generalized anti fuzzy bi-ideals in ordered semigroups

ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2010, Vol. 31, No. 1, pp. 65–76. c© Pleiades Publishing, Ltd., 2010.

Generalized Anti Fuzzy Bi-Ideals in Ordered Semigroups

Young Bae Jun1*, Asghar Khan2**, Muhammad Shabir3***, and Seok Zun Song4

(Submitted by M.M. Arslanov)1Department of Mathematics Education Gyeongsang National University Chinju 660-701, Korea

2Department of Mathematics COMSATS Institute of IT Abbottabad, Pakistan3Department of Mathematics Quaid-i-Azam University Islamabad, Pakistan

4Department of Mathematics Cheju University, 690-756, KoreaReceived November 12, 2009

Abstract—Using the notion of anti fuzzy points and its besideness to and non-quasi-coincidentwith a fuzzy set, new concepts of an anti fuzzy bi-ideals in ordered semigroups are introduced andtheir inter-relations and related properties are investigated.

DOI: 10.1134/S1995080210010105

Key words and phrases: besides to, non-quasi coincidence with, (α, β)-anti fuzzy bi-ideal.

1. INTRODUCTION

The concept of a fuzzy set was first initiated by Zadeh [8]. Since then it has become a vigorous areaof research in engineering, medical science, social science, physics, statistics, graph theory, etc. In thispaper, we introduce the concept of an anti fuzzy bi-ideal of ordered semigroups by using the notionof anti fuzzy points and besideness to and non-quasi-coincidence with a fuzzy set, and investigatetheir inter-relations and related properties.

2. PRELIMINARIES

An ordered semigroup is a structure (S, ·,≤) satisfying the following conditions:(OS1) (S, ·) is a semigroups,(OS2) (S,≤) is a poset,(OS3) (∀a, b, x ∈ S)(a ≤ b =⇒ xa ≤ xb and ax ≤ bx).Let (S, ·,≤) be an ordered semigroup. For A ⊆ S, we denote

(A] := {t ∈ S|t ≤ h for some h ∈ A}.

For A,B ⊆ S, we denote, AB := {ab|a ∈ A, b ∈ B}. Let A,B ⊆ S. Then A ⊆ (A], (A](B] ⊆ (AB],and ((A]] = (A] [5].

Let S be an ordered semigroup and ∅ �= G ⊆ S. Then G is called a subsemigroup of S if G2 ⊆ G.A subsemigroup G of an ordered semigroup S is called a bi-ideal of S if (1) GSG ⊆ G and (2)(∀x, y ∈ S)(∀y ∈ G) (x ≤ y =⇒ x ∈ G) [5].

Definition 2.1. A fuzzy subset A of S is called an anti fuzzy bi-ideal of S if it satisfies:(i) (∀x, y ∈ S)(x ≤ y =⇒ A(x) ≤ A(y)),(ii) (∀x, y ∈ S)(A(xy) ≤max{A(x),A(y)}),

*E-mail: [email protected]**E-mail: [email protected]

***E-mail: [email protected]

65

66 JUN et al.

(iii) (∀a, x, y ∈ S)(A(xay) ≤max{A(x),A(y)}).Proposition 2.2. Let (S, ·,≤) be an ordered semigroup and A is a fuzzy subset of S. Then A is

an anti fuzzy bi-ideal of S if and only if the complement Ac of A is a fuzzy bi-ideal of S.Proof. =⇒. Suppose that A is an anti fuzzy bi-ideal of S. Let x, y ∈ S be such that x ≤ y. Then

Ac(x) = 1 −A(x) ≥ 1 −A(y) = Ac(y).

For a, x, y ∈ S, we have

Ac(xy) = 1 −A(xy) ≥ 1 − max{A(x),A(y)} = min{1 −A(x), 1 −A(y)} = min{Ac(x),Ac(y)},

Ac(xay) = 1 −A(xay) ≥ 1 − max{A(x),A(y)} = min{1 −A(x), 1 −A(y)} = min{Ac(x),Ac(y)}.

Hence Ac is a fuzzy bi-ideal of S.⇐=. Assume that Ac is a fuzzy bi-ideal of S. Let x, y ∈ S be such that x ≤ y. Then

A(x) = 1 −Ac(x) ≤ 1 −Ac(y) = A(y).

For a, x, y ∈ S we have

A(xy) = 1 −Ac(xy) ≤ 1 − min{Ac(x),Ac(y)} = max{1 −Ac(y), 1 −Ac(y)} = max{A(y),A(y)},

A(xay) = 1 −Ac(xay) ≤ 1 − min{Ac(x),Ac(y)} = max{1 −Ac(y), 1 −Ac(y)}= max{A(y),A(y)}.

Hence A is an anti fuzzy bi-ideal of S. �Given a fuzzy subset A of a set S and for every t ∈ [0, 1], the subsets

C(A; t) := {x ∈ S|A(x) ≤ t} and O(A; t) := {x ∈ S|A(x) < t}are called the closed t-cut and open t-cut of A, respectively.

Theorem 2.3. A fuzzy subset A of an ordered semigroup S is an anti fuzzy bi-ideal of S if andonly if C(A; t) is a bi-ideal of S.

