider the following two cases:

The idea of non-quasi-coincidence of an anti fuzzy point with a fuzzy set play a vital role to generate some different types of fuzzy subalgebras, called (α, β)∗ fuzzy subalgebras, for α, β ∈ {≺, Υ, ≺ ∨Υ, ≺ ∧Υ}. Notations: The following notations will be used in this paper:

Case (i): Assume that γ(x ¯ y) > max (γ(x), γ(y), 0.5). Then γ(x ¯ y) > max(γ(x), γ(y)). Take s such that γ(x¯y) > s > max(γ(x), γ(y)). Then xs ≺ γ, ys ≺ γ, but (x ¯ y)s ≺ ∨Υγ, a contradiction. Case (ii): Assume that γ(x¯y) > 0.5. Then x0.5 , y0.5 ≺ γ but (x ¯ y)0.5 ≺ ∨Υγ, a contradiction.

(i) max(γ(x), γ(y)) > 0.5. (ii) max(γ(x), γ(y)) ≤ 0.5.

806

World Appl. Sci. J., 7(7): 805-811, 2009

Conversely let xs , yt ≺ γ, then γ(x) ≤ s, γ(y) ≤ t. Now, we have

Theorem 12 Let γ be a fuzzy set in a K-algebra K. Then L(γ; t), t ∈ [0.5, 1) is a subalgebra of K if and only if

γ(x ¯ y) ≤ max(γ(x), γ(y), 0.5) ≤ max(s, t, 0.5).

min(γ(x¯y), 0.5) ≤ max(γ(x), γ(y)), ∀ x, y ∈ G. (A)

If max(s, t) < 0.5, then γ(x ¯ y) ≤ 0.5 ⇒ γ(x ¯ y) + max(s, t) < 1. On the other hand, if max(s, t) ≥ 0.5, then γ(x ¯ y) ≤ max(s, t). Hence (x ¯ y)max(s,t) ≺ ∨Υγ. u t

Proof: Suppose that L(γ; t) is a subalgebra of K. Let x, y ∈ G, min(γ(x¯y), 0.5) > max(γ(x), γ(y)) = t, then t ∈ [0.5, 1), γ(x ¯ y) > t, x ≺ L(γ; t) and y ≺ L(γ; t). Since x, y ≺ L(γ; t) and L(γ; t) is a subalgebra of K, so x ¯ y ≺ L(γ; t) or γ(x ¯ y) ≤ t, a contradiction with γ(x ¯ y) > t. Conversely, suppose that condition (A) holds. Assume that t ∈ [0.5, 1), x, y ≺ L(γ; t). Then

Theorem 10 Let γ be an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K. (i) If there exists x ∈ G such that γ(x) ≤ 0.5, then γ(e) ≤ 0.5. (ii) If γ(e) > 0.5, then γ is an (≺, ≺)∗ -fuzzy subalgebra of K.

0.5

Proof: (i) Let x ∈ G such that γ(x) ≤ 0.5. −1

γ(x

≤

) = max(γ(x), 0.5) = 0.5.

≥

min(γ(x ¯ y), 0.5)

⇒

γ(x ¯ y) ≤ t,

Theorem 13 Let G0 ⊃ G1 ⊃ · · · ⊃ Gn = G be a strictly decreasing chain of a subalgebras of a K-algebra K, then there exists (≺, ≺ ∨Υ)∗ -fuzzy subalgebra γ of K whose level subalgebras are precisely the members of the chain with γ1 = G0 .

γ(x ¯ x)) = γ(x.x−1 ) max(γ(x), γ(x−1 ), 0.5) = 0.5.

(ii) γ(e) > 0.5 ⇒ γ(x) > 0.5 for all x ∈ G. Thus we conclude that:

Proof: Let {ti : ti ∈ [0.5, 1), i = 1, 2, · · · , n} be such that t1 > t2 > t3 > · · · > tn . Let γ : G → [0, 1] defined by  t, if x = 0,     n, if x = 0, x ∈ G0     t1 , if x ∈ G1 \ G0 , γ(x) = t2 , if x ∈ G2 \ G1 ,    ..   .    tn , if x ∈ Gn \ Gn−1 .

