COMMUTATIVE FUZZY IDEALS OF BCH-ALGEBRAS AND OTHER SUPERIOR LEVELS OF FUZZYFICATION

AND OTHER SUPERIOR LEVELS OF FUZZYFICATION

MUHAMMAD ZULFIQAR

Communicated by the former editorial board

In this paper, we introduce the concepts of commutative (∈,∈ ∨qk)-fuzzy ideals and commutative (¯∈,∈ ∨¯ q¯k)-fuzzy ideals of BCH-algebras and investigate some of their properties. The concept of a commutative fuzzy ideal with thresholds, commutative fuzzyfying ideal and commutative t-implication-based fuzzy ideal are introduced and studied.

AMS 2010 Subject Classification: 03G10, 03B05, 03B52, 06F35.

Key words: BCH-algebra, commutative (∈,∈ ∨qk)-fuzzy ideal, commutative (¯∈,∈ ∨¯ q¯k)-fuzzy ideal, commutative fuzzy ideal with threshold;

commutative t-implication-based fuzzy ideal.

1. INTRODUCTION

The theory of BCH-algebras was first introduced by Hu and Li in [13]

and gave examples of proper BCH-algebras [14]. The theories were further enriched by many authors (see [1, 6–8, 11, 25, 28–30]). The concept of a fuzzy set, which was introduced by Zadeh in his definitive paper [36] of 1965, was applied by many researchers to generalize some of the basic concepts of al- gebras. The fuzzy algebraic structures play a vital role in Mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, real analysis, measure theory etc. Chang applied it to the topological spaces in [5]. Das and Rosenfeld applied it to the fundamental theory of fuzzy groups in [9, 27]. In [15], Hong et al. applied the concept to BCH-algebras and studied fuzzy dot subalgebras of BCH-algebras. Jun, gives characterizations of BCI/BCH-algebras in [17]. In 2001, Junet al. discussed on imaginable T-fuzzy subalgebras and imaginable T-fuzzy closed ideals of BCH- algebras [18]. Kim [22] studied intuitionistic (T, S)-normed fuzzy closed ideals of BCH-algebras. In [20], Jun et al. discussed N-structures applied to closed ideals in BCH-algebras. Jun and Park investigated filters of BCH-algebras based on bipolar-valued fuzzy sets in [19]. In [10], Dudek and Rousseau, give

MATH. REPORTS16(66),3(2014), 331–372

the idea of set-theoretic relations and BCH-algebras with trivial structure. In [21], Kazanciet al. studied soft set and soft BCH-algebras. Yin studied fuzzy dot ideals and fuzzy dot H-ideals of BCH-algebras in [32]. In [31], Saeid et al.

discussed fuzzy n-fold ideals in BCH-algebras.

In 1971, Rosenfeld’s laid the foundation of fuzzy groups in [27]. Murali defined the concept of belongingness of a fuzzy point to a fuzzy set under a natural equivalence on a fuzzy set in [24]. In [26], the idea of fuzzy point and its belongingness to and quasi-coincidence with a fuzzy set were used to define (α, β)-fuzzy subgroups, where α, β ∈ {∈, q,∈ ∨q,∈ ∧q} with α 6=∈ ∧q. This concept was further discussed by Yuan et al. [35]. In particular, (∈,∈ ∨q)- fuzzy subgroup is an important and useful generalization of the Rosenfeld’s fuzzy subgroups. The (∈ ∨q)-level subsets was discussed in [2]. In [3], Bhakat introduced the concept of (∈,∈ ∨q)-fuzzy normal, quasi-normal and maximal subgroups. In [4], Bhakat and Das studied (∈,∈ ∨q)-fuzzy subgroups.

In this paper, we show that a fuzzy setµof a BCH-algebra X is a commu- tative fuzzy ideal of X if and only ifµt6=φis a commutative ideal of X. We also prove that a fuzzy setµof a BCH-algebra X is a commutative (∈,∈ ∨q_k)-fuzzy ideal of X if and only if it satisfies (K) and (L), where

(K) µ(0)≥µ(x)∧^1−k₂ ,

(L) µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)^1−k₂ , for all x, y, z ∈X.

We further show that a fuzzy set µof a BCH-algebra X, the following are equivalent:

(U) µis a commutative (∈,∈ ∨q_k)-fuzzy ideal of X.

(V) [µ]^k_t 6=φ ⇒ [µ]BX, for all t∈(0,1].

We call [µ]^k_t a commutative (∈ ∨q

k)-level ideal of µ.

We prove that if {µ_i|i∈Λ} be a family of commutative (∈,∈ ∨q_k)-fuzzy ideals of a BCH-algebra X, thenµ=∩_i∈Λµ_i is a commutative (∈,∈ ∨q_k)-fuzzy ideal of X.

In Section 2, we recall some ideal and define commutative ideal of BCH- algebra; in Section 3, we review some fuzzy logic concepts and define com- mutative fuzzy ideal and discuss some of their level ideal; in Section 4, we define (∈,∈ ∨q_k)-fuzzy subalgebra, (∈,∈ ∨q_k)-fuzzy ideal and commutative (∈,∈ ∨q_k)-fuzzy ideal and investigate some of their properties; in Section 5, we define the concepts of (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal and commutative (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal of BCH-algebras and investigate some of their properties; in Section 6, we define the concepts of fuzzy ideal with thresholds and commutative fuzzy ideal with thresholds. Finally, in Section 7, we consider the concepts of fuzzifying ideal, commutative fuzzifying ideal, t-implication-based fuzzy ideal and com- mutative t-implication-based fuzzy ideal of BCH-algebras. In particular, we

discuss the implication operators by Lukasiewicz system of continuous-valued logic.