Proof. Suppose that A is an anti fuzzy bi-ideal of S. Let x, y ∈ S be such that x ≤ y. If y ∈ C(A; t)then A(y) ≤ t. Since x ≤ y and A is an anti fuzzy bi-ideal of S, we have A(x) ≤ A(y), and henceA(x) ≤ t =⇒ x ∈ C(A; t). Let x, y ∈ C(A; t) and a ∈ S. Then A(x) ≤ t, and A(y) ≤ t. By (ii) and (iii) ofDefinition 2.1, we have

A(xy) ≤ max{A(x),A(y)} ≤ t, A(xay) ≤ max{A(x),A(y)} ≤ t.

Hence xy ∈ C(A; t) and xay ∈ C(A; t). Thus C(A; t) is a bi-ideal of S.Conversely, assume that C(A; t) is a bi-ideal of S. Let x, y ∈ S be such that x ≤ y. If A(y) <

A(x) then there exists t ∈ (0, 1) such that A(y) ≤ t < A(x). Then y ∈ C(A; t) but x /∈ C(A; t). Thisimpossible and hence A(y) ≤ A(x) for all x, y ∈ S with x ≤ y. Suppose that (ii) is not valid. Then

max{A(x),A(y)} ≤ t < A(xy),

for some x, y ∈ S and t ∈ (0, 1). Then x, y ∈ C(A; t) but xy /∈ C(A; t). This is a contradiction. HenceA(xy) ≤ max{A(x),A(y)} for all x, y ∈ S. Suppose that (iii) is false. Then

max{A(x),A(y)} ≤ t < A(xay),

for some a, x, y ∈ S and t ∈ (0, 1). Then x, y ∈ C(A; t) but xay /∈ C(A; t). This is a contradiction.Therefore A(xay) ≤ max{A(x),A(y)} for all a, x, y ∈ S. �

Example 2.4. Consider the set S = {a, b, c, d, e} with the following multiplication table andorder relation “≤”

(a, e), (b, d), (b, e), (c, e), (d, e)}≤:= {(a, a), (b, b), (c, c), (d, d), (e, e), (a, c), (a, d).

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 31 No. 1 2010

GENERALIZED ANTI FUZZY Bi-IDEALS 67

· a b c d e

a a d a d d

b a b a d d

c a d c d e

d a d a d d

e a d c d e

Then S is an ordered semigroup (see [5]) and {a}, {a, b, d} and {a, b, c, d} are bi-ideals of S. Define afuzzy subset A : S −→ [0, 1] by

A(a) = 0.2, A(b) = 0.4, A(c) = 0.5, A(d) = 0.3, A(e) = 0.6.

Then

C(A; t) =

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

S if t ∈ [0.6, 1){a, b, c, d} if t ∈ [0.5, 0.6){a, b, d} if t ∈ [0.4, 0.5){a} if t ∈ [0.2, 0.3)∅ if t ∈ [0, 0.2).

Then C(A; t) is a bi-ideal and by Theorem 2.3, A is an anti fuzzy bi-ideal of S.A fuzzy subset A of a set S of the form

A(y) =

{t ∈ [0, 1) if y = x,

1 otherwise,

is called an anti fuzzy point with support x and value t and is denoted byt

x. A fuzzy subset A of S is

said to be non-unit if there exists x ∈ S such that A(x) < 1.

For an anti fuzzy pointt

xand a fuzzy subset A in a set S, Jun and Song [4] gave meaning to the

symbolt

xαA, where α ∈ {[∈], [q], [∈] ∨ [q], [∈] ∧ [q]}.

To say thatt

x[∈]A(resp.

t

x[q]A) means that A(x) ≤ t(resp. A(x) + t < 1), and in this case,

t

xis said

to be beside to (resp. be non-quasi-coincident with) a fuzzy subset A. To say thatt

x[∈] ∨ [q]A (resp.

t

x[∈] ∧ [q]A) means that

t

x[∈]A or

t

x[q]A (resp.

t

x[∈]A and

t

x[q]A). To say that

t

xαA means that

t

xαA

does not hold.

3. GENERALIZED ANTI FUZZY BI-IDEALS

In what follows let S denote an ordered semigroup unless otherwise specified.Definition 3.1. A fuzzy subset A of S is called an ([∈], [∈])-fuzzy bi-ideal of S if it satisfies:

(i) (∀x, y ∈ S)(∀t ∈ [0, 1))(

x ≤ y,t

y[∈]A =⇒ t

x[∈]A

)

,

(ii) (∀x, y ∈ S)(∀t1, t2 ∈ [0, 1))(

t1x

[∈]A,t2y

[∈]A =⇒max{t1, t2}xy

[∈]A)

,

(iii) (∀x, y, a ∈ S)(∀t1, t2 ∈ [0, 1))(

t1x

[∈]A,t2y

[∈]A =⇒max{t1, t2}xay

[∈]A)

.


68 JUN et al.

Theorem 3.2. A fuzzy subset A of S is an ([∈], [∈])-fuzzy bi-ideal of S if and only if it satisfies:(i) (∀x, y ∈ S)(x ≤ y =⇒ A(x) ≤ A(y)),(ii) (∀x, y ∈ S)(A(xy) ≤max{A(x),A(y)}),(iii) (∀x, y, a ∈ S)(A(xay) ≤max{A(x),A(y)}).Proof. Assume that A satisfies the conditions (i), (ii) and (iii). Let x, y ∈ S and t ∈ [0, 1) be such that

x ≤ y andt

y[∈]A. Using (i), we have A(x) ≤ A(y) ≤ t, and so

t

x[∈]A. Let a, x, y ∈ S and t1, t2 ∈ [0, 1)

be such thatt1x

[∈]A andt2y

[∈]A. Then A(x) ≤ t1 and A(y) ≤ t2, which implies from (ii) and (iii) that

A(xy) ≤ max{A(x),A(y)} ≤ max{t1, t2},A(xay) ≤ max{A(x),A(y)} ≤ max{t1, t2}.