γ(x ¯ y) ≤ max(γ(x), γ(y)), ∀ x, y ∈ G. Hence γ is an (≺, ≺)∗ -fuzzy subalgebra of K. u t Theorem 11 Let γ be a fuzzy set of K. Then γ is an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K if and only if the nonempty set L(γ; t), t ∈ [0, 1) is a subalgebra of K. Proof: Assume that γ is an (≺, ≺ ∨Υ)∗ fuzzy subalgebra of K and let t ∈ [0, 1). If x, y ∈ L(γ; t), then γ(x) ≤ t and γ(y) ≤ t. Thus, it follows that

Let x, y ∈ G. If x ¯ y ∈ G0 , then from Theorem 3.3, we have γ(x ¯ y) ≤ 0.5 ≤ max(γ(x), γ(y), 0.5).

γ(x ¯ y) ≤ max(γ(x), γ(y), 0.5) ≤ max(t, 0.5) = t,

On the other hand, If x ¯ y ∈ / G0 , then there exists i, 1 ≤ i ≤ n such that x ¯ y ∈ Gi \ Gi−1 so that γ(x ¯ y) = ti . Now there exists j(≥ i) such that x ∈ Gj or y ∈ Gj . If x, y ∈ Gk (k < i), then Gk is a subalgebra of K, x ¯ y ∈ Gk which contradicts x ¯ y ∈ / Gi−1 . Thus

and so x ¯ y ∈ L(γ; t). This shows that L(γ; t) is a subalgebra of K. Conversely, let γ be a fuzzy set such that L(γ; t) = {x ∈ G : γ(x) ≤ t} is a subalgebra of K, for all t ∈ [0, 1). If there exist x, y ∈ G such that γ(x ¯ y) > max(γ(x), γ(y), 0.5), then we can take t ∈ (0.5, 1) such that γ(x ¯ y) > t > max(γ(x), γ(y), 0.5).

γ(x ¯ y) ≤ ti ≥ tj ≤ max(γ(x), γ(y), 0.5). Hence γ is an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K. It follows from the contradiction of γ that γ1.0 = G0 , γti = Gi for i = 1, 2, · · · , n. This completes the proof. u t

Thus x, y ∈ L(γ; t) and t > 0.5, and but x¯y ∈ / L(γ; t), which contradicts to the assumption that all L(γ; t) are subalgebras. Therefore,

Definition 14 An (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K is called proper if Im γ has at least two elements. Two (≺, ≺ ∨Υ)∗ - fuzzy subalgebras γ1 and γ2 are said to be equivalent if they have the same family of level subalgebras.

γ(x ¯ y) ≤ max(γ(x), γ(y), 0.5). Hence γ is an (≺, ≺ ∨Υ)∗ fuzzy subalgebra of K.

t ≥ max(γ(x), γ(y))

and so x ¯ y ≺ L(γ; t). This shows that L(γ; t) is a subalgebra of K. u t

Hence γ(e) =

>

u t 807

World Appl. Sci. J., 7(7): 805-811, 2009

Theorem 15 A proper (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K such that cardinality of {γ(x) : γ(x) > 0.5} ≤ 2 can be expressed as the union of two proper non-equivalent (≺, ≺ ∨Υ)∗ -fuzzy subalgebras of K.

The verification is analogous for other condition and we omit the details. By Theorem 9, it follows that γ is an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K. u t ∗ Remark 1. If {γi : i ∈ Λ} is a family of (≺, ≺ ∨Υ) S fuzzy subalgebras of K. Is γ = i∈Λ γi is an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K? When? The following example shows that it is not an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra in general.

Proof: Let γ be a proper (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K with {γ(x) : γ(x) > 0.5} = t1 , t2 , · · · , tn } where t1 < t2 < · · · < tn and n ≥ 2. Then γ0.5 ⊆ γt1 ⊆ · · · ⊆ γtn = G

