BCI-Algebras

International Journal of Fuzzy Systems, Vol. 14, No. 1, March 2012

175

Another Generalization of Fuzzy BCK/BCI-Algebras A. Borumand Saeid, M. Kuchaki Rafsanjani, and D. R. Prince Williams Abstract1 In this paper a new generalization of the fuzzy BCK/BCI-algebras of type (α , β )* are introduced. Then we state and prove some theorems which determine the relationships between these notions and sub algebras of BCK/BCI-algebras. Characterizations of (α , β )* -interval-valued fuzzy sub algebras in BCK/BCI-algebras are given. Keywords: Besides to, non-quasi coincident with, (α , β )* -interval-valued fuzzy sub algebra.

1. Introduction The concept of fuzzy sets was first initiated by Zadeh [9]. Since then it has become a vigorous area of research in engineering, medical science, social science, physics, statistics, graph theory, etc. After the introducing of fuzzy sets by Zadeh [9], there have been a number of generalizations of this fundamental concept. A new type of fuzzy subgroup, that is, the (∈,∈ ∨q ) -fuzzy subgroup, was introduced by Bhakat and Das in [2] by using the combined notions of “belongingness” and “quasi-coincidence” of fuzzy points and fuzzy sets. In fact, the (∈,∈ ∨q ) -fuzzy subgroup is an important generalization of Rosenfeld's fuzzy subgroup. Now, it is natural to investigate some similar types of generalizations of the existing fuzzy subsystems by considering some other structures [1, 3, 5, 6]. Fuzzy BCK-algebra is studied in some papers. In [4], A. Borumand Saeid and Y. B. Jun introduced new definition of fuzzy BCK/BCI-algebras by considering two relations (called besideness and non quasi-coincidence) between an anti-fuzzy point and a fuzzy set. In this paper, we generalized the notion of fuzzy sub Corresponding Author: Arsham Borumand Saeid is with Dept. of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran. E-mail: [email protected] Marjan Kuchaki Rafsanjani is with Dept. of Computer Science, Shahid Bahonar University of Kerman, Kerman, Iran. E-mail: [email protected] D. R Prince Williams is with Dept. of Information Technology, College of Applied Sciences, Post Box: 135, Sohar-311, Sultanate of Oman. E-mail: [email protected]

algebra of BCK/BCI-algebras which is defined in [4] and the notion of (α , β )* -interval-valued fuzzy sub algebra of a BCK/BCI-algebra are introduced where (α , β ) ∈ {∠, ϒ, ∠ ∨ ϒ, ∠ ∧ ϒ} , β ≠ ∠ ∧ ϒ .

2. Preliminaries Definition 2.1 [7]: Let X be a non-empty set with a binary operation “*” and a constant “0”. Then (X, *, 0) is called a BCI-algebra if it satisfies the following conditions: (i) ((x * y ) *(x * z )) *(z * y ) = 0 , (ii) (x *(x * y )) * y = 0, (iii) x * x = 0 , (iv) x * y = 0 and y * x = 0 .imply x = y , for all x , y ,z ∈X . If a BCI-algebra X satisfies 0* x = 0 , for all x ∈ X , then we say that X is a BCK-algebra. A nonempty subset S of X is called a sub algebra of X if x * y ∈ S for all x , y ∈ S . We refer the reader to [7] for further information regarding BCK/BCI-algebras. A fuzzy set A in X of the form

⎧⎪t ∈ [0,1) y = x A ( y ) := ⎨ (1) ⎪⎩1 y ≠ x is called an anti-fuzzy point with support x and value t and is denoted by x t . A fuzzy set A in X is said to be non-unit if there exists x ∈ X such that A (x ) < 1. A fuzzy set A in a BCK/BCI-algebra X is called anti-fuzzy sub algebra of X if it satisfies (∀x , y ∈ X ) (A (x * y ) ≤ max{A (x ), A ( y )}. (2) Proposition 2.2 [4]: Let A be a fuzzy set in X . Then A is an anti-fuzzy sub algebra of X if and only if L (A ; t ) := {x ∈ X A (x ) ≤ t } is a sub algebra of X , for all t ∈ [0,1). In [8], Pu and Liu initiated “belong to” relation

(∈) and the “quasi coincident with” relation (q ) be-

tween a fuzzy point and a fuzzy set. In [4], the “beside to” relation (∠) and the “non quasi coincident” relation

© 2012 TFSA


176

( ϒ) between an anti-fuzzy point and a fuzzy set were introduced. Definition 2.3 [4]: An anti-fuzzy point x t is said to beside to (resp. be non-quasi coincident with) a fuzzy set A , denoted by x t ∠A (resp. x t ϒA ), if A (x ) ≤ t

Now we generalize the notion of non-quasi-coincidence and introduce the notion of (α , β )* -interval-valued fuzzy BCK-algebras Let x ∈ X and t ∈ D [0,1]. An interval-valued fuzzy set of the form

⎧⎪t y = x A ( y ) := ⎨ (5) ⎪⎩[1,1] y ≠ x where [0, 0] ≤ t ≤ [1,1]. is called interval-valued (briefly, i-v) anti fuzzy point and denoted by x t . An i-v fuzzy set A in X is called an i-v anti-fuzzy (∀x , y ∈ X ) (∀t 1 , t 2 ∈ [0,1)) sub algebra of X if it satisfies (∀x , y ∈ X )(A (x * y ) ≤ r max{A (x ), A ( y )}. (6) (x t1 , y t 2 ∠A ⇒ (x * y ) max{t1 ,t 2 } ∠A ). (3) Note that if A is a fuzzy set in X such that Definition 3.1: An i-v anti-fuzzy point x t . is said to A (x ) ≥ 0.5, for all x ∈ X , then the set beside to (resp. be non-quasi coincident with) a fuzzy set A denoted by x t ∠A (resp. x t ϒA ), if A (x ) ≤ t {x t x t ∠ ∧ ϒA } is empty. Definition 2.5 [4]: A fuzzy set A in a (resp. A ( x ) + t < [1,1] ]). We say that ∠ (resp. ϒ ) is