Hencemax{t1, t2}

xy[∈]A and

max{t1, t2}xay

[∈]A.

Conversely, assume that a fuzzy subset A of S is an ([∈], [∈])-fuzzy bi-ideal of S. Let x, y ∈ S be

such that x ≤ y. If A(y) < A(x), then there exists t ∈ (0, 1) such that A(y) ≤ t < A(x). Thust

y[∈]A,

butt

x[∈]A. This is impossible, and therefore A(x) ≤ A(y) for all x, y ∈ S with x ≤ y. Suppose that (ii)

is not valid. Thenmax{A(x),A(y)} ≤ t < A(xy)

for some x, y ∈ S and t ∈ (0, 1). It follows thatt

x[∈]A, and

t

y[∈]A, but

t

xy[∈]A. This is a contradicition.

Hence A(xy) ≤ max{A(x),A(y)} for all x, y ∈ S. Finally assume that (iii) is false. Then

max{A(x),A(y)} ≤ t < A(xay),

for every a, x, y ∈ S and t ∈ (0, 1). Hencet

x[∈]A, and

t

y[∈]A, but

t

xay[∈]A. This is a contradiction.

Therefore A(xay) ≤ max{A(x),A(y)} for all a, x, y ∈ S. �Remark 3.3. From Theorem 3.2, it follows that every anti fuzzy bi-ideal of S is an ([∈], [∈])-

fuzzy bi-ideal of S. But the converse is not true.Example 3.4. Consider the ordered semigroup given in Example 2.4, and define a fuzzy subset

A as follows:

A(a) = 0.2, A(b) = 0.4, A(c) = 0.8, A(d) = 0.6, A(e) = 0.5.

Then A is an ([∈], [∈])-fuzzy bi-ideal of S. But A is not an anti fuzzy bi-ideal of S, because

0.6 = A(d) = A(ab) ≤ max{A(a),A(b)} = max{0.2, 0.4} = 0.4.

Definition 3.5. A fuzzy subset A of S is called an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S if it satisfies:

(i) (∀x, y ∈ S)(∀t ∈ [0, 1))(

x ≤ y,t

y[∈]A =⇒ t

x[∈] ∨ [q]A

)

,

(ii) (∀x, y ∈ S)(∀t1, t2 ∈ [0, 1))(

t1y

[∈]A,t2x

[∈]A =⇒ max{t1, t2}xy

[∈] ∨ [q]A)

,

(iii) (∀a, x, y ∈ S)(∀t1, t2 ∈ [0, 1))(

t1y

[∈]A,t2x

[∈]A =⇒ max{t1, t2}xay

[∈] ∨ [q]A)

.

Theorem 3.6. A fuzzy subset A of S is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S if and only if itsatisfies:

(i) (∀x, y ∈ S)(x ≤ y =⇒ A(x) ≤ max{A(y), 0.5}),(ii) (∀x, y ∈ S)(A(xy) ≤ max{A(x),A(y), 0.5}),(iii) (∀a, x, y ∈ S)(A(xay) ≤ max{A(x),A(y), 0.5}).



Proof. Suppose that A satisfies the conditions (i), (ii) and (iii). Let x, y ∈ S, x ≤ y and t ∈ [0, 1) be

such thatt

y[∈]A. Then A(y) ≤ t, by using (i), we have

A(x) ≤ max{A(y), 0.5} ≤ max{t, 0.5}.

Thus A(x) ≤ t or A(x) ≤ 0.5, according to t > 0.5 or t ≤ 0.5. Thust

x[∈] ∨ [q]A. Let a, x, y ∈ S and

t, r ∈ [0, 1) be such thatt

x[∈]A and

r

y[∈]A. Then A(x) ≤ t and A(y) ≤ r and by using (ii) we have

A(xy) ≤ max{A(x),A(y), 0.5} ≥ max{t, r, 0.5},

If max{t, r} < 0.5 then A(xy) ≤ 0.5 and so A(xy) + max{t, r} < 0.5 + 0.5 = 1, and we havemax{t, r}

xy[q]A. If max{t, r} ≥ 0.5 then A(xy) ≤ max{A(x),A(y)} and so

max{t, r}xy

[∈]A and hence

max{t, r}xy

[∈] ∨ [q]A. Let a, x, y ∈ S and t, r ∈ [0, 1) be such thatt

x[∈]A and

r

y[∈]A. Then A(x) ≤ t

and A(y) ≤ r and by using (iii) we have

A(xay) ≤ max{A(x),A(y), 0.5} ≥ max{t, r, 0.5}.

If max{t, r} < 0.5 then A(xay) ≤ 0.5 and so A(xay) + max{t, r} < 0.5 + 0.5 = 1, and we havemax{t, r}

xay[q]A. If max{t, r} ≥ 0.5 then A(xay) ≤ max{A(x),A(y)} and so

max{t, r}xay

[∈]A and hence

max{t, r}xay

[∈] ∨ [q]A.