Example 18 Consider the K-algebra K=G, ·, ¯, e), where G = {e, a, a2 , a3 } is the cyclic group of order 4 and ¯ is given by the following Cayley’s table:

is the chain of (≺, ≺ ∨Υ)∗ - subalgebras of γ. Define two fuzzy sets γ1 , γ2 ≥ γ defined by  t1 , if x ∈ γt1 ,     t2 , if x ∈ γt2 \ γt1 , µ1 (x) = ..  .    tn , if x ∈ γtn \ γtn−1 ,  γ(x), if x ∈ γ0.5 ,     if x ∈ γt2 \ γ0.5 ,   n, t3 , if x ∈ γt3 \ γt2 , µ2 (x) =  ..   .    tn , if x ∈ γtn \ γtn−1 , respectively, where t3 > n > t2 . Then γ1 and γ2 are (≺, ≺ ∨Υ)∗ -fuzzy subalgebras of K with

¯ e a a2 a3

γt0.5 ⊆ γt2 ⊆ · · · ⊆ γtn

1 =

Theorem 16 Let {γi : i ∈ Λ} beSa family of (≺, ≺)∗ fuzzy subalgebra of K. Then γ = i∈Λ γi is an (≺, ≺)∗ fuzzy subalgebra of K.

max(γ1 (a3 ), γ2 (a3 ))

= (γ1 ∪ γ2 )(a3 ) = (γ1 ∪ γ2 )(a1 ¯ a2 ) ≤ min((γ1 ∪ γ2 )(a), (γ1 ∪ γ2 )(a2 ), 0.5)

Proof: Let xs ≺ γ and yt ≺ γ, where s, t ∈ [0, 1). Then γ(x) ≤ s and γ(y) ≤ t. Thus we have γi (x) ≤ s and γi (y) ≤ t for all i ∈ Λ. Hence γi (x ¯ y) ≤ max(s, t). Therefore, γ(x ¯ y) ≤ max(s, t), which implies that (x ¯ y)max{s,t} ≺ γ. This completes the proof. u t

= min(0, 0, 0.5) = 0. Theorem 19 Let {γi : i ∈ Λ} be a family of (≺, ≺ ∨Υ)∗ -fuzzy subalgebras of K such S that γi ⊆ γj or γj ⊆ γi for all i, j ∈ Λ . Then γ := i∈Λ γi is an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of L.

Theorem 17 Let {γi : i ∈ Λ} be a family T of (≺, ≺ ∨Υ)∗ -fuzzy subalgebras of K. Then γ := i∈Λ γi is an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K.

Proof: By Theorem 9, we have γ(x ¯ y) max(γ(x), γ(y), 0.5), and hence

≤

γ(x ¯ y) =

inf γi (x ¯ y) inf max{γi (x), γi (y), 0.5}

= max{inf γi (x), inf γi (y), 0.5} i∈Λ i∈Λ \ \ = max{ γi (x), γi (y), 0.5}

=

sup max{γi (x), γi (y), 0.5} i∈Λ

i∈Λ

i∈Λ

sup γi (x ¯ y) i∈Λ

i∈Λ

≤

≤

a3 a a2 a3 e

then both γ1 and γ2 are (≺, ≺ ∨Υ)∗ -fuzzy subalgebras of K, but γ1 ∪ γ2 is not an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K since

being respectively chains of (Υ, ≺ ∨Υ)∗ -fuzzy subalgebras. Hence γ can be expressed as the union of two proper non-equivalent (≺, ≺ ∨Υ)∗ -fuzzy subalgebras of K. u t

=

a2 a2 a3 e a

   0.3 if x = e, 1 if x = a3 , γ2 (x) :=   0 if x ∈ {a, a2 }.

and

γ(x ¯ y)

a a3 e a a2

If we define fuzzy sets γ1 , γ2 : G → [0, 1] by putting    0.6 if x = e, 1 if x = a3 , γ1 (x) :=   0 if x ∈ {a, a2 }.

γt1 ⊆ γt2 ⊆ · · · ⊆ γtn

Proof: By Theorem 9, we have γ(x ¯ y) max(γ(x), γ(y), 0.5), and hence

e e a a2 a3

max{sup γi (x), sup γi (y), 0.5} i∈Λ i∈Λ [ [ = max{ γi (x), γi (y), 0.5}

i∈Λ

i∈Λ

= max{γ(x), γ(y), 0.5}.

= 808

i∈Λ

max{γ(x), γ(y), 0.5}.