(resp. A (x ) + t < 1. ). We say that (∠) (resp. ( ϒ) ) is a beside to (resp. non-quasi coincident with) relation between anti-fuzzy points and fuzzy sets. Proposition 2.4 [4]: Let A be a fuzzy set in a BCK/BCI-algebra X . Then A satisfies the condition (2) if and only if it satisfies the following condition.

BCK/BCI-algebra X is called an (α , β )* -fuzzy sub algebra of X , where α ≠ ∠ ∧ ϒ if it satisfies the following implication (∀x , y ∈ X )

(∀t 1 , t 2 ∈ [0,1))

(x t1α A , y t 2 α A ⇒ (x * y ) max{t1 ,t 2 } β A ).

(4)

An interval-valued fuzzy set [10] (briefly, i-v fuzzy set) A defined on X is given by L U A = {(x ,[ μ A (x ), μA (x )]) x ∈ X } Briefly, denoted by A = [ μ , μ ] where μ and μ L A

U A

L A

U A

are any two

fuzzy sets in X such that μAL (x ) ≤ μUA (x ), for all

x ∈X .

a beside to relation (resp. non-quasi coincident with relation) between i-v anti-fuzzy points and i-v fuzzy sets. If x t ∠A or x t ϒA (resp. x t ∠A and x t ϒA ), we say that x t ∠ ∨ ϒA (resp. x t ∠ ∧ ϒA ). Proposition 3.2: Let A be an i-v fuzzy set in X . Then A satisfies in (6) if and only if it satisfies the following condition.

(∀x , y ∈ X )

(∀[0, 0] ≤ t 1 , t 2 ≤ [1,1])

(x t1 , y t 2 ∠A ⇒ (x * y ) r max{t1 ,t 2 } ∠A ).

(7)

Proof: Assume that A satisfies the condition (6). Let x , y ∈ X and [0, 0] ≤ t 1 , t 2 ≤ [1,1] satisfy x t1 , y t 2 ∠A .

A (x ) = [ μAL (x ), μUA (x )] for all x ∈ X and Then A (x ) ≤ t 1 and A (x ) ≤ t 2 . By (6) D [0,1] denotes the family of all closed sub-intervals of A (x * y ) ≤ r max{A (x ), A ( y )} ≤ r max{t 1 , t 2 } [0,1]. It is clear that A (x ) ∈ D [0,1], for all Hence (x * y ) r max{t1 ,t 2 } ∠A . x ∈ X . Therefore the i-v fuzzy set A is given Conversely, suppose that the condition (7) is valid. by A = {( x , A ( x )) x ∈ X } where A : X → D [0,1]. ∠A and y ∠A for all x , y ∈ X , it Since x Let

Now we define refined maximum (briefly, rmax) and order on elements D1 = [a1 , b1 ] and D 2 = [a2 , b 2 ] of

D [0,1] as: r max(D1 , D 2 ) = [max{a1 , a2 }, max{b1 , b 2 }] D1 ≤ D 2 ⇔ a1 ≤ a2 ∧ b1 ≤ b 2 Similarly we can define ≥ and = .

A (x )

follows from (7)

A (y )

that (x * y ) r max{A ( x ), A ( y )} ∠A , so that

A (x * y ) ≤ r max{A (x ), A ( y )}. Note that if A is an i-v fuzzy set in X such that A (x ) ≥ [0.5, 0.5] for all x ∈ X , then the set {x t x t ∠ ∧ ϒA } is empty. In what follows let α and

3. Generalized fuzzy BCK/BCI-algebras

β denote any one of ∠, ϒ, ∠ ∨ ϒ or ∠ ∧ ϒ unless otherwise specified. To say that x t α A means that x t α A

From now on (X ,*, 0) is a BCK/BCI-algebra, unless otherwise is stated.

does not hold. Definition 3.3: An i-v fuzzy set A in X is called an (α , β )* -interval-valued fuzzy sub algebra of X , where

A. Borumand Saeid, M. Kuchaki Rafsanjani, and D. R. Prince Williams: Another Generalization of Fuzzy BCK/BCI-Algebras

177

α ≠ ∠ ∧ ϒ if it satisfies the following implication: (∀x , y ∈ X ) (∀[0, 0] ≤ t 1 , t 2 ≤ [1,1])

an For

A of X may not be * (∠, ∠ ∨ ϒ) -interval-valued fuzzy sub algebra. * (x t1 , y t 2 α A ⇒ (x * y ) r max{t1 ,t 2 } β A ). (8) example, the (∠, ∠ ∨ ϒ) -interval-valued fuzzy Example 3.4 [3]: Let X = {0, a , b , c } be a set with the algebra A of X in Example 3.4 is not (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of following Cayley table: since a[0.4,0.5] ϒA and b[0.05,0.07] ϒA but * 0 a b c

0 0 a b c

a a 0 c b

b b c 0 a

bra

c c b a 0

the first diagram shows the relationship between (α , β )* -interval-valued fuzzy sub algebras of X , where α , β are one of ∠ and ϒ. Also we have the second diagram.

