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Iranian Journal of Fuzzy Systems Vol. 1, No. 2, (2004) pp. 69-84 69

SOME QUOTIENTS ON A BCK-ALGEBRA GENERATED BY A

FUZZY SET

A. HASANKHANI AND H. SAADAT

Abstract. First we show that the cosets of a fuzzy idea µin a BCK-algebra

Xform another BCK-algebra X

µ(called the fuzzy quotient BCK-algebra of

Xby µ). Also we show that X

µis a fuzzy partition of X, and prove several

some isomorphism theorems. Moreover we prove that if the associated fuzzy

similarity relation of a fuzzy partition Pof a commutative BCK-algebra is

compatible, then Pis a fuzzy quotient BCK-algebra. Finally we deﬁne the

notion of a coset of a fuzzy ideal and an element of a BCK-algebra and prove

related theorems.

1. Introduction

In 1966, the notion of a BCK-algebra was introduced by Y. Imai and K. Iseki

[3]. Zadeh in 1965 [13] introduced the notion of fuzzy subset of a nonempty set A

as a function from Ato [0,1]. Ougen Xi extended these ideas to BCK-algebra [11].

In this paper the notions of fuzzy quotient BCK-algebra induced by fuzzy ideals,

and the concept of a quotient algebra of a BCK-algebra, generated by a fuzzy ideal

and an element are deﬁned and then related theorems are proved.

2. Preliminaries

Deﬁnition 2.1. [3, 6] (a) A BCK-algebra is a nonempty set Xwith a binary

operation ”*” and a constant 0 satisfying the following axioms:

(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 that x=y

(v) 0 ∗x= 0 ,forall x, y , z ∈X·

(b) A nonempty set Aof a BCK-algebra is said to be an ideal of Xif the following

conditions hold:

(i) 0 ∈A

(ii)x∈X , y ∗x∈Aimply that y∈A , forall y∈X

(c) A BCK-algebra Xis said to be commutative if x∗(x∗y) = y∗(y∗x), for all

x, y ∈X.x∗(x∗y) is denoted by x∧y

Received: June 2003; Accepted: November 2003

Key words and Phrases: Fuzzy similarity relations, Fuzzy partitions, Fuzzy quotient, Fuzzy

ideal, cosets, quotient algebra.

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70 A. Hasankhani and H. Saadat

Lemma 2.2. [7] Let Xbe a BCK-algebra. Then,

(i)x∗0 = x , ∀x∈X

(ii) [(y1∗x)∗(y2∗x)] ∗(y1∗y2) = 0 ,∀x, y1, y2∈X

(iii) (x∗y)∗z= (x∗z)∗y , ∀x, y, z ∈X

(iv) (x∗y)∗x= 0 ∀x, y ∈X

(v) (x∧y)∗x= (x∧y)∗y , ∀x, y ∈X

Deﬁnition 2.3. [9, 13] (i) For r∈[0,1] fuzzy point xris deﬁned to be fuzzy subset

of Xsuch that

xr(y) = rif y=x

0 if y6=x

(ii) If µ,ηare two fuzzy subsets of X. Then

µ⊆η⇔µ(x)≤η(x),∀x∈X

Deﬁnition 2.4. [11] A fuzzy sunset µof a BCK-algebra Xis a fuzzy ideal if it

satisﬁes

(i)µ(0) = 1 ,∀x∈X

(ii)µ(x)≥min{µ(x∗y), µ(y)},∀x, y ∈X

Lemma 2.5. [2] Let Xbe a BCK-algebra and µa fuzzy ideal of X. Then

(i)µ(x∗y)≥min{µ(x∗z), µ(y)(z∗y)},∀x, y, z ∈X

(ii)if x∗y= 0 then µ(x)≥µ(y),∀x, y ∈X·

Deﬁnition 2.6. Let µbe a fuzzy subset of Xand α∈[0,1]. Then by a level subset

µαof µwe mean the set {x∈X:µ(x)≥α}.