≤

World Appl. Sci. J., 7(7): 805-811, 2009

It is not difficult to see that sup max{γi (x), γi (y), 0.5} ≥ i∈Λ

[

max(γi (x), γi (y), 0.5).

i∈Λ

Suppose that sup max{γi (x), γi (y), 0.5} 6= i∈Λ

[

max(γi (x), γi (y), 0.5),

i∈Λ

then there exists s such that

Theorem 23 Let f : K1 → K2 be an epimorphism of K-algebras. If γ is an anti fuzzy subalgebra with thresholds (m1 , m2 ) in K2 , then f −1 (γ) is an anti fuzzy subalgebra with thresholds (m1 , m2 ) in K1 , where

supi∈ΛSmax{γi (x), γi (y), 0.5} > s > i∈Λ max(γi (x), γi (y), 0.5). Since γi ⊆ γj or γj ⊆ γi for all i, j ∈ Λ, there exists k ∈ Λ such that s > max(γk (x), γk (y), 0.5). On the other hand, max(γi (x), γi (y), 0.5) > s for all i ∈ Λ, a contradiction. Hence

f −1 (γ)(x) = γ(f (x)) f or all x ∈ K1 . Proof: Let x, y ∈ K1 , then

supi∈Λ max{γ i (x), γS i (y), 0.5} S = max( i∈Λ γi (x), i∈Λ γi (y), 0.5)

min(f −1 (γ)(x ¯ y), m1 ) = min(γ(f (x ¯ y)), m1 ) = min(γ(f (x) ¯ f (y)), m1 ) ≤ max(γ(f (x)), γ(f (y)), m2 ) = max(f −1 (γ)(x), f −1 (γ)(y), m2 ).

= max{γ(x), γ(y), 0.5} The verification is analogous for other condition and we omit the details. By Theorem 9, it follows that γ is an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K. u t

The verification is analogous for other condition and we omit the details. Hence f −1 (γ) is an anti fuzzy subalgebra with thresholds (m1 , m2 ) in K1 . u t

Remark 20 From above discussion we know that: L(γ; t) is a subalgebra of K when (i) t ∈ [0, 1) (ii) t ∈ [0, 0.5) (iii) t ∈ [0.5, 1). An obvious question is whether µ is a kind of anti fuzzy subalgebra or not when L(γ; t)(6= ∅)(e.g. m1 , m2 ∈ [0, 1] and (m1 < m2 ). Based on this discussion, We can extend the concept of an anti fuzzy K-subalgebra with thresholds in the following way:

Theorem 24 Let f : K1 → K2 be an onto homomorphism of K-algebras. If γ is an anti fuzzy subalgebra with thresholds (m1 , m2 ) in K1 , then f (γ) is an anti fuzzy subalgebra with thresholds (m1 , m2 ) in K2 , where f (γ)(y) := inf{γ(x) | f (x) = y f or all y ∈ K2 }. Proof:Let y1 , y2 ∈ K2 . Then

Definition 21 Let m1 , m2 ∈ [0, 1] and m1 < m2 . If γ is a fuzzy set of a algebra K, then γ is called an anti fuzzy subalgebra with thresholds (m1 , m2 ) if

min(f (γ)(y1 ¯ y2 ), m1 ) = min(inf{γ(x1 ¯ x2 ), m1 ) | f (x1 ¯ x2 ) = y1 ¯ y2 }, m1 ) = inf{min(γ(x1 ¯ x2 ), m1 ) | f (x1 ¯ x2 ) = y1 ¯ y2 } ≤ inf{max(γ(x1 ), γ(x2 ), m1 ) | f (x1 ) = y1 , f (x2 ) = y2 } = max(inf{γ(x1 ) | f (x1 ) = y1 }, inf{γ(x2 ) | f (x2 ) = y2 }, m2 ) = max(f (γ)(y1 ), f (γ)(y2 ), m2 ).

(5) min(γ(e), m1 ) ≤ max(γ(x), m2 ) (6) min(γ(x ¯ y), m1 ) ≤ max(γ(x), γ(y), m2 ) for all x, y ∈ G. Theorem 22 A fuzzy set γ of algebra K is an anti fuzzy subalgebra with thresholds (m1 , m2 ) of K if and only if L(γ; t)(6= ∅), for t ∈ (m1 , m2 ], is a subalgebra of K.