(α , α ∨ β )* (α , β )*

Theorem 3.5: Every (∠ ∨ ϒ, ∠ ∨ ϒ) -interval-valued fuzzy sub algebra of X is an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X .

(α , α )*

*

(α , α ∧ β )*

Proof: Let A be an (∠ ∨ ϒ, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . Let x , y ∈ X and satisfy

x t ∠A

and

1

(∠ ∨ ϒ, ∠ ∨ ϒ)*

y t ∠A .

(∠ ∨ ϒ, ∠)*

(∠ ∨ ϒ, ϒ)*

2

Then x t ∠ ∨ ϒA and y t ∠ ∨ ϒA , which imply that 1

X,

(a *b ) rmax {[0.4,0.5],[0.05,0.07]} = c[0.4,0.5] ∠ ∨ ϒA . Theorem 3.6: Let A be an i-v fuzzy set in X . Then

Then (X ,*, 0) is a BCI-algebra. Let A be an i-v fuzzy set in X defined by A (0) = [0.4, 0.5] A (a ) = [0.3, 0.4] and A (b ) = A (c ) = [0.7, 0.9]. It is A routine to verify that is an * (∠, ∠ ∨ ϒ) -interval-valued fuzzy sub algebra of X .

[0, 0] ≤ t 1 , t 2 ≤ [1,1]

sub an

2

(x * y ) rmax {t ,t } ∠ ∨ ϒA .

Hence

A

is

an

1 2

(∠ ∨ ϒ, ∠ ∧ ϒ)*

(∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . The converse of Theorem 3.5 is not true in general. For example, the (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra A of X in Example 3.4 is not an (∠ ∨ ϒ, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of

X since a[0.5,0.6]∠ ∨ ϒA and c[0.04,0.07]∠ ∨ ϒA , but (a *c ) rmax {[0.5,0.6],[0.04,0.07]} = b[0.5,0.6] ∠ ∨ ϒA . Obviously any (∠, ∠) -interval-valued fuzzy sub *

algebra is an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra, but the converse is not true. For example, In Example 3.4 A is an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . But, is not an (∠, ∠)* -interval-valued fuzzy sub algebra of X since

Proposition 3.7: Let A be an i-v fuzzy set in X which is non-unit. If A is an (α , β )* -interval-valued fuzzy sub algebra of X , then A (0) < [1,1]. Proof: Assume that A (0) = [1,1]. Since A is non-unit, there exists x ∈ X , such that

A (x ) = t < [1,1]. If α = ∠ or α = ∠ ∨ ϒ, then x t α A , but (x * x ) rmax {t ,t } = 0t β A . This is a contradiction.

If

α = ϒ,

x 0α A

then

A (x ) + 0 = t + 0 = t < [1,1].

But

because

( x * x) rmax{0,0} =

0 0 β A, which is a contradiction. Hence A (0) < [1,1]. Proposition 3.8: Let A be an i-v fuzzy set in X . If A is an (∠, ∠)* -interval-valued fuzzy sub algebra of X , a[0.38,0.42]∠A and a[0.34,0.45]∠A , but then A (0) ≤ A (x ) , for all x ∈ X . (a * a ) rmax {[0.38,0.42],[0.34,0.45]} = 0[0.38,0.45] ∠A . Proof: Since x * x = 0 , for all x ∈ X . Then we get Also an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub alge- that A (0) = A (x * x ) ≤ rmax (A (x ), A (x )) = A (x ) .


178

For an i-v fuzzy set A in X , we denote

X := {x ∈ X ∣A (x ) < [1,1]}. Theorem 3.9: Let A be an i-v fuzzy set in X which is non-unit. If A is an (α , β )* -interval-valued fuzzy sub algebra of X where (α , β ) is one of the follow*

is non-unit. If A is an (α , β )* -interval-valued fuzzy sub algebra of X where (α , β ) is one of the following:

• (∠, ∠ ∧ ϒ), • (∠, ∠ ∨ ϒ) , • ( ϒ, ∠ ∨ ϒ ) , • ( ϒ, ∠ ∧ ϒ), ing: • ( ∠ ∨ ϒ, ∠ ∨ ϒ ) , • ( ∠ ∨ ϒ, ∠ ∧ ϒ ) , • (∠, ∠), • (∠, ϒ) , • ( ϒ, ∠), • ( ϒ, ϒ ) then the set X * is a sub algebra of X then the set X * is a sub algebra of X . Proof: By Theorem 3.6, it is enough to prove for the * Proof: (i) Assume that A is an (∠, ∠) - inter- cases: (i) (∠, ∠ ∨ ϒ) , (ii) ( ϒ, ∠ ∨ ϒ). (i) Let x , y ∈ X * . Then A (x ), A ( y ) < [1,1] , and val-valued fuzzy sub algebra of X . Let x , y ∈ X * . Then A (x ) < [1,1] and A ( y ) < [1,1]. Assume that so A ( x ) = t 1 and A ( y ) = t 2 for some [0, 0] ≤ t1 , t2 A (x * y ) = [1,1]. Note that x A ( x ) ∠A and < [1,1]. It follows that x ∠A and y ∠A so that t1 t2 y A ( y ) ∠A . But, since A( x * y ) = [1,1] > rmax{ A( x), (x * y ) ∠ ∨ ϒA , i.e., (x * y ) ∠A or A( y )}, we get that (x * y ) rmax {A ( x ),A ( y )} ∠A . This is a contradiction, and so A (x * y ) < [1,1] which shows

rmax {t1 ,t 2 }

rmax {t1 ,t 2 }

(x * y ) rmax {t ,t } ϒA .

