Some quotients on a Bck-algebra generated by a fuzzy set

Abstract

First we show that the cosets of a fuzzy idea µ in a BCK-algebra X form another BCK-algebra X µ (called the fuzzy quotient BCK-algebra of X by µ). Also we show that X µ is 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 P of a commutative BCK-algebra is compatible, then P is 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.

Full text

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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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. 
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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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