Deﬁnition 2.7. Let Xand Ybe two sets, and fa function of Xinto Y. Let µ

and ηbe fuzzy subsets of Xand Y, respectively. Then f(µ) the image of µunder

f, is a fuzzy subset of Y:

f(µ)(y) = (sup

f(x)=y

µ(x) if f−1(y)6= Φ

0 if f−1(y) = Φ ,

for all y∈Y,f−1(η) the pre-image of ηunder f, is a fuzzy subset of Xsuch that

f−1(η)(x) = η(f(x)) ,∀x∈X·

Lemma 2.8. [11] (i)Let µbe a fuzzy ideal of BCK-algebra X. For all α∈[0,1],

if µα6= Φ, then µαis an ideal of X.

(ii)Let f:X→X0be an epimorphism of BCK-algebra and µ0a fuzzy ideal of X0.

Then f−1(µ0)is a fuzzy ideal of X.

Deﬁnition 2.9. [10] Let Xbe a nonempty set and Ra fuzzy subset of X×X.

Then Ris called a fuzzy similarity relation on Xif

(i)R(x, x) = 1 ,∀x∈X

(ii)R(x, y) = R(y, x)

(iii)R(x, z)≥min{R(x, y), R(y, z)} ·

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Some Quotients on a BCK-algebra Generated by a Fuzzy set 71

Deﬁnition 2.10. [8, 10, 12] A fuzzy partition of a set Xis a subset Pof [0,1]X

whose members satisfy the following conditions:

(i) Every N∈Pis normalized; i.e.N(x) = 1, for at least one N∈X;

(ii) For each x∈X, there is exactly one N∈Pwith N(x) = 1;

(iii) If M, N ∈Pand, x, y ∈Xare such that M(x) = N(y) = 1, Then

M(y) = N(x) = sup{min{M(z), N (z)}:z∈X} ·

Given a fuzzy partition Pof Xand element x∈X, we denote the unique element

of Pwith value 1 at xby [x]p. It is called the fuzzy similarity class of x.

Lemma 2.11. [10, 12] A canonical one-to-one correspondence between fuzzy par-

tition and fuzzy similarity relations is deﬁned by sending a fuzzy partition Pof X

to its fuzzy similarity relation RP∈[0,1]X×X, where for all x, y ∈X, we have

RP(x, y) = [x]P(y).

The inverse correspondence is deﬁned by sending a fuzzy similarity relation R

on Xto its fuzzy partition PR⊆[0,1]Xgiven by PR={Rhxi:x∈X}, where

Rhxiis the fuzzy subset of Xdeﬁned for all y∈Xby Rhxi(y) = R(x, y).

Lemma 2.12. [10] Let Rbe a fuzzy similarity relation on X, and a, b ∈X. Then

Rhai=Rhbi ⇔ R(a, b) = 1 ·

Deﬁnition 2.13. Let Xand X0be general sets, f:X→X0a function, and µ

a fuzzy subset of X, If f(x) = f(y) implies that µ(x) = µ(y), then µis called

f-invariant.

Theorem 2.14. [4] Let Abe an ideal of X. The relation ∼Aon Xis deﬁned by

x∼Ay⇔x∗y∈A , y ∗x∈A·

i)The relation ∼Ais an equivalence relation.

ii)Let Cxbe the equivalence class of xand X

A={Cx:x∈X}.

Then (X

A, o, Co), is a BCK-algebra where Cxocy=Cx∗y,∀x, y ∈X.

Deﬁnition 2.15. [7] A BCK-algebra Xis called bounded if there is an element 1

of Xsuch that x∗1 = 0 for all x∈X.

Lemma 2.16. [7] Let Xbe a bounded and commutative BCK-algebra then

(i) (x∧y)∧z=x∧(y∧z)for all x, y, z ∈X

(ii)x∧1 = 1 ∧x=x

Deﬁnition 2.17. [1] A fuzzy ideal µof a BCK-algebra Xis said to be prime if:

µ(x∧y) = µ(x) or µ(x∧y) = µ(y),for all x, y ∈X·

3. Fuzzy cosets

From now on, Xis a BCK-algebra and µis a fuzzy ideal of X.

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72 A. Hasankhani and H. Saadat

Deﬁnition 3.1. Let x∈X. Then the fuzzy subset µxwhich is deﬁned by

µx(y) = min{µ(x∗y), µ(y∗x)}

is called a fuzzy coset of µ. The set of all fuzzy cosets of µis denoted by X

µ.