(x * y ) rmax {t ,t } ∠ A

If

1 2

then

1 2

(ii) Assume that A is an (∠, ϒ)* -interval-valued

A (x * y ) ≤ rmax {t 1 , t 2 } < [1,1] and thus x * y ∈ X * . A( x * y ) ≤ A( x * y ) If (x * y ) rmax {t ,t } ϒA , then

fuzzy sub algebra of X . Let x , y ∈ X . Then A (x ) < [1,1] and A ( y ) < [1,1]. If A (x * y ) = [1,1], thus

A (x ) < [1,1] and A ( y ) < [1,1] , which imply that x 0 ϒA and y 0 ϒA .

that x * y ∈ X . Hence X *

*

is a sub algebra of X . *

A (x * y ) + rmax {A (x ), A ( y )} ≥ 1.

Hence (x * y ) rmax {A ( x ),A ( y )} ϒA , which is a contradiction

since

x A ( x ) ∠A

and

y A ( y ) ∠A .

1 2

+ rmax{t1 , t2 } < [1,1]. Hence x * y ∈ X * .

x , y ∈X

(ii) Let

*

. Then

Since A is an ( ϒ, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra,

(x * y ) 0 = (x * y ) rmax {0,0} ∠ ∨ ϒA ,

i.e.,

Thus

(x * y ) 0 ∠A or (x * y ) 0 ϒA . If (x * y ) 0 ∠A , then A (x * y ) < [1,1], and so x * y ∈ X * . Therefore X * A (x * y ) = [0, 0] < [1,1]. If (x * y ) ϒA , then 0 is a sub algebra of X . A (x * y ) = A (x * y ) + [0, 0] < [1,1]. Therefore (iii) Assume that A is an ( ϒ, ∠)* -interval-valued * x * y ∈ X . This completes the proof. fuzzy sub algebra of X . Let x , y ∈ X * . Then Theorem 3.11: Let A be an i-v fuzzy set in X which A (x ) < [1,1] and A ( y ) < [1,1]. Thus x 0 ϒA and is non-unit. Then every ( ϒ, ϒ)* -interval-valued fuzzy y 0 ϒA . If A (x * y ) = [1,1], then A( x * y ) = [1,1] > sub algebra of X is a constant on X * . [0, 0] = rmax{0, 0}.

Therefore

(x * y ) rmax {0,0} ∠A ,

which is a contradiction. Hence A (x * y ) < [1,1], and so x * y ∈ X * . (iv) Assume that A is an ( ϒ, ϒ)* -interval-valued fuzzy sub algebra of X and x , y ∈ X * . Then A (x ) < [1,1] and A ( y ) < [1,1]. If A (x * y ) = [1,1], then

A (x * y ) + rmax {0, 0} = [1,1]

Proof: Let A be an ( ϒ, ϒ)* -interval-valued fuzzy sub algebra of X which is non-unit. Assume that A is not constant on X * . Then there exists y ∈ X * such that t y = A ( y ) ≠ A (0) = t 0 . Then either t y > t 0 or

t y < t 0.

If

t y < t0,

A( y ) + ([1,1] − t0 ) =

then

t y + [1,1] − t0 < [1,1] and so y [1,1]−t ϒA . Since 0

and

so

(x * y ) rmax {0,0} ϒA . This is impossible, and hence

A (x * y ) < [1,1], i.e., x * y ∈ X * . This completes the proof. Corollary 3.10: Let A be an i-v fuzzy set in X

which

A ( y * y ) + ([1,1] − t 0 ) = A (0) + [1,1] − t 0 = t 0 + [1,1] − t 0 = [1,1], we have ( y * y ) rmax {[1,1]−t

0 ,[1,1]−t 0 }

ϒA . This is a contra-


diction.

Now

assume

t y > t 0.

that

t1 < [1,1] − t y < t2
[0.5, 0.5] for all x ∈ X .

we get ( y *0) rmax {t

1 ,t 2 }

(9)

Proof: Let A be a non-unit ( ϒ, ϒ)* -interval-valued fuzzy sub algebra of X . Then by Proposition 3.7 and Theorems 3.9 and 3.11 we get that A (x ) < [1,1] , for all

x ∈ X and X

*

1 2

1 2

is a sub algebra of X

and

Since

x ∈X

A *

is not constant on

X * , there exists

such that t x = A ( x ) ≠ A (0) = t 0 . Then ei-

ther t 0 > t x

or t 0 < t x . For the first case, choose

δ < [0.5, 0.5] such that t x + δ < [1,1] < t 0 + δ . It

follows that x δ ϒA , ⎧⎪A (0) x ∈ X * A (x * x ) = A (0) = t 0 > δ = rmax {δ , δ }, (10) A ( y ) := ⎨ [1,1] otherwise ⎪⎩ A (x * x ) + rmax {δ , δ } = A (0) + δ = t 0 + δ > [1,1] Conversely, let S be a sub algebra of X which satisfy (9). Assume that x s ϒA and y r ϒA for some so that (x * x ) rmax {δ ,δ } ∠ ∨ ϒA . This is a contradiction. [0, 0] ≤ s , r < [1,1] . Then A (x ) + s < [1,1] and A( y ) For the second case, we can choose δ < [0.5, 0.5] such + r < [1,1], then A (x ) ≠ [1,1] and A ( y ) ≠ [1,1] . that t x + δ > [1,1] > t 0 + δ . Then 0 ϒA and δ Thus x , y ∈ S and so x * y ∈ S . It follows that x 1 ϒA , but (x *0) rmax {1,δ } = x 1 ∠ ∨ ϒA since A (x * y ) + rmax {s , r } = t + rmax {s , r } < [1,1] so that (x * y ) rmax {s , r } ϒA . Therefore A is a non-unit A (x ) > [0.5, 0.5] > δ and A (x ) + δ = t x + δ > [1,1]. This

leads

to

a

contradiction.