Conversely, let x, y ∈ S and x ≤ y. We consider the following cases:a) A(y) > 0.5,b) A(y) ≤ 0.5.Case a): Let x, y ∈ S and x ≤ y. Assume that max{A(y), 0.5} < A(x), which implies that A(y) <

A(x). Choose t such that A(y) ≤ t < A(x). Thent

y[∈]A but

t

x[∈]A and so

t

x[∈] ∨ [q]A. This is a

contradiction.Case b): Let x, y ∈ S and x ≤ y. Assume that max{A(y), 0.5} < A(x). Then A(x) > 0.5 and so

0.5y

[∈]A but0.5x

[∈]A and so0.5x

[∈] ∨ [q]A. This is a contradiction. Hence A(x) ≤ max{A(y), 0.5} for

all x, y ∈ S with x ≤ y.Let x, y ∈ S and we consider the following cases:a) max{A(x),A(y)} ≤ 0.5,b) max{A(x),A(y)} > 0.5.

Case a): Let x, y ∈ S be such that max{A(x),A(y), 0.5} < A(xy). Then A(xy) > 0.5,0.5x

[∈]A and

0.5y

[∈]A but0.5xy

[∈] ∨ [q]A. This is a contradiction.

Case b): Let x, y ∈ S be such that max{A(x),A(y), 0.5} < A(xy). Then max{A(x),A(y)} <

A(xy). Choose r ∈ [0, 1) such that max{A(x),A(y), 0.5} ≤ r < A(xy). Thenr

x[∈]A and

r

y[∈]A but

r

xy[∈]A and so

r

xy[∈] ∨ [q]A. This is a contradiciton. Hence A(xy) ≤ max{A(x),A(y), 0.5} for all

x, y ∈ S.Let a, x, y ∈ S and we consider the following cases:a) max{A(x),A(y)} ≤ 0.5,b) max{A(x),A(y)} > 0.5.


70 JUN et al.

Case a): Let a, x, y ∈ S be such that max{A(x),A(y), 0.5} < A(xay). Then A(xay) > 0.5,0.5x

[∈]A

and0.5y

[∈]A but0.5xay

[∈]A and so0.5xay

[∈] ∨ [q]A. This is a contradiction.

Case b): Let a, x, y ∈ S be such that max{A(x),A(y), 0.5} < A(xay). Then max{A(x),A(y)} <

A(xay). Choose s ∈ [0, 1) such that max{A(x),A(y), 0.5} ≤ s < A(xay). Thenr

x[∈]A and

r

y[∈]A but

r

xay[∈]A and

r

xay[∈] ∨ [q]A. This is a contradiciton. Hence A(xay) ≤ max{A(x),A(y), 0.5} for all

a, x, y ∈ S. �Remark 3.7. From Theorem 3.6, and Remark 3.3, it follows that every anti fuzzy bi-ideal of S

is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S. But the converse is not true.Example 3.8. Consider the ordered semigroup given in Example 2.4, and define a fuzzy subset

A as follows:

A(a) = 0.2, A(b) = 0.4, A(c) = 0.8, A(d) = 0.6, A(e) = 0.5.

Then A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S. But A is not an anti fuzzy bi-ideal of S, because

0.6 = A(d) = A(ab) ≤ max{A(a),A(b)} = max{0.2, 0.4} = 0.4.

In the following Theorem we provide a condition for an ([∈], [∈] ∨ [q])-fuzzy bi-ideal to an ([∈], [∈])-fuzzy bi-ideal.

Theorem 3.9. Let A be an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S such that A(x) > 0.5 for all x ∈ S.Then A is an ([∈], [∈])-fuzzy bi-ideal of S.

Proof. Let x, y ∈ S, x ≤ y and t ∈ [0, 1) be such thatt

y[∈]A. Then A(y) ≤ t and by (i) of Theorem

3.6, it follows that

A(x) ≤ max{A(y), 0.5} ≤ A(y) ≤ t,

and sot

x[∈]A. Let a, x, y ∈ S and r, s ∈ [0, 1) be such that

r

x[∈]A. and

s

x[∈]A. Then A(x) ≤ r and

A(y) ≤ s. By (ii) and (iii) of Theorem 3.6, it follows that

A(xy) ≤ max{A(x),A(y), 0.5} ≤ max{r, s},A(xay) ≤ max{A(x),A(y), 0.5} ≤ max{r, s}.

Hencemax{r, s}

xy[∈]A and

max{r, s}xay

[∈]A and therefore A is an ([∈], [∈])-fuzzy bi-ideal of S. �

For any fuzzy subset A of S and t ∈ [0, 1), we denote

Q(A; t) :={

x ∈ S

∣∣∣∣t

x[q]A

}

and [A; t] :={

x ∈ S

∣∣∣∣t

x[∈] ∨ [q]A

}

.

Obviously [A; t] = C(A; t) ∪ Q(A; t).In the following Theorem we provide another characterization of ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S by

using the set [A; t].Theorem 3.9. A fuzzy subset A of S is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S if and only if [A; t]

is a bi-ideal of S for all t ∈ [0, 1).We call [A; t] an ([∈] ∨ [q])-level bi-ideal of A.Proof. =⇒. Suppose that A is an ([∈], [∈]∨ [q])-fuzzy bi-ideal of S. Let x, y ∈ S, x ≤ y and t ∈ [0, 1)

be such that y ∈ [A; t]. Thent

y[∈]∨ [q]A and so A(y) ≤ t or A(y) + t < 1. Since A is an ([∈], [∈]∨ [q])-

fuzzy bi-ideal of S and x ≤ y we have

A(x) ≤ max{A(y), 0.5},then we have the following cases:



Case 1 A(y) ≤ t. If t ≤ 0.5, then A(x) ≤ 0.5 and so A(x) + t < 0.5 + 0.5 = 1. Hencet

x[q]A If t > 0.5

then A(x) ≤ max{A(y), 0.5} ≤ t and sot

x[∈]A.