The verification is analogous for other condition and we omit the details. Hence f (γ) is an anti fuzzy subalgebra with thresholds (m1 , m2 ) in K2 . u t

Proof: Assume that γ is an anti fuzzy subalgebra with thresholds (m1 , m2 ) of K. Let x, y ∈ L(γ; t). Then γ(x) ≤ t and γ(y) ≤ t, t ∈ (m1 , m2 ]. Then it follows that min(γ(x ¯ y), m1 ) ≤ =

Conversely, assume that γ is a fuzzy set such that L(γ; t) 6= ∅ is a subalgebra of K for m1 , m2 ∈ [0, 1] and m1 < m2 . Suppose that min(γ(x ¯ y), m1 ) > max(γ(x), γ(y), m2 ) = t, then γ(x ¯ y) > t, x ∈ L(γ; t) , y ∈ L(γ; t), t ∈ (m1 , m2 ]. Since x, y ∈ L(γ; t) and L(γ; t) are subalgebras, x ¯ y ∈ L(γ; t), i.e., γ(x ¯ y) ≤ t. This is a contradiction. The verification is analogous for other condition and we omit the details. This completes the proof. u t

(≺, ≺ ∨Υm )∗ − FUZZY K-SUBALGEBRA Let m be an element of (0, 1]unless otherwise specified. By xt Υm γ, we mean γ(x) + t + m < 1. The notation xt ≺ ∨ Υm γ means xt ≺ γ or xt Υm γ.

max(γ(x), γ(y), m2 ) t =⇒ γ(x ¯ y) ≤ t,

and hence x ¯ y ∈ L(γ; t). This shows that L(γ; t) is a subalgebra of K.

Definition 25 A fuzzy set γ in G is called an (≺, ≺ ∨Υm )∗ −fuzzy K-subalgebra of K if the following conditions are satisfied: 809

World Appl. Sci. J., 7(7): 805-811, 2009

(7) xs ≺ γ ⇒ es ≺ ∨Υm γ,

i.e., (a ¯ b) 1−m Υm γ. Hence (a ¯ b) 1−m ≺ ∨Υm γ, a 2 2 contradiction. So (B) is valid. Conversely, assume that γ satisfies (B). Let x, y ∈ G and t1 , t2 ∈ [0, 1) be such that xt1 ≺ γ and yt2 ≺ γ. Then

(8) xs , yt ≺ γ ⇒ (x ¯ y)max(s,t) ≺ ∨Υm γ for all x, y ∈ G, s, t ∈ [0, 1).

1−m 1−m ) ≥ max(t1 , t2 , ). 2 2

Note that an (≺, ≺ ∨Υm )∗ −fuzzy K-subalgebra with m = 0 is called (≺, ≺ ∨Υ)∗ −fuzzy K-subalgebra.

γ(x¯y) ≤ max(γ(x), γ(y),

Example 26 Consider the K-algebra K = (G, ·, ¯, e), where G = {e, a, a2 , a3 , a4 } is the cyclic group of order 5 and ¯ is given by the following Cayley’s table:

Assume that t1 ≥ 1−m or t2 ≥ 1−m 2 2 . Then γ(x) ≤ max(t1 , t2 )), which implies that (x ¯ y)max(t1 ,t2 ) ≺ γ. 1−m Now suppose that t1 < 1−m 2 and t2 < 2 . Then γ(x¯ 1−m y) ≤ 2 , and thus

¯ e a a2 a3 a4

e e a a2 a3 a4

a a4 e a a2 a3

a2 a3 a4 e a a2

a3 a2 a3 a4 e a

a4 a a2 a3 a4 e

γ(x ¯ y) + max(t1 , t2 ) >

1−m 1−m + = 1 − m, 2 2

i.e., (x ¯ y)max(t1 ,t2 ) Υm γ. Hence (x ¯ y)max(t1 ,t2 ) Υm γ, and consequently, γ is an (≺, ≺ ∨Υm )∗ -fuzzy subalgebra of K. u t Taking m = 0 in Theorem 28, we obtain the following corollary.

Define a fuzzy set γ in G defined by ½ 0.07, if x =e, γ(x) = 0.18, otherwise.