Therefore

( ϒ, ϒ)* -interval-valued fuzzy sub algebra of X . A (x ) ≤ [0.5, 0.5] for some x ∈ X . We now show Theorem 3.13: Let S be a sub algebra of X and A that A (0) ≤ [0.5, 0.5]. Assume that A(0) = t0 > be an i-v fuzzy set in X such that [0.5, 0.5]. Since there exists x ∈ X such that (i) (∀x ∈ X 5 S ) (A (x ) = [1,1]), (ii) (∀x ∈ S ) (A (x ) ≤ [0.5, 0.5]). A (x ) = t x ≤ [0.5, 0.5], we have t 0 > t x . Choose * Then A is an ( ϒ, ∠ ∨ ϒ) -interval-valued fuzzy sub t 1 < t 0 such that t x + t 1 < [1,1] < t 0 + t 1. Then algebra of X . A (x ) + t 1 = t x + t 1 < [1,1], and so x t ϒA . Now we 1 Proof: Let x , y ∈ X and [0, 0] ≤ t 1 , t 2 < [1,1] be such that x t ϒA 1

and y t ϒA , that is, A( x) + t1 < 2

[1,1] and A ( y ) + t 2 < [1,1]. If x * y ∉ S , then x ∈ X 5 S or y ∈ X 5 S , i.e., A (x ) = [1,1] or

get that

A (x * x ) + rmax {t 1 , t 1} = A (0) + t 1 = t 0 + t 1 > [1,1], A (x * x ) = A (0) = t 0 > t 1 = rmax {t 1 , t 1}.


180

Hence ( x * x ) rmax {t

1 ,t 1 }

= rmax{ A( x), A( y )} ≤ rmax{t1 , t2 }.

∠ ∨ ϒA , which is a contradic-

tion. Therefore A (0) ≤ [0.5, 0.5]. Finally suppose that

t x = A (x ) > [0.5, 0.5] for some x ∈ X * . Let t such that [0, 0] < t < [0.5, 0.5]

be

and t x > [0.5, 0.5]

+ t . Therefore A (x ) + [0, 0] < [1,1] and

A(0) +

This

is

a

contradiction, and rmax {A (x ), A ( y )} ≤ [0.5, 0.5]. It follows that

A (x * y ) + rmax {t 1 , t 2 } < 2A (x * y ) ≤ 2rmax {A (x ), A ( y ),[0.5, 0.5]} ≤ [1,1] so that ( x * y)rmax{t ,t } ϒA. Hence (x * y ) rmax {t ,t } ∠ ∨ ϒA ,

([0.5, 0.5] − t ) < [1,1] which imply that x 0 ϒA and and consequently 0([0.5,0.5]−t ) ϒA . But (x *0) rmax (0,[0.5,0.5]−t ) = x ([0.5,0.5]−t )

1 2

and so A ( x ) > [0.5, 0.5] − t

and A( x) + [0.5, 0.5] −

so

1 2

is an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . Theorem 3.16: For any subset S of X , let χS de-

A

t > [1,1] , thus (x *0) rmax {[0,0],[0.5,0.5]−t } ∠ ∨ ϒA , which note the characteristic function of S . Then the function χSc : X → [0,1] defined by χSc (x ) = 1 − χS (x ) for is a contradiction. Hence A (x ) ≤ [0.5, 0.5] . We give a characterization of an (∠, ∠ ∨ ϒ)* - inter- all x ∈ X is an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy val-valued fuzzy sub algebra. sub algebra of X iff S is a sub algebra of X . Theorem 3.15: Let A be an i-v fuzzy set in X . Then Proof: Assume that χ c is an (∠, ∠ ∨ ϒ)* -interS A is an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra val-valued fuzzy sub algebra of X and x , y ∈ S . of X iff it satisfies the following inequality Then χSc (x ) = [1,1] − χS (x ) = [0, 0] and χ Sc ( y ) = [1,1] − (∀x , y ∈ X ) c c A (x * y ) ≤ rmax {A (x ), A ( y ),[0.5, 0.5]} (11) χ S ( y ) = [0, 0]. Hence x 0 ∠χS and y 0 ∠χS , which c Proof: Assume that A is an (∠, ∠ ∨ ϒ)* -inter- imply that ( x * y ) 0 = ( x * y ) rmax {[0,0],[0,0]} ∠ ∨ ϒχS . val-valued fuzzy sub algebra of X . Let x , y ∈ X be Thus χSc (x * y ) ≤ [0, 0] or χSc (x * y ) + [0, 0] < [1,1]. such that rmax {A (x ), A ( y )} > [0.5, 0.5]. Then If χ c (x * y ) ≤ [0, 0], then [1,1] − χ (x * y ) = [0, 0], S S A (x * y ) ≤ rmax {A (x ), A ( y )}. If it is not true, then i.e., χS (x * y ) = [1,1]. Hence x * y ∈ S . If A (x * y ) < t < rmax {A (x ), A ( y )} for some χSc (x * y ) + [0, 0] < [1,1], then χS (x * y ) > [0, 0]. [0.5, 0.5] < t < [1,1]. It follows that x t ∠A and Thus χS (x * y ) = [1,1], and so x * y ∈ S . Therefore y t ∠A , but (x * y ) rmax {t ,t } = (x * y )t ∠ ∨ ϒA S is a sub algebra of X . which is a contradiction. Hence A( x * y ) ≤ Conversely, suppose that S is a sub algebra of rmax{ A( x), A( y )} whenever rmax{ A( x), A( y )} > X and x , y ∈ X . If x , y ∈ S , then x * y ∈ S , [0.5, 0.5]. If rmax {A (x ), A ( y )} ≤ [0.5, 0.5], then thus χSc (x * y ) = rmax {χSc (x ), χSc ( y )} x 0.5∠A and y 0.5∠A which imply that ( x * y )0.5 = ≤ rmax {χSc (x ), χSc ( y ),[0.5, 0.5]}. ( x * y ) rmax{0.5,0.5} ∠ ∨ ϒA. Therefore A (x * y ) ≤ [0.5, 0.5] because if A (x * y ) > [0.5, 0.5], then A( x * y ) + If any one of x and y does not belong to S , c c [0.5, 0.5] > [0.5, 0.5] + [0.5, 0.5] = [1,1], a contradic- then χS (x ) = [1,1] or χS ( y ) = [1,1]. Hence tion. Hence A (x * y ) ≤ rmax {A (x ), A ( y ),[0.5, 0.5]} χSc (x * y ) ≤ rmax {χSc (x ), χSc ( y )} for all x , y ∈ X . ≤ rmax {χSc (x ), χSc ( y ),[0.5, 0.5]}. Conversely, assume that A satisfies (11). Let Using Theorem 3.15, we know that χSc is an x , y ∈ X and [0, 0] ≤ t 1 , t 2 < [1,1] be such that (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . x t ∠A and y t ∠A . Then A (x ) ≤ t 1 and Theorem 3.17: An i-v fuzzy set A in X is an 1