Case 2 A(y) + t < 1. If t ≤ 0.5 then

A(x) ≤ max{A(y), 0.5} ≤ max{1 − t, 0.5} = 1 − t,

and so A(x) < 1 − t =⇒ A(x) + t < 1. Hencet

x[q]A. If t > 0.5 then

A(x) ≤ max{A(y), 0.5} ≤ max{1 − t, 0.5} = 0.5 ≤ t,

and sot

x[∈]A. Thus in both cases, we have

t

x[∈] ∨ [q]A and hence x ∈ [A; t]. Let x, y ∈ S and t ∈ [0, 1)

be such that x, y ∈ [A; t]. Thent

x[∈] ∨ [q]A and

t

y[∈] ∨ [q]A. Hence A(x) ≤ t or A(x) + t < 1 and

A(y) ≤ t or A(y) + t < 1. Since A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S, we have

A(xy) ≤ max{A(x),A(y), 0.5}.

Then we have the following cases:Case 1 Let A(x) < t and A(y) < t. If t ≤ 0.5. Then

A(xy) ≤ max{A(x),A(y), 0.5} = 0.5

and hencet

xy[q]A. If t > 0.5. Then

A(xy) ≤ max{A(x),A(y), 0.5} ≤ t

and sot

xy[∈]A. Hence

t

xy[∈] ∨ [q]A.

Case 2 Let A(x) ≤ t and A(y) + t < 1. If t ≤ 0.5, then

A(xy) ≤ max{A(x),A(y), 0.5} = max{A(y), 0.5}max{1 − t, 0.5} = 1 − t,

so A(xy) + t < 1 and hencet

xy[q]A. If t > 0.5. Then

A(xy) ≤ max{A(x),A(y), 0.5} ≤ max{t, 1 − t, 0.5} = t,

and hencet

xy[∈]A. Thus

t

xy[∈] ∨ [q]A.

Case 3 Let A(x) + t < 1 and A(y) ≤ t. If t ≥ 0.5, then

A(xy) ≤ max{A(x),A(y), 0.5} ≤ max{A(x), 0.5} = max{1 − t, 0.5} = 1 − t,

so A(xy) + t < 1 and hencet

xy[q]A. If t < 0.5, then

A(xy) ≤ max{A(x),A(y), 0.5} ≤ max{1 − t, t, 0.5} = t,

and so A(xy) ≤ t =⇒ t

xy[∈]A. Hence

t

xy[∈] ∨ [q]A.

Case 4 Let A(x) + t < 1 and A(y) + t < 1. If t ≤ 0.5, then

A(xy) ≤ max{A(x),A(y), 0.5} ≤ max{1 − t, 0.5} = 1 − t,

and so A(xy) + t < 1, hencet

xy[q]A. If t > 0.5 then

A(xy) ≤ max{A(x),A(y), 0.5} ≤ max{1 − t, 0.5} = 0.5 ≤ t,

hencet

xy[∈]A. Thus

t

xy[∈] ∨ [q]A. Therefore in any case we have

t

xy[∈] ∨ [q]A and hence xy ∈ [A; t].


72 JUN et al.

Now, let a, x, y ∈ S and t ∈ [0, 1) be such that x, y ∈ [A; t]. Thent

x[∈]∨ [q]A and

t

y[∈]∨ [q]A. Hence

A(x) ≤ t or A(x) + t < 1 and A(y) ≤ t or A(y) + t < 1. Since A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal ofS, we have

A(xay) ≤ max{A(x),A(y), 0.5}.

Then we have the following cases:

Case 1 Let A(x) < t and A(y) < t. If t ≤ 0.5. Then

A(xay) ≤ max{A(x),A(y), 0.5} = 0.5

and hencet

xay[q]A. If t > 0.5. Then

A(xay) ≤ max{A(x),A(y), 0.5} ≤ t

and sot

xay[∈]A. Hence

t

xay[∈] ∨ [q]A.

Case 2 Let A(x) ≤ t and A(y) + t < 1. If t ≤ 0.5, then

A(xay) ≤ max{A(x),A(y), 0.5} = max{A(y), 0.5}max{1 − t, 0.5} = 1 − t,

so A(xay) + t < 1 and hencet

xay[q]A. If t > 0.5. Then

A(xay) ≤ max{A(x),A(y), 0.5} ≤ max{t, 1 − t, 0.5} = t,

and hencet

xay[∈]A. Thus

t

xay[∈] ∨ [q]A.

Case 3 Let A(x) + t < 1 and A(y) ≤ t. If t ≥ 0.5, then

A(xay) ≤ max{A(x),A(y), 0.5} ≤ max{A(x), 0.5} = max{1 − t, 0.5} = 1 − t,

so A(xay) + t < 1 and hencet

xay[q]A. If t < 0.5, then

A(xay) ≤ max{A(x),A(y), 0.5} ≤ max{1 − t, t, 0.5} = t,

and so A(xay) ≤ t =⇒ t

xay[∈]A. Hence

t

xay[∈] ∨ [q]A.

