Intuitionistic Q-Fuzzy Subalgebras

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International Mathematical Forum, 1, 2006, no. 24, 1167 - 1174

Intuitionistic Q-Fuzzy Subalgebras

of BCK/BCI-Algebras

Eun Hwan RohDepartment of Mathematics Education

Chinju National University of EducationJinju 660-756, Korea

[email protected]

Kyung Ho Kim and Jong Geol LeeDepartment of Mathematics, Chungju National University

Chungju 380-702, [email protected]

Abstract. The intuitionistic Q-fuzzification of the concept of subalgebras inBCK/BCI -algebra is considered, and some related properties are investigated.

Keywords: BCK/BCI-algebra, Q-fuzzy set, Intuitionistic Q-fuzzy set, in-tuitionistic Q-fuzzy subalgebra, preimage

Mathematics Subject Classification: 06F35, 03G25, 03E72

1. Introduction

The notion of BCK-algebras was proposed by Imai and Iseki in 1966. Inthe same year, Iseki [4] introduced the notion of a BCI-algebra which is ageneralization of a BCK-algebra. After the introduction of the concept of fuzzy

sets by Zadeh [6], several researches were conducted on the generalization of thenotion of fuzzy sets. The idea of “intuitionistic fuzzy set” was first publishedby Atanassov [1, 2], as a generalization of the notion of fuzzy set. In this paper,using the Atanassov’s idea, we establish the intuitionistic Q-fuzzification of theconcept of subalgebras in BCK/BCI-algebras, and investigate some of theirproperties.

2. Preliminaries

In this section we include some elementary aspects that are necessary forthis paper.

Recall that a BCI-algebra is an algebra (X, ∗, 0) of type (2, 0) satisfyingthe following axioms:

(I) ((x ∗ y) ∗ (x ∗ z )) ∗ (z ∗ y) = 0,(II) (x ∗ (x ∗ y)) ∗ y = 0,(III) x ∗ x = 0, and(IV) x ∗ y = 0 and y ∗ x = 0 imply x = y



1168 E. H. Roh, K. H. Kim and J. G. Lee

for every x,y,z ∈ X . A BCI-algebra X satisfying the condition(V) 0 ∗ x = 0 for all x ∈ X

is called a BCK-algebra . In any BCK/BCI-algebra X one can define a partialorder “≤” by putting x ≤ y if and only if x ∗ y = 0.

A BCK/BCI-algebra X has the following properties:(2.1) x ∗ 0 = x,

(2.2) (x ∗ y) ∗ z = (x ∗ z ) ∗ y,(2.3) x ≤ y implies that x ∗ z ≤ y ∗ z and z ∗ y ≤ z ∗ x,(2.4) (x ∗ z ) ∗ (y ∗ z ) ≤ x ∗ y

for all x,y,z ∈ X .A non-empty subset S of a BCK/BCI-algebra X is called a subalgebra of X

if x ∗ y ∈ S whenever x, y ∈ S . A mapping f : X → Y of BCK/BCI-algebrasis called a homomorphism if f (x ∗ y) = f (x) ∗ f (y) for all x, y ∈ X . Let X

be a BCK/BCI-algebra. A fuzzy set f in X , i.e., a mapping f : X → [0, 1], iscalled a fuzzy subalgebra of X if f (x ∗ y) ≥ f (x) ∧ f (y) for all x, y ∈ X . Notethat if f is a fuzzy subalgebra of a BCK/BCI-algebra X , then f (0) ≥ f (x) for

all x ∈ X .

3. Intuitionistic Q-fuzzy subalgebras

In what follows, let Q and X denote a set and a BCK/BCI-algebra, respec-tively, unless otherwise specified. A mapping H : X × Q → [0, 1] is called aQ-fuzzy set in X .

A Q-fuzzy set H : X × Q → [0, 1] is called a fuzzy subalgebra of X overQ (briefly, Q-fuzzy subalgebra of X ) if H (x ∗ y, q ) ≥ H (x, q ) ∧ H (y, q ) for allx, y ∈ X and q ∈ Q.