2

* A ( y ) ≤ t 2 . Suppose that A (x * y ) > rmax {t 1 , t 2 }. If (∠, ∠ ∨ ϒ) -interval-valued fuzzy sub algebra of X iff the set rmax {A (x ), A ( y )} > [0.5, 0.5] then L (A ; t ) := {x ∈ X ∣A (x ) ≤ t },[0.5, 0.5] ≤ t < [1,1] A( x * y ) ≤ rmax{ A( x), A( y ),[0.5, 0.5]} is a sub algebra of X .


Proof: Assume that A is an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X and let x , y ∈ L (A ; t ). Then A (x ) ≤ t and A ( y ) ≤ t ,

algebra of X . Conversely,

nonempty level set L ( A ; t ) is a sub algebra of X . Theorem 3.19: Let A be an i-v fuzzy set in X . Then A is a non-unit ( ϒ, ϒ)* -interval-valued fuzzy sub al-

by Theorem 3.12, we have

181

L (A ; A (0)) = X * and 0 ∈ L (A ; A (0)) , X * is a sub algebra of X and A and so x t ∠A and y t ∠A . It follows from Theorem is non-unit. Now let x ∈ X * . Then A (x ) ≥ A (0) and 3.15 that A (x ) > [0, 0] . Since L (A ; A (x )) =/ ∅ , so A (x * y ) ≤ rmax {A (x ), A ( y ),[0.5, 0.5]} L (A ; A (x )) is a sub algebra of X . Then ≤ rmax {t ,[0.5, 0.5]} = t 0 ∈ L (A ; A (x )) implies that A (0) ≥ A (x ) . Hence so that x * y ∈ L ( A ; t ). Hence L (A ; t ) is a sub A (x ) = A (0) , for all x ∈ X * . Therefore algebra of X . ⎧A (0) if x ∈ X * Conversely, let A be an i-v fuzzy set in X such A (x ) = ⎨ otherwise ⎩ [1,1] that the set L ( A ; t ) := {x ∈ X ∣ A (x ) ≤ t } is a sub * algebra of X for all [0.5, 0.5] ≤ t < [1,1]. If there Hence by Theorem 3.12, A is an ( ϒ, ϒ) -interexist x , y ∈ X such that A( x * y ) > rmax{ A( x), val-valued fuzzy sub algebra of X . * A( y ),[0.5, 0.5]}, then we can take [0, 0] < t < [1,1] Theorem 3.20: Every ( ϒ, ϒ) -interval-valued fuzzy sub algebra is an (∠, ∠)* -interval-valued fuzzy sub algesuch that rmax {A (x ), A ( y ),[0.5, 0.5]} < t < A ( x * y ). bra. A be a non-unit 3.21: Let Thus x , y ∈ L ( A ; t ) and t > [0.5, 0.5], and so Theorem * x * y ∈ L (A ; t ), i.e., A (x * y ) ≤ t . This is a con- ( ϒ, ∠ ∨ ϒ) -interval-valued fuzzy sub algebra of X . tradiction. Therefore A( x * y ) ≤ rmax{ A( x), A( y ), Then the nonempty level set L (A ; t ) is a sub algebra [0.5, 0.5]} for all x , y ∈ X . Using Theorem 3.15, we of X , for all [0.5, 0.5] ≤ t ≤ [1,1] . * conclude that A is an (∠, ∠ ∨ ϒ)* -interval-valued Proof: If A is a constant on X , then by Theorem 3.12, A is an ( ϒ, ϒ)* -interval-valued fuzzy sub algefuzzy sub algebra of X . Proposition 3.18: Let A be an i-v fuzzy set in X . bra. Thus by Theorem 3.19 we have the nonempty level Then A is an (∠, ∠)* -interval-valued fuzzy sub al- set L (A ; t ) is a sub algebra of X , for * gebra of X if and only if for all [0, 0] ≤ t ≤ [1,1] , the [0, 0] ≤ t ≤ [1,1] . If A is not a constant on X , then since