Corollary 29 A fuzzy set γ in K is an (≺, ≺ ∨Υ)∗ fuzzy subalgebra of K if and only if it satisfies the following assertion for all x, y ∈ G:

By routine computations, we can easily check that γ is an (≺, ≺ ∨Υ0.4 )∗ -fuzzy subalgebra of K.

γ(x ¯ y) ≤ max{γ(x), γ(y), 0.5}.

The following proposition is trivial. Theorem 30 Let γ be a fuzzy set of a K-algebra K. Then the non-empty level set L(γ; t) is a subalgebra of K for all t ∈ [ 1−m 2 , 1) if and only if γ satisfies the following assertion for all x, y ∈ G:

Proposition 27 Every (≺, ≺)∗ -fuzzy subalgebra is an (≺, ≺ ∨Υm )∗ -fuzzy subalgebra. Theorem 28 A fuzzy set γ in K is an (≺, ≺ ∨Υm )∗ fuzzy subalgebra of K if and only if it satisfies the following assertion for all x, y ∈ G: γ(x ¯ y) ≤ max{γ(x), γ(y),

1−m }. 2

min(γ(x ¯ y),

Proof: Let t ∈ [ 1−m 2 , 1) such that L(γ; t)(6= ∅) is a subalgebra of K. Assume that min(γ(x ¯ y), 1−m 2 ) > max(γ(x), γ(y)) = t for some x, y ∈ G, then t ∈ [ 1−m 2 , 1), γ(x) > t, x ∈ L(γ; t) and y ∈ L(γ; t). Since x, y ∈ L(γ; t), L(γ; t) is a subalgebra of K, so x ¯ y ∈ L(γ; t), a contradiction. The proof of the sufficiency part is straightforward and is hence omitted. This completes the proof. u t Taking m = 0 in Theorem 30, we obtain the following corollary.

(B)

Proof: Let γ be an (≺, ≺ ∨qm )∗ -fuzzy subalgebra of K. Assume that (B) is not valid. Then there exist a, b ∈ G such that γ(a ¯ b) > max{γ(a), γ(b)), If max(γ(a), γ(b) > max(γ(a), γ(b)). Thus

1−m 2 ,

1−m }. 2

then γ(a ¯ b)

γ(a ¯ b) > t ≥ max{γ(a), γ(b)}

>

Corollary 31 Let γ be a fuzzy set of a K-algebra K. Then the non-empty level set L(γ; t) is a subalgebra of K for all t ∈ [0.5, 1) if and only if γ satisfies the following assertion for all x, y ∈ G:

for some t ∈ [0, 1).

It follows that at ≺ γ and bt ≺ γ, but (a ¯ b)t ≺γ, a contradiction. Moreover, γ(a ¯ b) + t > 2t > 1 − m, and so (a ¯ b)t qm γ. Consequently (a ¯ b)t ≺ ∨Υm γ, a contradiction. On the other hand, If max(γ(a), γ(b)) ≤ 1−m 1−m 1−m and γ(a ¯ b) > 2 , then γ(a) ≤ 2 , γ(b) ≤ 2 1−m 1−m ≺ γ, b 1−m ≺ γ, but (a ¯ b) 1−m ≺γ. . Thus a 2 2 2 2 Also γ(a ¯ b) +

1−m ) ≤ max(γ(x), γ(y). 2

min(γ(x ¯ y), 0.5) ≤ max(γ(x), γ(y). Theorem 32 Let γ be an (≺, ≺ ∨Υm )∗ -fuzzy subalgebra of K. (i) If there exists x ∈ G such that γ(x) ≤ γ(e) ≤ 1−m 2 .

1−m 1−m 1−m > + = 1 − m, 2 2 2 810

1−m 2 ,

then

World Appl. Sci. J., 7(7): 805-811, 2009

(ii) If γ(e) > of K.