Lemma 3.2. Let µbe a fuzzy relation on Xwhich is deﬁned by

µ(x, y) = µx(y),∀x, y ∈X·

Then µis a fuzzy similarity relation on X.

Proof. Clearly the conditions (i) and (ii) of Deﬁnition 2.9 hold. Now by Lemma

2.5 (i), for all x, y, z ∈X,

µ(x∗z)≥min{µ(x∗y), µ(y∗z)}, µ(z∗x)≥min{µ(z∗y), µ(y∗x)}

Therefore the condition (iii) of Deﬁnition 2.9 holds.

Remark 3.3. Clearly µhxi=µx,∀x∈X.

Lemma 3.4. Let x, y1, y2∈Xand µy1=µy2. Then

µx∗y1=µx∗y2, µy1∗x=µy2∗x

Proof. Since µy1=µy2, then by Lemma 2.12, we get that µ(y1∗y2) = µ(y2∗y1) = 1.

On the other hand, from Deﬁnition 2.1 (a) (i) and Lemma 2.5 (ii) we obtain that:

µ((x∗y1)∗(x∗y2)) ≥µ(y2∗y1)·

Thus µ((x∗y1)∗(x∗y2)) = 1. Similarly µ((x∗y2)∗(x∗y1)) = 1.

Consequently µ(x∗y1, x∗y2) = 1 and hence by Remark 3.3 and Lemma 2.12 we have

µx∗y1=µx∗y2. Similarly, by Lemma 2.2 (ii) we can show that µy1∗x=µy2∗x.

Lemma 3.5. Let x, y, x0, y0∈X, µx=µx0and µy=µy0. Then µx∗y=µx0∗y0.

Proof. By Lemma 3.4 µx∗y=µx0∗yand µx0∗y=µx0∗y0. Therefore µx∗y=µx0∗y0.

Theorem 3.6. (X

µ, O, µ0)is a BCK-algebra where

O:X

µ×X

µ→X

µ

(µx, µy)7→ µx∗y·

Proof. The proof follows from Lemma 3.5.

Theorem 3.7. X

µis a fuzzy partition of X.

Proof. The proof follows from Lemmas 3.2 and 2.11.

Theorem 3.8. There exists an ideal Kof X

µsuch that

(X

µ)

K'X

µα

for all α∈[0,1].

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Some Quotients on a BCK-algebra Generated by a Fuzzy set 73

Proof. Let α∈[0,1]. By Lemma 2.8 (i), µαis an ideal of X. Deﬁne ϕ:X

µ→X

µα

by ϕ(µx) = Cxfor all x∈X. If µx=µy, then by Lemma 2.12 µ(x, y) = 1 and

hence µ(x∗y) = µ(y∗x) = 1 ≥α, which implies that x∗y∈µαand y∗x∈µα.

hence Cx=Cy. Thus ϕis well-deﬁned. Clearly ϕis an epimorphism. Now let

K=Kerϕ. The theorem is proved.

Deﬁnition 3.9. By µ∗, we mean the set {x∈X:µ(x)=1}. Clearly µ∗is an

ideal of X.

Theorem 3.10. X

µ'X

µ∗.

Proof. It is enough to show that the epimorphism ϕ, deﬁned in the proof of theorem

3.8, is one-to-one. To do this, Let Cx, Cy∈X

µ∗be such that Cx=Cyfor x, y ∈X.

Then x∗y∈µ∗and y∗x∈µ∗. In other words, µ(x∗y) = µ(y∗x) = 1 and hence

by Remark 3.3 and Lemmas 2.12 and 3.2, µx=µy.

Theorem 3.11. Let fbe a BCK-homomorphism from Xonto X0and µan f-

invariant fuzzy ideal of Xsuch that µ∗⊆Kerf . Then X

µ'X0.

Proof. Deﬁne g:X

µ→X0by g(µx) = f(x). By Lemmas 2.12 and 3.2, we have for

all x, x0∈X

µx=µx0⇒x∗x0, x0∗x∈µ∗⇒x∗x0, x0∗x∈Kerf ⇒f(x) = f(x0)

Therefore gis well-deﬁned. Clearly gis an epimorphism.