Definition 3.1. let Q and X denote a set and a BCK/BCI-algebra, respec-tively. An intuitionistic Q-fuzzy set (IQFS for short) A is an object having theform

A = {(x, μA

(x, q ), γ A

(x, q )) : x ∈ X, q ∈ Q}

where the functions μA

: X × Q → [0, 1] and γ A

: X × Q → [0, 1] denotethe degree of membership (namely μ

A(x, q )) and the degree of nonmembership

(namely γ A

(x, q )) of each element (x, q ) ∈ X × Q to the set A, respectively,and 0 ≤ μ

A(x, q ) + γ

A(x, q ) ≤ 1 for all x ∈ X and q ∈ Q.

For the sake of simplicity, we shall use the symbol A = (μA

, γ A

) for the IQFSA = {(x, μ

A(x, q ), γ

A(x, q )) : x ∈ X, q ∈ Q}.

Definition 3.2. An IQFS A = (μA

, γ A

) in X is called an intuitionistic Q-fuzzy subalgebra of X if

(IQF1) μA(x∗y, q ) ≥ μA(x, q )∧μA(y, q ) and γ A(x∗y, q ) ≤ γ A(x, q )∨γ A(y, q )for all x, y ∈ X and q ∈ Q.



INTUITIONISTIC Q-FUZZY SUBALGEBRAS 1169

Example 3.3. Let X = {0, a , b , c} be a BCK-algebra with the following Cay-ley table:

∗ 0 a b c

0 0 0 0 0a a 0 0 ab b a 0 bc c c c 0

Define an IQFS = (μA, γ A) in X as follows: for every q ∈ Q,

μA(0, q ) = μA(b, q ) = 0.6, μA(a, q ) = μA(c, q ) = 0.2

and

γ A(0, q ) = γ A(b, q ) = 0.3, γ A(a, q ) = γ A(c, q ) = 0.7

It is easy to verify that A = (μA, γ A) is an intuitionistic Q-fuzzy subalgebra of X .

Example 3.4. Consider a BCI-algebra X = {0, x} with Cayley table as fol-

lows (Iseki [2]):∗ 0 x

0 0 xx x 0

Let Q = {1, 2} and let A = (μA, γ A) be an intuitionistic Q-fuzzy set in X defined by

μA(0, 1) = μA(0, 2) = 1, μA(x, 1) = 0.8, μA(x, 2) = 0.5

and

γ A(0, 1) = γ A(0, 2) = 0, γ A(x, 1) = 0.1, γ A(x, 2) = 0.2.

It is easy to verify that A = (μA, γ A) is an intuitionistic Q-fuzzy subalgebra of X .

Proposition 3.5. If A = (μA

, γ A

) in X is an intuitionistic fuzzy Q-subalgebra of X , then μ

A(0, q ) ≥ μ

A(x, q ) and γ

A(0, q ) ≤ γ

A(x, q ) for all x ∈ X and q ∈ Q.

Proof. Let x ∈ X and q ∈ Q. Then μA

(0, q ) = μA

(x ∗ x, q ) ≥ μA

(x, q ) ∧μA

(x, q ) = μA

(x, q ) and γ A

(0, q ) = γ A

(x ∗ x, q ) ≤ γ A

(x, q ) ∨ γ A

(x, q ) = γ A

(x, q ).

Proposition 3.6. Let A = (μA, γ A) be an intuitionistic Q-fuzzy subalgebra of X . Define an intuitionistic Q-fuzzy set B = (μB, γ B) in X by

μB(x, q ) =μA(x, q )

μA(0, q ), γ B(x, q ) =

γ A(x, q )

γ A(0, q )

for all x ∈ X and q ∈ Q. Then B = (μB, γ B) is an intuitionistic Q-fuzzy subalgebra of X .




Proof. Let x, y ∈ X and q ∈ Q. Then

μB(x ∗ y, q ) = μA(x∗y,q)μA(0,q)

≥ 1μA(0,q)

{μA(x, q ) ∧ μA(y, q )}

= {μA(x,q)μA(0,q)

∧ μA(y,q)μA(0,q)

} = μB(x, q ) ∧ μB(y, q )

and

γ B(x ∗ y, q ) =γ A(x∗y,q)

γ A(0,q) ≤1

γ A(0,q){γ A(x, q ) ∨ γ A(y, q )}= {γ A(x,q)

γ A(0,q)∨ γ A(y,q)

γ A(0,q)} = γ B(x, q ) ∨ γ B(y, q ).