⎧α if x ∈ X * A (x ) = ⎨ otherwise ⎩[1,1] gebra of X if and only if L (A ; A (0)) = X * and for where α ≤ [0.5, 0.5] . Now we show that the nonempty all [0, 0] ≤ t ≤ [1,1] , the nonempty level set L ( A ; t ) level set L (A ; t ) is a sub algebra of X for is a sub algebra of X . [0.5, 0.5] ≤ t ≤ [1,1] . If t = [1,1] , then it is clear that * Proof: Let A be a non-unit ( ϒ, ϒ) -interval-valued L (A ; t ) is a sub algebra of X . Now let fuzzy sub algebra of X . Then by Theorem 3.12 we [0.5, 0.5] ≤ t < [1,1] and x , y ∈ L (A ; t ) . Then have

A (x ), A ( y ) ≤ t < [1,1] imply that x , y ∈ X * . Thus x ∗ y ∈ X * and so A (x ∗ y ) ≤ [0.5, 0.5] ≤ t . Thereotherwise fore x ∗ y ∈ L (A ; t ) . that L (A ; A (0)) = X * . Let Theorem 3.22: Let A be a non-unit i-v fuzzy set of [0, 0] ≤ t ≤ [1,1] . Then X , L (A ;[0.5, 0.5]) = X * and the nonempty level set . If t = [1,1] , then it is clear L (A ; t ) is a sub algebra of X , for all Now, let [0, 0] ≤ t < [1,1] . [0, 0] ≤ t ≤ [1,1] . Then A is an ( ϒ, ∠ ∨ ϒ)* -inter-

⎧A (0) A (x ) = ⎨ ⎩ [1,1] So it is easy to check

x , y ∈ L (A ; t ) , for A (x ) ≤ t and A ( y ) ≤ t that x ∗ y ∈ L (A ;[1,1]) .

if x ∈ X

*

x , y ∈ X * and so x ∗ y ∈ X * . Hence val-valued fuzzy sub algebra of X . A (x ∗ y ) = A (0) ≤ t . Therefore L (A ; t ) is a sub Proof: Since A =/ [1,1] we get that X * =/ ∅ . Thus by

Then


182

hypothesis we have L (A ;[0.5, 0.5]) =/ ∅ and so X * is a sub algebra of X . Also A (x ) ≤ [0.5, 0.5] , for all

x ∈ X and A (x ) = [1,1] , if x ∈/ X * . Therefore by Theorem 3.21, A is an ( ϒ, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . Theorem 3.23: Let A be an ( ϒ, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . Then for all [0.5, 0.5] ≤ t ≤ [1,1] , the nonempty level set L (A ; t ) is a sub algebra of X . Conversely, if the nonempty level set A is a sub algebra of X , for all [0, 0] ≤ t ≤ [1,1] , then A is an ( ϒ, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . Proof: Let A be an ( ϒ, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . If t = [1,1] , then L (A ; t ) is a sub algebra of X . Now let L (A ; t ) =/ ∅ , [0, 0] ≤ t < [1,1] x , y ∈ L (A ; t ) . Then and A (x ), A ( y ) ≤ t . Thus by hypothesis we have A (x * y ) ≤ rmax (A (x ), A ( y ),[0.5, 0.5]) *

≤ rmax (t ,[0.5, 0.5]) ≤ t Therefore L (A ; t ) is a sub algebra of X . Conversely, let x , y ∈ X . Then we

have

A (x ), A ( y ) ≤ rmax (A (x ), A ( y ),[0.5, 0.5]) = t 0 Hence x , y ∈ L (A ; t 0 ) , for [0, 0] ≤ t 0 ≤ [1,1] and so

x ∗ y ∈ L( A; t0 ). Therefore

A( x * y ) ≤ t0 = rmax( A( x),

A( y ),[0.5, 0.5]) , then A is an ( ϒ, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . For any fuzzy set A in X and [0, 0] ≤ t < [1,1],

then x t ∠ ∨ ϒA or y t ∠ ∨ ϒA , which is a contra-

A( x * y ) ≤ rmax{t , [0.5, 0.5]} = t and so x * y ∈ L (A ; t ) ⊆ [A ]t . If t < [0.5, 0.5], then A( x * y ) ≤ rmax{t ,[0.5, 0.5]} = [0.5, 0.5] A( x * y ) + t < [0.5, 0.5] + and thus [0.5, 0.5] = [1,1]. Hence (x * y )t ϒA , and so x * y ∈ At ⊆ [A ]t . Therefore [A ]t is a sub algebra of X . Conversely, let A be an i-v fuzzy set in X and [0, 0] ≤ t < [1,1] be such that [A ]t is a sub algebra of X. Let rmax {A (x ), A ( y ),[0.5, 0.5]} < t < A ( x * y ) for some [0.5, 0.5] < t < [1,1]. Then x, y ∈ L( A; t ) ⊆ [ A]t , which implies that x * y ∈ [A ]t . Hence A (x * y ) ≤ t or A (x * y ) + t < [1,1], which is a contradiction. Therefore A( x * y ) ≤ rmax{ A( x), A( y ), [0.5, 0.5]} for all x , y ∈ X . By Theorem 3.15, we know that A is an (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X . i ∈ Λ} be a family of Theorem 3.25: Let {A i ∣