Now let µx∈Kerg. Then f(x) = f(0) = 0. Since µis f-invariant, hence µ(x) =

µ(0). From Deﬁnition 2.1 (a) (v) and Lemma 2.2 (i) we obtain that µ(x∗0) =

µ(0 ∗x) = µ(0) = 1.

Hence, µ(x, 0) = 1, which implies that µx=µ0, by Lemma 2.12. Thus Kerg =

{µ0}, and hence gis one-to-one.

Theorem 3.12. Let fbe a BCK-homomorphism from Xonto X0and µ∗=Kerf .

Then X

µ'X0·

Proof. Since X

Kerf 'X0, we conclude that X

µ∗'X0. Also by theorem 3.10

X

µ'X

µ∗. Thus X

µ∼

=X0.

Lemma 3.13. Let Qµ:X→X

µbe a function deﬁned by Qµ(x) = µx. Then

(i) 0 Qµis an epimorphism

(ii)if µ=χ{0}, then Qµis an isomorphism. in other words,

X'X

µ

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74 A. Hasankhani and H. Saadat

Proof. (i) The proof is easy.

(ii) If µx=µy, for x, y ∈X, then µ(x∗y) = µ(y∗x) = 1. Thus x∗y=y∗x= 0.

Hence x=y, therefore Qµis one-to-one.

Theorem 3.14. Let fbe a BCK-homomorphism from Xinto X0,µa fuzzy ideal

of Xand µ0a fuzzy ideal of X0such f(µ)⊆µ0. Then there is a homomorphism

of BCK-algebras f∗:X

µ→X0

µ0such that f∗Qµ=Qµ0f. In another words, the

following diagram is commutative.

Xf

→X0

↓Qµ↓Qµ0

X

µ

→

f∗X0

µ0

Proof. Deﬁne f∗:X

µ→X0

µ0by f∗(µx) = µ0

f(x). At ﬁrst we show that f∗is well-

deﬁned. To do this let µx1=µx2. Then by Lemma 2.12 µ(x1, x2) = 1, and hence

µ(x1∗x2) = µ(x2∗x1) = 1. Now we have

µ0(f(x1)∗f(x2)) = µ0(f(x1∗x2))

=f−1(µ0)(x1∗x2)

≥µ(x1∗x2),since f(µ)⊆µ0

= 1 ·

Similarly µ0(f(x2)∗f(x1)) = 1, thus µ0

f(x1)=µ0

f(x2)by Lemma 2.12. It is easily

seen that f∗is a homomorphism and f∗Qµ=Qµ0f.

Theorem 3.15. (Isomorphism theorem) Let f:X→X0be an epimorphism of

BCK-algebras, and µ0a fuzzy ideal of X0. Then

X

f−1(µ0)'X0

µ0·

Proof. By Lemma 2.8 (ii), µ=f−1(µ0) is a fuzzy ideal of X. Since fis onto, then

f(µ) = f(f−1(µ0)) = µ0·

By Theorem 3.14, the mapping f∗is a homomorphism. Clearly f∗is onto. To

show that f∗is one-to-one, suppose that µa∈Kerf ∗, for a∈Xthen we have

µ0

0=f∗(µa) = µ0

f(a)it follows that µ0(f(a)∗0) = 1. In other words µ0(f(a)) = 1.

Hence µ(a∗0) = µ(a) = (f−1(µ0))(a) = µ0(f(a)) = 1. On the other hand 1 =

µ(0) = µ(0 ∗a). Consequently µa=µ0. This completes the proof.

Corollary 3.16. (Homomorphism Theorem). Let f:X→X0be an epimorphism

of BCK-algebras. Then X

f−1(χ{0})'X0.

Proof. The proof follows from Theorem 3.15 and Lemma 3.13 (ii).

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Some Quotients on a BCK-algebra Generated by a Fuzzy set 75

Deﬁnition 3.17. A fuzzy similarity relation Ron Xis said to be compatible if for

each x, y, z ∈Xwe have:

R(x∗z, y ∗z)≥R(x, y) and R(z∗x, z ∗y)≥R(x, y)

Theorem 3.18. Let Rbe a compatible fuzzy similarity on X. Then Rh0iis a fuzzy

ideal of X.