Hence B = (μB, γ B) is an intuitionistic Q-fuzzy subalgebra of X .

Let A = (μA, γ A) be an IQFS in a set X and let α, β ∈ [0, 1] be such thatα + β ≤ 1. Then we define the set

X (α,β )A = {x ∈ X | μA(x, q ) ≥ α, γ A(x, q ) ≤ β, q ∈ Q}.

Theorem 3.7. Let A = (μA, γ A) be an intuitionistic Q-fuzzy subalgebra of X.

Then X (α,β )A is a subalgebra of X for every (α, β ) ∈ Im(μA) × Im(γ A) with

α + β ≤ 1.Proof. Let x, y ∈ X

(α,β )A and q ∈ Q. Then μA(x, q ) ≥ α, γ A(x, q ) ≤ β, μA(y, q ) ≥

α, γ A(y, q ) ≤ β which imply that

μA(xy,q ) ≥ μA(x, q ) ∧ μA(y, q ) ≥ α

γ A(xy,q ) ≤ γ A(x, q ) ∨ γ A(y, q ) ≤ β.

Thus xy ∈ X (α,β )A Therefore X

(α,β )A is a subalgebra of X.

For the convenience of notation, we denote

(. . . ((x ∗ y1) ∗ y

2) ∗ . . . ) ∗ y

n= x ∗

ni=1

yi, and

yn ∗ (yn−1 ∗ (· · · ∗ (y1 ∗ x) · · · )) = x ∗ni=1

yi.

Proposition 3.8. Let A = (μA

, γ A

) be an intuitionistic Q-fuzzy subalgebra of a BCK-algebra X and let x1, x2, · · · , xn be arbitrary elements of X . If there

exists k ∈ {1, 2, · · · , n} such that xk = x1, then μA

(x1 ∗ni=2

xi, q ) ≥ μA

(x, q )

and γ A

(x1 ∗n

i=2

xi, q ) ≤ γ A

(x, q ) for all x ∈ X and q ∈ Q.

Proof. Let k be a fixed number in {1, 2, · · · , n} such that xk = x1. Using (III),(V) and (2.2), one can deduce

μA

(x1 ∗ni=2

xi, q ) = μA

(0, q ) and γ A

(x1 ∗ni=2

xi, q ) = γ A

(0, q ).




It follows from Proposition 3.5 that

μA

(x1 ∗ni=2

xi, q ) ≥ μA

(x, q ) and γ A

(x1 ∗ni=2

xi, q ) ≤ γ A

(x, q )

for all x ∈ X and q ∈ Q.

Proposition 3.9. Let A = (μA

, γ A

) be an intuitionistic Q-fuzzy subalgebra of a BCK-algebra X. Then

(i) μA

(x ∗2k−1i=1

x, q ) ≥ μA

(x, q ) and γ A

(x ∗2k−1i=1

x, q ) ≤ γ A

(x, q ) for all x ∈

X, q ∈ Q and for k = 1, 2, · · · .

(ii) μA

(x ∗2ki=1

x, q ) = μA

(x, q ) and γ A

(x ∗2ki=1

x, q ) = γ A


X, q ∈ Q and for k = 0, 1, 2, · · · .

(iii) μA

(x ∗n−1i=1

x, q ) ≥ μA

(x, q ) and γ A

(x ∗n−1i=1

x, q ) ≤ γ A


X, q ∈ Q and for n = 1, 2, · · · .

Proof. It follows immediately from (III), (2.1) and Proposition 3.5.

Theorem 3.10. If {Ai : i ∈ Λ} is an arbitrary family of intuitionistic Q-fuzzy subalgebras of X , then ∩Ai is an intuitionistic Q-fuzzy subalgebra of X where ∩Ai = {((x, ∧μ

Ai(x, q ), ∨γ

Ai(x, q )) : x ∈ X, q ∈ Q}.