Case 4 Let A(x) + t < 1 and A(y) + t < 1. If t ≤ 0.5, then

A(xay) ≤ max{A(x),A(y), 0.5} ≤ max{1 − t, 0.5} = 1 − t,

and so A(xay) + t < 1, hencet

xay[q]A. If t > 0.5 then

A(xay) ≤ max{A(x),A(y), 0.5} ≤ max{1 − t, 0.5} = 0.5 ≤ t,

hencet

xay[∈]A. Thus

t

xay[∈] ∨ [q]A. Therefore in any case we have

t

xy[∈] ∨ [q]A and hence xay ∈

[A; t]. Thus [A; t] is a bi-ideal of S.

Conversely, assume that A is a fuzzy subset of S and t ∈ [0, 1) be such that [A; t] is a bi-ideal of S. Ifpossible, let max{A(y), 0.5} ≤ t < A(x) for some t ∈ [0, 1). Then y ∈ C(A; t) ⊆ [A; t]. Since x ≤ y and[A; t] is a bi-ideal of S, we have x ∈ [A; t]. Then A(x) ≤ t of A(x) + t < 1. This is a contradition. Thus

A(x) ≤ max{A(y), 0.5} for all x, y ∈ S with x ≤ y.

Let x, y ∈ S and t ∈ [0, 1) be such that

max{A(x),A(y), 0.5} ≤ t < A(xy),



then x, y ∈ C(A; t) ⊆ [A; t]. Since [A; t] is a bi-ideal of S, we have xy ∈ [A; t]. Then A(xy) ≤ t orA(xy) + t < 1. This is a contradiction. Hence A(xy) ≤ max{A(x),A(y), 0.5} for all x, y ∈ S. Finally,let a, x, y ∈ S and t ∈ [0, 1) be such that

max{A(x),A(y), 0.5} ≤ t < A(xay),

then x, y ∈ C(A; t) ⊆ [A; t]. Since [A; t] is a bi-ideal of S, we have xay ∈ [A; t]. Then A(xay) ≤ tor A(xay) + t < 1. This is a contradiction. Hence A(xay) ≤ max{A(x),A(y), 0.5} for all x, y ∈ S.Therefore A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S. �

Theorem 3.10. Every ([∈]∨ [q], [∈]∨ [q])-fuzzy bi-ideal is an ([∈], [∈]∨ [q])-fuzzy bi-ideal of S.Proof. Let A be an ([∈] ∨ [q], [∈] ∨ [q])-fuzzy bi-ideal of S. Let x, y ∈ S with x ≤ y and t ∈ [0, 1)

be such thatt

y[∈]A. Then

t

y[∈] ∨ [q], [∈] ∨ [q]A. Since A is an ([∈] ∨ [q], [∈] ∨ [q])-fuzzy bi-ideal of S,

and x ≤ y we havet

x[∈] ∨ [q]A. Let x, y ∈ S and r, s ∈ [0, 1) be such that

t

x[∈]A, and

t

y[∈]A. Then

r

x[∈] ∨ [q]A, and

s

y[∈] ∨ [q]A and hence

max{r, s}x

[∈] ∨ [q]A. Finally, let a, x, y ∈ S and r, s ∈ [0, 1) be

such thatt

x[∈]A, and

t

y[∈]A. Then

t

x[∈] ∨ [q]A, and

t

y[∈] ∨ [q]A and we have

max{r, s}xay

[∈] ∨ [q]A.

Therefore A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S. �The converse of Theorem 3.10 is not true in general. Consider the ordered semigroup given in

Example 2.4, and define a fuzzy subset A : S −→ [0, 1] by

A(x) = 0.2, A(b) = 0.4, A(c) = 0.5, A(d) = 0.3, A(e) = 0.6.

Then A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S but A is not an ([∈] ∨ [q], [∈] ∨ [q])-fuzzy bi-ideal,

because0.22a

[∈]A and0.42

b[∈]A but

max{0.22, 0.42}ab

=0.42d

[∈]A.

Theorem 3.11. Every ([∈], [∈])-fuzzy bi-ideal is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S.Proof. Straightforward. �Theorem 3.12. A fuzzy subset A of S is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S if and only if

C(A; t) is a bi-ideal of S for all t ∈ [0.5, 1).Proof. Suppose that A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S. Let x, y ∈ S with x ≤ y and t ∈

[0.5, 1) be such that y ∈ C(A; t). Then A(y) ≤ t. Since A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal we have,

A(x) ≤ max{A(y), 0.5} ≤ t,

and so x ∈ C(A; t). Let x, y ∈ C(A; t). Then A(x) ≤ t and A(y) ≤ t. By hypothesis,

A(xy) ≤ max{A(x),A(y), 0.5} ≤ t,

and so xy ∈ C(A; t). Finally, let a, x, y ∈ S and t ∈ [0.5, 1) be such that x, y ∈ C(A; t). It follows from(iii) of Theorem that