Proof. Clearly Rh0i(0) = 1. Now let x, y ∈Xwe have

Rh0i(x) = R(0, x)≥min{R(0, x ∗y), R(x∗y , x)},by Deﬁnition 2.9·(iii)

= min{R(0, x ∗y), R(x∗y , x ∗0)}by Lemma 2.2(i)

≥min{R(0, x ∗y), R(y , 0)}by Deﬁnition 3.17

= min{Rh0i(x∗y), Rh0i(y)} ·

Theorem 3.19. Let Xbe a commulative BCK-algebra, Pa fuzzy partition of X

such that its fuzzy similarity RP(see Lemma 2.11) is compatible. Then X

RPh0i=P.

Proof. For simplicity of notation, we will denote RPh0iby η. At ﬁrst we show that

P⊆X

η. To do this, let M∈P. Then by Deﬁnition 2.10 (i) there exists x∈X

such that M(x) = 1. On the other hand, for all y∈X, [y]P(y) = 1. Thus by

Deﬁnition 2.1- (iii) and 3.17 we have:

M(y) = [y]P(x) = RP(x, y) = RP(y, x)≤RP(y∗y , x ∗y) = RP(0, x ∗y),

and also

M(y) = Rp(x, y)≤Rp(x∗x, y ∗x) = RP(0, y ∗x)·

Therefore

(1) M(y)≤ηx(y),∀y∈X·

On the other hand we obtain that

ηx(y)≤RP(0, x ∗y)≤RP(x∗0, x ∗(x∗y)) = RP(x, x ∗(x∗y))

and

ηx(y)≤RP(0, y ∗x)≤RP(y∗0, y ∗(y∗x)) = RP(y, y ∗(y∗x)) ·

Since Xis commutative, it follows that

ηx(y)≤min{RP(x, x ∧y), RP(x∧y, y)} ≤ RP(x, y)

Hence

(2) ηx(y)≤RP(x, y) = [y]P(x) = M(y),∀y∈X·

From (1) and (2) we obtain that

M=ηx,∃x∈X·

Thus

(3) P⊆X

η·

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76 A. Hasankhani and H. Saadat

Now let ηx∈X

η. Then by Deﬁnition 2.10 (ii), there exists N∈Psuch that

N(x) = 1. As we have proved, N=ηx, which implies that

(4) X

η⊆P·

Now the proof follows from (3) and (4).

4. Cosets of a BCK-algebra generated by a fuzzy ideal and an element

Deﬁnition 4.1. Let a∈X. We deﬁne the relation ”∼a” on Xas follows.

x∼ay⇔µ(x∗y)≥µ(a), µ(y∗x)≥µ(a) for all x, y ∈X

Theorem 4.2. x∼ayis an equivalence relation on X.

Proof. By Deﬁnition 2.1 and 2.4 (i), ”∼a” is reﬂexive and clearly ”∼a” is symmetric.

Now we prove that ”∼a” is transitive. To do this let x, y, z ∈X,x∼ayand y∼az.

Then we have

µ((x∗z)∗(x∗y)) ≥min{µ(((x∗z)∗(x∗y)) ∗(y∗z)), µ(y∗z)}

by Deﬁnition 2.4(ii)

= min{µ(0), µ(y∗z)},by Deﬁnition 2.1(i)

=µ(y∗z),by Deﬁnition 2.1(i)

≥µ(a),since y∼az·.

Hence:

µ(x∗z)≥min{µ((x∗z)∗(x∗y)), µ(x∗y)}

= min{µ(0), µ(a)},since x∼ay

Therefor

µ(x∗z)≥µ(a)·

Similarly by Lemma 2.2 (ii), we can show that

µ(z∗x)≥µ(a)·

Hence x∼az.

Deﬁnition 4.3. Let a∈X. For x∈X, the equivalence class of xwith respect to

”∼a” is denoted by Cx(a, µ) and it is called the coset of xin Xand generated by

aand µ.

Remark 4.4. The set of all cosets generated by aand µis denoted by CX(a, µ).