Proof. Let x, y ∈ X, q ∈ Q. Then

∧μAi

(x ∗ y, q ) ≥ ∧(μAi

(x, q ) ∧ μAi

(y, q )) = (∧μAi

(x, q )) ∧ (∧μAi

(y, q ))

and

∨γ Ai

(x ∗ y, q ) ≤ ∨(γ Ai

(x, q ) ∨ γ Ai

(y, q )) = (∨γ Ai

(x, q )) ∨ (∨γ Ai

(y, q )).

Hence ∩Ai = (∧μAi, ∨γ Ai ) is an intuitionistic Q-fuzzy subalgebra of X .

Theorem 3.11. If an IQFS A = (μA

, γ A

) in X is an intuitionistic Q-fuzzy subalgebra of X , then so is A, where A = {((x, q ), μ

A(x, q ), 1−μ

A(x, q )) :

x ∈ X, q ∈ Q}.

Proof. It is sufficient to show that μA

satisfies the second condition in (IQF1).Let x, y ∈ X and q ∈ Q. Then

μA

(x ∗ y, q ) = 1 − μA

(x ∗ y, q ) ≤ 1 − (μA

(x, q ) ∧ μA

(y, q ))= (1 − μ

A(x, q )) ∨ (1 − μ

A(y, q ))

= μA

(x, q ) ∨ μA

(y, q ).

Hence A is an intuitionistic Q-fuzzy subalgebra of X .


, γ A

) in X is an intuitionistic Q-fuzzy subalgebra of X , then the sets

X μ := {x ∈ X : μA

(x, q ) = μA

(0, q )}




and X γ := {x ∈ X : γ

A(x, q ) = γ

A(0, q )}

are subalgebras of X for all q ∈ Q.

Proof. Let x, y ∈ X μ and q ∈ Q. Then μA

(x, q ) = μA

(0, q ) = μA

(y, q ), and so

μA

(x ∗ y, q ) ≥ μA

(x, q ) ∧ μA

(y) = μA

(0, q ).

By using Proposition 3.5, we know that μA(x ∗ y, q ) = μA(0, q ) or equivalentlyx ∗ y ∈ X μ. Now let x, y ∈ X γ . Then

γ A

(x ∗ y, q ) ≤ γ A

(x, q ) ∨ γ A

(y, q ) = γ A

(0, q ),

and by applying Proposition 3.5, we conclude that γ A

(x ∗ y, q ) = γ A

(0, q ) andhence x ∗ y ∈ X γ .

Definition 3.13. Let A = (μA

, γ A

) be an IQFS in X and let α ∈ [0, 1]. Thenthe set μ≥

A,α:= {x ∈ X : μ

A(x, q ) ≥ α, q ∈ Q} is called a μ-level α-cut and

γ ≤A,α

:= {x ∈ X : γ A

(x, q ) ≤ α, q ∈ Q} is called a γ -level α-cut of A.


, γ A

) in X is an intuitionistic Q-fuzzy

subalgebra of X , then the μ-level α-cut and γ -level α-cut of A are subalgebras of X for every α ∈ [0, 1] such that α ∈ Im(μ

A) ∩ Im(γ

A), which are called a

μ-level subalgebra and a γ -level subalgebra respectively.

Proof. Let x, y ∈ μ≥A,α

and q ∈ Q. Then μA

(x, q ) ≥ α and μA

(y, q ) ≥ α.

It follows that μA

(x ∗ y, q ) ≥ μA

(x, q ) ∧ μA

(y, q ) ≥ α so that x ∗ y ∈ μ≥A,α

.

Hence μ≥A,α

is a subalgebra of X . Now let x, y ∈ γ ≤A,α

. Then γ A

(x ∗ y, q ) ≤

γ A

(x, q ) ∨ γ A

(y, q ) ≤ α and so x ∗ y ∈ γ ≤A,α

. Therefore γ ≤A,α

is a subalgebra of X .

Theorem 3.15. Let A = (μA

, γ A

) be an IQFS in X such that the sets μ≥A,α

and γ ≤A,α

are subalgebras of X . Then A = (μA

, γ A

) is an intuitionistic Q-fuzzy subalgebra of X .