A(xay) ≤ max{A(x),A(y), 0.5} ≤ t,

hence xay ∈ C(A; t). Therefore C(A; t) is bi-ideal of S.Conversely, assume that C(A; t) is a bi-ideal of S for all t ∈ [0.5, 1). Let x, y ∈ S with x ≤ y. If

possible, let there exists t ∈ [0.5, 1) such that

max{A(y), 0.5} ≤ t < A(x),

then A(y) ≤ t and we have y ∈ C(A; t). Since x ≤ y, we have x ∈ C(A; t). This is a contradiction. HenceA(x) ≤ max{A(y), 0.5} for all x, y ∈ S with x ≤ y. Let x, y ∈ S and t ∈ [0.5, 1) be such that

max{A(x),A(y), 0.5} ≤ t < A(xy),

then x, y ∈ C(A; t). Since C(A; t) is a bi-ideal we have xy ∈ C(A; t). Then A(xy) ≤ t. This is a contra-diction. Hence A(xy) ≤ max{A(x),A(y), 0.5} for all x, y ∈ S. Finally, let a, x, y ∈ S and t ∈ [0.5, 1) besuch that

max{A(x),A(y), 0.5} ≤ t < A(xay),


74 JUN et al.

then x, y ∈ C(A; t) and hence xay ∈ C(A; t). Then A(xay) ≤ t. This is a contradiction.Hence A(xay) ≤ max{A(x),A(y), 0.5} for all a, x, y ∈ S. Thus A is an ([∈], [∈] ∨ [q])-fuzzybi-ideal of S. �

The following example shows that there exists an ([∈], [∈]∨ [q])-fuzzy bi-ideal of S such that Q(A; t)is not a bi-ideal of S for some t ∈ [0, 1).

Example 3.13. Consider the ordered semigroup S = {a, b, c, d, e} as given in Example 2.4, anddefine a fuzzy subset A by:

A(a) = 0.2, A(b) = 0.3, A(c) = 0.8, A(e) = 0.5, A(d) = 0.6.

Then Q(A; t) = {a, b, d, e} for all 0.5 < t ≤ 0.6. Since0.3a

[∈]A and0.6b

[∈]A. But

max{0.3, 0.6}ab

=0.6d

[q]A.

Definition 3.14. A fuzzy subset A of S is called an ([q], [q])-fuzzy bi-ideal of S if it satisfies:

(i) (∀x, y ∈ S)(t ∈ [0, 1))(x ≤ y,t

y[q]A =⇒ t

x[q]A),

(ii) (∀x, y ∈ S)(r, s ∈ [0, 1))(r

x[q]A,

s

y[q]A =⇒ max{r, s}

xy[q]A),

(iii) (∀a, x, y ∈ S)(r, s ∈ [0, 1))(r

x[q]A,

s

y[q]A =⇒ max{r, s}

xay[q]A).

Theorem 3.17. Let B be a bi-ideal of S and let A be a fuzzy subset of S such that(i) (∀x ∈ S\B)(A(x) = 1),(ii) (∀x ∈ B)(A(x) ≤ 0.5).Then A is both a ([q], [∈] ∨ [q])-fuzzy bi-ideal and an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S.

Proof. Let x, y ∈ S, x ≤ y and t ∈ [0, 1) be such thatt

y[q]A. Then y ∈ B. Since x ≤ y ∈ B, then

x ∈ B. If t ≥ 0.5 then A(x) ≤ 0.5 ≤ t and hencet

x[∈]A. If t < 0.5 then A(x) + t < 0.5 + 0.5 = 1 and

sot

x[q]A. Hence

t

x[∈]∨ [q]A for all x, y ∈ S with x ≤ y. Let x, y ∈ S and r, s ∈ [0, 1) be such that

t

x[q]A

andt

y[q]A. Then x, y ∈ B and hence xy ∈ B. If max{r, s} ≥ 0.5 then A(xy) ≤ 0.5 ≤ max{r, s} and so

max{r, s}xy

[∈]A. If max{r, s} < 0.5 then A(xy) + max{r, s} < 0.5 + 0.5 = 1, and hencemax{r, s}

xy[q]A.

Thusmax{r, s}

xy[∈] ∨ [q]A. Let a, x, y ∈ S and r, s ∈ [0, 1) be such that

t

x[q]A and

t

y[q]A. Then

x, y ∈ B and hence xay ∈ B. If max{r, s} ≥ 0.5 then A(xay) ≤ 0.5 ≤ max{r, s} and somax{r, s}

xay[∈]

A. If max{r, s} < 0.5 then A(xay) + max{r, s} < 0.5 + 0.5 = 1, and hencemax{r, s}

xay[q]A. Thus

max{r, s}xay

[∈] ∨ [q]A. Therefore A is an ([q], [∈] ∨ [q])-fuzzy bi-ideal of S.

Let x, y ∈ S, x ≤ y and t ∈ [0, 1) be such thatt

y[∈]A. ThenA(y) ≤ t and so y ∈ B. Since x ≤ y ∈ B,

then x ∈ B. If t ≥ 0.5 then A(x) ≤ 0.5 ≤ t and hencet

x[∈]A. If t < 0.5 then A(x) + t < 0.5 + 0.5 = 1

and sot

x[q]A. Hence

t

x[∈] ∨ [q]A for all x, y ∈ S with x ≤ y. Let x, y ∈ S and r, s ∈ [0, 1) be such that

t

x[∈]A and

t

y[∈]A. Then A(x) ∈ t and A(y) ≤ t and so x, y ∈ B and hence xy ∈ B. If max{r, s} ≥ 0.5



then A(xy) ≤ 0.5 ≤ max{r, s} and somax{r, s}

xy[∈]A. If max{r, s} < 0.5 then A(xy) + max{r, s} <

0.5 + 0.5 = 1, and hencemax{r, s}

xy[q]A. Thus

max{r, s}xy

[∈] ∨ [q]A. Let a, x, y ∈ S and r, s ∈ [0, 1)

be such thatt

x[∈]A and

t

y[∈]A. Then A(x) ∈ t and A(y) ≤ t and so x, y ∈ B and hence xay ∈ B.