Corollary 4.5. CX(a, µ)is a partition for X.

Proposition 4.6. Let a, x ∈X. Then a∈Cx(a, µ)if and only if

Cx(a, µ) = C0(a, µ)·

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Some Quotients on a BCK-algebra Generated by a Fuzzy set 77

Proof. Let Cx(a, µ) = C0(a, µ). By Deﬁnition 2.1 (iv) and Lemma 2.2. (i) it follows

that a∈C0(a, µ). Hence a∈Cx(a, µ). Conversely, let a∈Cx(a, µ). Then we have

µ(x∗0) = µ(x)≥min{µ(x∗a), µ(a)} ≥ µ(a)·

On the other hand

µ(0 ∗x) = µ(0) ≥µ(a)·

Hence 0 ∼ax, therefore C0(a, µ) = Cx(a, µ).

Proposition 4.7. For all x, y, a ∈X,x∗y, y ∗x∈C0(a, µ)if and only if

Cx(a, µ) = Cy(a, µ)·

Proof. The proof follows from Deﬁnition 2.1 (iii) and Lemma 2.2. (i).

Proposition 4.8. y∈Cx(0, µ)implies that µ(x) = µ(y).

Proof. The proof follows from Lemma 2.5 (ii), and Deﬁnition 2.4 (ii).

Proposition 4.9. For a∈X,C0(a, µ)is a subalgebra and an ideal of X.

Proof. Let x, y ∈C0(a, µ). Then by Lemma 2.2 (iv) and 2.5 (ii) we have µ(x∗y)≥

µ(x)≥µ(a) and µ(y∗x)≥µ(y)≥µ(a). Since

µ((x∗y)∗0) = µ(x∗y), µ((y∗x)∗0) = µ(y∗x),

hence x∗y∈C0(a, µ). Therefore C0(a, µ) is a subalgebra of X. From Deﬁnition

2.5 and some calculation we get that, C0(a, µ) is an ideal of X.

Lemma 4.10. For all x, a, b ∈X,

(i) Cx(a∧b, µ)⊆Cx(a, µ)∩Cx(b, µ),

(ii) if µis a fuzzy prime ideal of X, then Cx(a∧b, µ) = Cx(a, µ)∩Cx(b, µ).

Proof. (i) From Lemma 2.2. (v) and 2.5 (ii) we can prove (i).

(ii) follows from Deﬁnition 2.17.

Theorem 4.11. Let Xbe a bounded, commutative BCK-algebra, µa fuzzy prime

ideal of X,x∈X, and Cx(X, µ) = {Cx(a, µ) : a∈X}. Deﬁne the operation ”.”

On Cx(X, µ)as follows: Cx(a, µ)·Cx(b, µ) = Cx(a∧b, µ). Then (Cx(X, µ),·)is a

monoid.

Proof. The proof follows from Lemmas 4.10 (ii) and 2.16 (ii).

Lemma 4.12. Let a, y1, y2∈Xand y1∼ay2. Then

x∗y1∼ax∗y2and y1∗x∼ay2∗x , for all x∈X·

Proof. Since y1∼ay2, we have µ(y1∗y2)≥µ(a) and µ(y2∗y1)≥µ(a). On the

other hand by Deﬁnition 2.1 (i) we get that

((x∗y1)∗(x∗y2)) ∗(y2∗y1) = 0 ·

Hence from Deﬁnition 2.4.

µ((x∗y1)∗(x∗y2)) ≥min{µ[((x∗y1)∗(x∗y2)) ∗(y2∗y1)], µ(y2∗y1)}

≥min{µ(0), µ(a)},since y1∼ay2=µ(a)

=µ(a)·

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78 A. Hasankhani and H. Saadat

Similarly we have µ((x∗y2)∗(x∗y1)) ≥µ(a). Consequently

x∗y1∼ax∗y2

Similarly from Lemma 2.2 (ii) we get that y1∗x∼ay2∗x.

Lemma 4.13. Let a∈X, We deﬁne ⊕:CX(a, µ)×CX(a, µ)→CX(a, µ)as

follows

Cx(a, µ)⊕Cy(a, µ) = Cx∗y(a, µ),∀x, y ∈X·

Then ⊕is an operation on CX(a, µ).