diction. If t ≥ [0.5, 0.5], then

(∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebras of X . Then A := ∩ A i is an (∠, ∠ ∨ ϒ)* -interval-valued i ∈Λ

fuzzy sub algebra of X . Proof: By Theorem

3.15

we

get that A i (x * y ) ≤ rmax {A (x ), A ( y ),[0.5, 0.5]}, and so

A (x * y ) = inf A i (x * y ) i ∈Λ

≤ inf rmax {A i ( x ), A i ( y ),[0.5, 0.5]} i ∈Λ

= rmax {inf A i (x ),inf A i ( y ),[0.5, 0.5]} i ∈Λ i ∈Λ At := {x ∈ X ∣ x t ϒA } [A ]t := {x ∈ X ∣ x t ∠ ∨ ϒA }. = rmax {A (x ), A ( y ),[0.5, 0.5]}. Obviously [A ]t = L (A ; t ) ∪ At . By Theorem 3.15, A is an (∠, ∠ ∨ ϒ)* -interTheorem 3.24: An i-v fuzzy set A in X is an val-valued fuzzy sub algebra of X . (∠, ∠ ∨ ϒ)* -interval-valued fuzzy sub algebra of X Theorem 3.26: Let {A ∣i ∈ Λ} be a family of i if and only if [A ]t is sub algebra of X , for all * (α , β ) -interval-valued fuzzy sub algebras of X . [0, 0] ≤ t < [1,1]. Then A := ∩ A i is an (α , β )* -interval-valued fuzzy * i ∈Λ Proof: Let A be an (∠, ∠ ∨ ϒ) -interval-valued sub algebra of X , where (α , β ) is one of the followfuzzy sub algebra of X and x , y ∈ [A ]t , for ing forms [0, 0] ≤ t < [1,1]. Then x t ∠ ∨ ϒA and y t ∠ ∨ ϒA , (ii) (i) (∠, ϒ) , (∠, ∠ ∧ ϒ) , that is, A (x ) ≤ t or A (x ) + t > [1,1], and (iv) (iii) ( ϒ, ∠) , ( ϒ, ∠ ∧ ϒ ) , A ( y ) ≤ t or A ( y ) + t > [1,1]. Since A( x * y ) ≤ (v) (∠ ∨ ϒ, ϒ) , (vi) ( ∠ ∨ ϒ, ∠ ∧ ϒ ) , rmax{ A( x), A( y ),[0.5, 0.5]} by Theorem 3.15, we (vii) (∠ ∨ ϒ, ∠) , (viii) ( ϒ, ∠ ∨ ϒ) , have A (x * y ) ≤ rmax {t ,[0.5, 0.5]}. If it is not true, we denote


(ix)

( ϒ, ϒ ) .

membership function AG .

Proof: We prove theorem for an ( ϒ, ϒ) -interval-valued fuzzy sub algebra. The proof of the other cases is similar. If there exists i ∈ Λ such that A i = [0, 0] , then *

A = [0, 0] . So A is an ( ϒ, ϒ)* -interval-valued fuzzy sub algebra. Let A i =/ [0, 0] for all i ∈ Λ . Then by Theorem 3.12 we have

⎧A (0) if x ∈ X i* A i (x ) = ⎨ i otherwise ⎩ [1,1] for all i ∈ Λ . So it is clear that ⎧A (0) if x ∈ ∩ X i* ⎪ i ∈Λ A (x ) = ⎨ ⎪⎩ [1,1] otherwise Since

∩X

* i

183

is a sub algebra of X , then by Theorem

i ∈Λ

3.12 A is an ( ϒ, ϒ)* -interval-valued fuzzy sub algebra of X . Theorem 3.27: Let {A i ∣ i ∈ Λ} be a family of

(i) If G is an (α , β )* -interval-valued fuzzy sub algebra of Y, then f −1 (G ) is an (α , β )* -interval-valued fuzzy sub algebra of X , (ii) Let f be epimorphism. If f −1 (G ) is an

(α , β )* -interval-valued fuzzy sub algebra of X , then G is an (α , β )* -interval-valued fuzzy sub algebra of Y Proof: (i) Let x t α A f −1 (G ) and y r α A f −1 (G ) , for [0, 0] ≤ t , r < [1,1] . Then by Lemma 3.28, we get that (f (x ))t α AG and (f ( y )) r α AG . Hence by hypothesis

(f (x ) ∗ f ( y )) rmax (t ,r ) β AG

.

Then

( f ( x ∗ y )) rmax ( t ,r β AG and so ( x ∗ y ) rmax ( t ,r ) β Af −1 (G ) . (ii) Let x , y ∈Y . Then by hypothesis there exist x ′, y ′ ∈ X such that f (x ′) = x and f ( y ′) = y . Assume

that

(f (x ′))t α AG

x t α AG and y r α AG , then and (f ( y ′)) r α AG . Thus x t′α A f −1 (G )

(∠, ∠)* -interval-valued fuzzy sub algebras of X . and yr′ α Af −1 (G ) and therefore ( x′ ∗ y′) rmax ( t ,r ) β Af −1 ( G ) . Then A := ∪ A i is an (∠, ∠)* -interval-valued fuzzy So (f (x ′ ∗ y ′)) rmax (t ,r ) β AG Then i ∈Λ

sub algebra of X . Proof: Let x t ∠A and y r ∠A , where [0, 0] ≤ t , r