Proof. We need to show that A = (μA

, γ A

) satisfies the condition (IQF1). If the first condition of (IQF1) is not true, then there exist x0, y0 ∈ X such thatμA

(x0 ∗ y0, q ) < μA

(x0, q ) ∧ μA

(y0, q ) for all q ∈ Q. Let

α0 :=1

2(μ

A(x0 ∗ y0, q ) + (μ

A(x0, q ) ∧ μ

A(y0, q ))) .

Then μA

(x0 ∗ y0, q ) < α0 < μA

(x0, q ) ∧ μA

(y0, q ), and so x0 ∗ y0 /∈ μ≥A,α0

, but

x0, y0 ∈ μ≥A,α0

. This leads to a contradiction. Now assume that the second

condition of (IQF1) does not hold. Then γ A

(x0 ∗ y0, q ) > γ A

(x0, q ) ∨ γ A

(y0, q )

for some x0, y0 ∈ X and q ∈ Q. Takingβ 0 :=

1

2(γ

A(x0 ∗ y0, q ) + (γ

A(x0, q ) ∨ γ

A(y0, q ))) ,

then γ A

(x0, q )∨γ A

(y0, q ) < β 0 < γ A

(x0 ∗y0, q ). It follows that x0, y0 ∈ γ ≤A,β0

and

x0 ∗ y0 /∈ γ ≤A,β0

. This leads to a contradiction. This completes the proof.




Theorem 3.16. Any subalgebra of X can be realized as both a μ-level sub-algebra and a γ -level subalgebra of some intuitionistic Q-fuzzy subalgebra of X .

Proof. Let S be a subalgebra of X and let μA and γ A be Q-fuzzy sets in X defined by

μA(x, q ) :=

α, if x ∈ S ,0, otherwise,

and γ A(x, q ) :=

β, if x ∈ S ,1, otherwise,

for all x ∈ X and q ∈ Q where α and β are fixed numbers in (0, 1) such thatα + β < 1. Let x, y ∈ X for all q ∈ Q. If x, y ∈ S , then x ∗ y ∈ S . HenceμA(x∗y, q ) = μA(x, q )∧μA(y, q ) and γ A(x∗y, q ) = γ A(x, q )∨γ A(y, q ). If at leastone of x and y does not belong to S , then at least one of μA(x, q ) and μA(y, q ) isequal to 0, and at least one of γ A(x, q ) and γ A(y, q ) is equal to 1. It follows thatμA(x ∗ y, q ) ≥ 0 = μA(x, q ) ∧ μA(y, q ) and γ A(x ∗ y, q ) ≤ 1 = γ A(x, q ) ∨ γ A(y, q ).Hence A = (μA, γ A) is an intuitionistic Q-fuzzy subalgebra of X . Obviously,μ≥A,α

= S = γ ≤A,β

. This completes the proof.

Definition 3.17. Let f be a map from a set X to a set Y . If A = (μA

, γ A

)and B = (μ

B, γ

B) are IQFSs in X and Y respectively, then the preimage of B

under f , denoted by f −1(B), is an IQFS in X defined by

f −1(B) = (f −1(μB

), f −1(γ B

)),

and the image of A under f , denoted by f (A), is an IQFS of Y defined by

f (A) = (f sup(μA

), f inf (γ A

)),

where

f sup(μA

)(y, q ) =

⎧⎨⎩

supx∈f −1(y)

μA

(x, q ), if f −1(y, q ) = ∅,

0, otherwise,and

f inf (γ A

)(y, q ) =

inf

x∈f −1(y)γ A

(x, q ), if f −1(y, q ) = ∅,

1, otherwise,

for each y ∈ Y and q ∈ Q.

Theorem 3.18. Let f : X → Y be a homomorphism of BCK/BCI-algebras.If B = (μB, γ B) is an intuitionistic Q-fuzzy subalgebra of Y , then the preim-age f −1(B) = (f −1(μB), f −1(γ B)) of B under f is an intuitionistic Q-fuzzy subalgebra of X .