1−m 2 ,

then γ is an anti fuzzy subalgebra

REFERENCES 1. Dar, K. H. and Akram, M., 2005. On a K-algebra built on a group, Southeast Asian Bulletin of Mathematics, 29(1): 41-49. 2. Dar, K. H. and Akram, M., 2004. Characterization of a K(G)-algebras by self maps, Southeast Asian Bulletin of Mathematics, 28:601-610. 3. Zadeh, L. A., 1965. Fuzzy sets, Information and Control, 8:338-353. 4. Murali, V., 2004. Fuzzy points of equivalent fuzzy subsets, Information Sciences, 158: 277-288. 5. Pu, P. M. and Liu, Y. M., 1980. Fuzzy topology, I. Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76(2): 571-599. 6. Bhakat, S. K. and Dass, P., 1996. (∈, ∈ ∨q)-fuzzy subgroup, Fuzzy Sets and Systems, 80: 359-368. 7. Akram, M., 2008. (∈, ∈ ∨q)-ideals of K-algebras, Ars Combinatoria, 89:191-204. 8. Akram, M., 2008. Redefined fuzzy Lie subalgebras, Quasigroups and Related Systems, 16(2):119-132. 9. Saeid, A. B. and Jun, Y. B., 2008. Redefined fuzzy subalgebras of BCK/BCI-algebras, Iranian Journal of Fuzzy Systems, 5(2):63-70. 10. Dar, K. H. and Akram, M., 2007. On K-homomorphisms of K-algebras, International Mathematical Forum, 46: 2283-2293. 11. Yuan, X., Zhang, C. and Rena, Y., 2003. Generalized fuzzy groups and many-valued implications, Fuzzy Sets and Systems, 138: 205-211. 12. Zimmermann, H. -J., 1985. Fuzzy Set Theory and its Applications, Kluwer- Nijhoff Publishing. 13. Akram, M, 2008. Bifuzzy ideals of K-algebras, WSEAS Transactions on Mathematics, 7(5):313322. 14. Biswas, R., 1990. Fuzzy subgroups and anti fuzzy subgroups, Fuzzy Sets and Systems, 44:121-124. 15. Meng, J. and Jun, Y. B., 1994. BCK-algebras, Kyung Moon Sa Co. Seoul, Korea. 16. Rosenfeld, A. 1971. Fuzzy groups, J. Math. Anal. Appl., 35:512-517. 17. Jun, Y.B., 2009. New types of fuzzy BCK-filters, Honam Mathematical Journal, 31(2):157-166. 18. Mohebbi, M., Mohammad, R., Akbarzadeh, T. and Fard, A. M., 2007. Microorganism DNA Pattern Search in a Multi-agent Genomic Engine Framework, World Applied Sciences Journal, 2(6):582586.

Proof: (i) Let x ∈ G such that γ(x) ≤ γ(x−1 ) = max(γ(x),

1−m 2 .

1−m 1−m )= . 2 2

Hence γ(e)

= γ(x ¯ x)) = γ(x.x−1 ) 1−m 1−m ≤ max(γ(x), γ(x−1 ), )= . 2 2

(ii) γ(e) > 1−m ⇒ γ(x) > 2 we conclude that:

1−m 2

for all x ∈ G. Thus

γ(x ¯ y) ≤ max(γ(x), γ(y)), ∀ x, y ∈ G. Hence γ is an anti fuzzy subalgebra of K. u t Taking m = 0 in Theorem 32, we obtain the following corollary. Corollary 33 Let γ be an (≺, ≺ ∨Υ)∗ -fuzzy subalgebra of K. (i) If there exists x ∈ G such that γ(x) ≤ 0.5, then γ(e) ≤ 0.5. (ii) If γ(e) > 0.5, then γ is an anti fuzzy subalgebra of K. CONCLUSIONS In handling information regarding various aspects of uncertainty, non-classical logic (a great extension and development of classical logic) is considered to be more powerful technique than the classical logic one. The non-classical logic, therefore, has nowadays become a useful tool in computer science. Moreover, non-classical logic deals with the fuzzy information and uncertainty. In this connection we have introduced a new fuzzy subalgebra and have presented some properties of fuzzy subalgebra of a K-algebra using the notion of an anti fuzzy point and its besideness to and non-quasi-coincidence with a fuzzy set. In our opinion the future study of fuzzy K-algebras can be connected with: investigating (α, β)- bifuzzy K-ideals. Our obtained results can probably be applied in: (1) engineering; (2) computer science: artificial intelligence, signal processing; (3) medical diagnosis. Acknowledgement: The authors are highly thankful to the referees for their valuable comments and suggestions for improving the paper. 811