Proof. Let Cx1(a, µ) = Cx2(a, µ) and Cy1(a, µ) = Cy2(a, µ). Then x1∼ax2and

y1∼ay2. Hence by Lemma 4.12 we have x1∗y1∼ax2∗y2. ALso y1∼ay2implies

that x1∗y1∼ax1∗y2. Therefore from Theorem 4.2 we have x1∗y1∼ax2∗y2. In

other words Cx1∗y1(a, µ) = Cx2∗y2(a, µ).

Theorem 4.14. Let a∈X. Then (CX(a, µ), C0(a, µ),⊕)is a BCK-algebra called

the quotient algebra generated by µand a.

Proof. Clearly the axioms (i),(ii),(iii),(v) of Deﬁnition 2.1 hold. Now let

Cx(a, µ)⊕Cy(a, µ) = Cy(a, µ)⊕Cx(a, µ) = C0(a, µ)

Then

Cx∗y(a, µ) = C0(a, µ) = Cy∗x(a, µ)·

Hence by proposition 4.7 we have:

Cx(a, µ) = Cy(a, µ)·

Theorem 4.15. Let f:X→X0be an epimorphism of BCK-algebras and µbe

f-invariant and µµ(a)⊆Kerf , for a∈X. Then Cx(a, µ)'X0.

Proof. Deﬁne ψ:CX(a, µ)→X0by, ψ(Cx(a, µ)) = f(x), for all x∈X.

Let Cx(a, µ) = Cy(a, µ), where x, y ∈X. Then x∗y , y ∗x∈µµ(a)⊆Kerf .

Hence f(x) = f(y). In other words ψis well deﬁned. Clearly ψis an epimorphism.

Now let f(x) = f(y), for x, y ∈X. Then f(x∗y) = f(y∗x) = f(0).

Since µis f-invariant, we have

µ(x∗y) = µ(0) ≥µ(a), µ(y∗x) = µ(0) ≥µ(a)·

Therefore Cx(a, µ) = Cy(a, µ), which implies that ψis one-to-one.

References

[1] A Hasankhani, F-Spectrum of a BCK-algebra, J. Fuzzy Math. Vol. 8, No. 1(2000), 1-11.

[2] C.S. Hoo, Fuzzy ideal of BCI and MV-algebra, Fuzzy sets and Systems 62 (1994), 111-114.

[3] Y. Imai, K. Iseki, On axiom systems of propositional calculi, XIV. Proc. Jopan Academy, 42

(1966), 19-22.

[4] K. Iseki, On ideals in BCK-algebra, Math. Seminar Notes, 3(1975), Kobe University.

[5] K. Iseki, Some properties of BCK-algebra, 2 (1975), xxxv, these notes.

[6] K. Iseki, S.Tanaka, Ideal theory ob BCK-algebra, Math. Japonica, 21 (1976), 351-366.

[7] K. Iseki, S. Tanaka, An introduction to the theory of BCK-algrba, Math. Japonica, 23 (1978),

1-26.

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www.SID.ir

Some Quotients on a BCK-algebra Generated by a Fuzzy set 79

[8] V. Hohle, Quotients with respect to similarity relations, Fuzzy sets and systems 27 (1988),

31-44.

[9] W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy sets and Systems 8(1982), 133-

139.

[10] S. Ovchinnikov, Similarity relations, fuzzy partitions, and fuzzy ordering, Fuzzy sets and

system, 40 (1991), 107-126.

[11] O. Xi, Fuzzy BCK-algebra, Math. Japonica, 36 (1991), 935-942.

[12] L.A. Zadeh, Similarity relations and fuzzy ordering, Inform. Sci. 3(1971), 177-200.

[13] L.A. Zadeh, Fuzzy sets, Information and control, 8(1965), 338-353.

A. Hasankhani∗, Department of Mathematics, Shahid Bahonar University of Kerman,

Kerman, Iran

E-mail address:abhasan@mail.uk.ac.ir

H. Saadat, Islamic Azad University Science and Research Campus, Kerman, Iran

E-mail address:saadat@iauk.ac.ir

∗Corresponding author

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