Proof. Assume that B = (μB, γ B) is an intuitionistic Q-fuzzy subalgebra of Y and let x1, x2 ∈ X and q ∈ Q. Then

f −1(μB)(x1 ∗ x2, q ) = μB(f (x1 ∗ x2), q ) = μB(f (x1) ∗ f (x2), q )≥ μB(f (x1), q ) ∧ μB(f (x2), q )= f −1(μB)(x1, q ) ∧ f −1(μB)(x2, q )




and

f −1(γ B)(x1 ∗ x2, q ) = γ B(f (x1 ∗ x2), q ) = γ B(f (x1) ∗ f (x2), q )≤ γ B(f (x1), q ) ∨ γ B(f (x2), q )= f −1(γ B)(x1, q ) ∨ f −1(γ B)(x2, q ).

Therefore f −1(B) = (f −1(μB), f −1(γ B)) is an intuitionistic Q-fuzzy subalgebraof X .

Theorem 3.19. Let f : X → Y be a homomorphism from a BCK/BCI-algebra X onto a BCK/BCI-algebra Y . If A = (μA, γ A) is an intuitionistic Q-

fuzzy subalgebra of X , then the image f (A) = (f sup(μA), f inf (γ A)) of A under f is an intuitionistic Q-fuzzy subalgebra of Y .

Proof. Let A = (μA, γ A) be an intuitionistic Q-fuzzy subalgebra of X and lety1, y2 ∈ Y and q ∈ Q. Noticing that

{(x1 ∗ x2, q ) | (x1, q ) ∈ f −1(y1, q ) and (x2, q ) ∈ f −1(y2, q )}⊆ {(x, q ) ∈ X × Q | (x, q ) ∈ f −1(y1 ∗ y2, q )},

we have

f sup(μA)(y1 ∗ y2, q )= sup{μA(x, q ) | (x, q ) ∈ f −1(y1 ∗ y2, q )}≥ sup{μA(x1 ∗ x2, q ) | (x1, q ) ∈ f −1(y1, q ) and (x2, q ) ∈ f −1(y2, q )}≥ sup{μA(x1, q ) ∧ μA(x2, q ) | (x1, q ) ∈ f −1(y1, q ) and (x2, q ) ∈ f −1(y2, q )}= sup{μA(x1, q ) | (x1, q ) ∈ f −1(y1, q )} ∧ sup{μA(x2, q ) | (x2, q ) ∈ f −1(y2, q )}= f sup(μA)(y1, q ) ∧ f sup(μA)(y2, q )

and

f inf (γ A)(y1 ∗ y2, q )= inf {γ A(x, q ) | (x, q ) ∈ f −1(y1 ∗ y2, q )}≤ inf {γ A(x1 ∗ x2, q ) | (x1, q ) ∈ f −1(y1, q ) and x2 ∈ f −1(y2, q )}≤ inf {γ A(x1, q ) ∨ γ A(x2, q ) | (x1, q ) ∈ f −1(y1, q ) and (x2, q ) ∈ f −1(y2, q )}

= inf {γ A(x1, q ) | (x1, q ) ∈ f −1

(y1, q )} ∨ inf {γ A(x2, q ) | (x2, q ) ∈ f −1

(y2, q )}= f inf (γ A)(y1, q ) ∨ f inf (γ A)(y2, q ).

Hence f (A) = (f sup(μA), f inf (γ A)) is an intuitionistic Q-fuzzy subalgebra of Y .

References

[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (No.1)(1986), 87-96.[2] K. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and

Systems 61 (1994), 137-142.[3] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and

Systems 88 (1997), 81-89.[4] K. Iseki, An algebra related with a propositional calculus, Proc. Japan.Acad. 42 (1966),

351-366.[5] K. Iseki, On BCI-algebras, Seminar Notes (presently Kobe J. Math) 8 (1980), 125-130.[6] Y. H. Yon and K. H. Kim, On intuitionistic fuzzy filters and ideals of lattices, Far East

J Math. Sci.(FJMS) 1 (3), (1999), 429-442.[7] L. A. Zadeh, Fuzzy sets , Inform. and Control, 8 (1965), 338-353.

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