If max{r, s} ≥ 0.5 then A(xay) ≤ 0.5 ≤ max{r, s} and somax{r, s}

xay[∈]A. If max{r, s} < 0.5 then

A(xay) + max{r, s} < 0.5 + 0.5 = 1, and hencemax{r, s}

xay[q]A. Thus

max{r, s}xay

[∈] ∨ [q]A. Therefore

A is an ([q], [∈] ∨ [q])-fuzzy bi-ideal of S. �Note that every ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S may not be ([q], [∈] ∨ [q])-fuzzy bi-ideal of S.Example 3.17. Consider an ordered semigroup S = {a, b, c, d, e} as given in Example 2.4, and

define an fuzzy subset A by:

A(a) = 0.2, A(b) = 0.4, A(c) = 0.5, A(d) = 0.3, A(e) = 0.6.

Then A is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal. But A is not a ([q], [∈] ∨ [q])-fuzzy bi-ideal of S. Since0.11a

[q]A and0.21

b[q]A. But

max{0.11, 0.21}ab

=0.21d

[∈]A.

Proposition 3.18. If {Ai}i∈I is a family of ([∈], [∈] ∨ [q])-fuzzy bi-ideals of S. Then⋂

i∈I

Ai is an

([∈], [∈] ∨ [q])-fuzzy bi-ideal of S, where⋂

i∈I

Ai(x) =∧

i∈I

Ai(x) for all x ∈ S.

Proof. Let x, y ∈ S be such that x ≤ y. Then

⋂

i∈I

Ai(x) =∧

i∈I

Ai(x) ≤∧

i∈I

{Ai(y) ∨ 0.5} =

{∧

i∈I

Ai(y) ∨ 0.5

}

=

{(⋂

i∈I

Ai

)

(y) ∨ 0.5

}

.

Let a, x, y ∈ S. Then

⋂

i∈I

Ai(xy) =∧

i∈I

Ai(xy) ≤∧

i∈I

{Ai(x) ∨ Ai(y) ∨ 0.5} =

{∧

i∈I

Ai(x) ∨∧

i∈I

Ai(y)

}

∨ 0.5

=

{(⋂

i∈I

Ai

)

(x) ∨(

⋂

i∈I

Ai

)

(y) ∨ 0.5

}

,

and⋂

i∈I

Ai(xay) =∧

i∈I

Ai(xay) ≤∧

i∈I

{Ai(x) ∨ Ai(y) ∨ 0.5} =

{∧

i∈I

Ai(x) ∨∧

i∈I

Ai(y)

}

∨ 0.5

=

{(⋂

i∈I

Ai

)

(x) ∨(

⋂

i∈I

Ai

)

(y) ∨ 0.5

}

.

Thus⋂

i∈I

Ai is an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S. �

Note that if {Ai}i∈I is a family of ([∈], [∈] ∨ [q])-fuzzy bi-ideals of S then⋃

i∈I

Ai may not be an

([∈], [∈] ∨ [q])-fuzzy bi-ideals of S.Example 3.19. Let S = {a, b, c, d, e} be an ordered semigroup, which is given in Example 2.4, and

let A1 and A2 be a fuzzy subset of S which are defined as follows:

A1(a) = 0.3, A1(b) = 0.5, A1(d) = 0.6, A1(c) = A1(e) = 0.7


76 JUN et al.

A2(a) = 0.5, A2(b) = A2(c) = A2(d) = 0.8, A2(e) = 0.75.

Then

C(A1; t) :=

{S if 0.7 ≤ t < 1{a, b, d} if 0.6 ≤ t < 0.7

C(A2; t) :=

{S if 0.8 ≤ t < 1{a} if 0.5 ≤ t < 0.8.

The union of A1 and A2, A1 ∪ A2 is given by:

A1 ∪ A2(a) = 0.5, A1 ∪A2(b) = A1 ∪ A2(c) = A1 ∪ A2(d) = 0.8, A1 ∪A2(a) = 0.75.

Then

C(A1 ∪ A2; t) :=

{S if 0.8 ≤ t < 1{a, e} if 0.5 ≤ t < 0.8.

Since {a, e} is not a bi-ideal of S, it follows from Theorem 3.6 that A1 ∪ A2 is not an ([∈], [∈] ∨ [q])-fuzzy bi-ideal of S.

ACKNOWLEDGEMENT

We would like to thank for financial support during this work under CIIT research grant program.

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Systems (to appear).4. Y. B. Jun and S. Z. Song, Generalized anti fuzzy subalgebras, East Asian Math. J. 22 (2), 195 (2006).5. A. Khan, Y. B. Jun and M. Shabir, Ordered semigroups characterized by their (∈,∈ ∨q)-fuzzy bi-ideals,

Bull. Malays. Math. Sci. Soc. (2) 32 (3), 391 (2009).6. P. M. Pu and Y. M. Liu, Fuzzy topology I, Neighbourhood structure of a fuzzy point and Moore-Smith

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