2.CRISP SETS – LEVEL 0

Throughout this paper X always means a BCH-algebra without any spec- ification. We also include some basic results that are necessary for this paper.

By a BCH-algebra [1], we mean an algebra (X, ∗, 0) of type (2, 0) satis- fying the axioms:

(BCH-I) x ∗ x = 0

(BCH-II) x ∗ y = 0 and y ∗x = 0 imply x = y (BCH-III) (x ∗ y)∗ z = (x∗ z)∗y

for all x, y, z ∈X. A BCH-algebra X is said to be a BCI-algebra if it satisfies the identity:

(BCI-I) ((x ∗ y)∗ (x∗ z)) ∗(z∗ y) = 0 for all x, y, z ∈X.

BCC-algebras (introduced by Komori [23]) are generalizations of BCK- algebras, weak BCC-algebras are generalizations of BCI-algebras. By many mathematicians, especially from China and Korea, weak BCC-algebras are called BZ-algebras ([12, 37, 38]). A weak BCC-algebra satisfying the identity

0∗ x = 0

is called a BCC-algebra. A weak BCC-algebra satisfying the identity (x ∗y) ∗ z = (x∗ z)∗y

is called a BCI-algebra. A weak BCC-algebra which is neither a BCI-algebra or a BCC-algebra is called proper. A BCC-algebra with the condition

(x ∗(x ∗y)) ∗ y = 0

is called a BCK-algebra. BCK-algebras and BCI-algebras are two important classes of logical algebras introduced by Imai and Iseki [16] in 1966. It is known that the class of BCK-algebras is a proper subclass of the class of BCI- algebras. Since then, a great deal of literature has been produced on the theory of BCK/BCI-algebras. In [13], Hu and Li introduced a wide class of abstract al- gebras called BCH-algebras based upon BCK/BCI-algebras, and subsequently gave examples of proper BCH-algebras [14]. For the general development of the BCK/BCI/BCH-algebras the subalgebras play a central role. It is known that every BCI-algebra is a BCH-algebra but not conversely. A BCH-algebra X is called proper if it is not a BCI-algebra. It is known that proper BCH-algebras exist. In any BCH / BCI-algebra X we can define a partial order≤by putting x≤y if and only if x ∗ y = 0.

Proposition 2.1 ([31]). In any BCH-algebra X, the following are true:

(1) x ∗ (x∗ y)≤y

(2) 0 ∗(x ∗y) = (0 ∗ x)∗ (0∗y) (3) x ∗ 0 = x

(4) x ≤0 implies x = 0 for all x, y ∈ X.

Definition 2.2 ([15]).A nonempty subset S of a BCH-algebra X is called a subalgebra of X if it satisfies

x∗y ∈S, for all x, y ∈S.

Definition 2.3 ([29]).A subset I of a BCH-algebra X is called an ideal of X if it satisfies (I1) and (I2), where

(I1) 0 ∈I,

(I2) x ∗ y∈I and y∈ I imply x∈I, for all x, y ∈X.

Definition 2.4.A subset I of a BCH-algebra X is called a commutative ideal of X, denoted by I BX, if it satisfies (I1) and (I3), where

(I1) 0 ∈I,

(I3) (x ∗ y) ∗ z ∈ I and z ∈ I imply x ∗ ((y ∗ (y ∗ x)) ∗ (0∗ (0 ∗ (x ∗ y)))) ∈I,

for all x, y, z ∈X.

Theorem 2.5. A commutative ideal of a BCH-algebra must be an ideal, but the converse does not hold.

Proof. Suppose I is a commutative ideal of X and for all x, y, z∈ X, we have

(x∗ y) ∗z ∈I and z ∈ I imply x∗ ((y∗ (y∗ x)) ∗ (0∗(0∗ (x∗ y))))∈ I.

Put y = 0 in above we get (x ∗ 0)∗ z ∈I and z∈I imply x ∗ ((0∗ (0∗ x)) ∗ (0∗(0∗ (x∗ 0))))∈I

x ∗ z ∈ I and z ∈ I imply x ∗ ((0 ∗ (0 ∗ x)) ∗ (0 ∗ (0 ∗ x))) ∈ I (by Proposition 2.1(3))

x ∗z ∈I and z ∈ I imply x∗ 0∈ I (BCH-I)

x ∗z ∈I and z ∈ I imply x∈I (by Proposition 2.1(3))

This means that I satisfy (I2). Combining (I1) implies that I is an ideal.

The last part is shown by the example.

Example 2.6. Suppose X = {0,1,2,3,4}, the operation is given by the table

* 0 1 2 3 4

0 0 0 0 0 0

1 1 0 1 0 1

2 2 2 0 2 0

3 3 1 3 0 3

4 4 4 4 4 0

Then (X, ∗, 0) is a BCH-algebra. It is easy to verify that {0,2} is an ideal but{0,1,3}is not commutative ideal.

Theorem 2.7. Let X be a BCH-algebra. Then an ideal I of X is a com- mutative ideal if and only if the condition

(for all x, y ∈X) (x ∗y∈ I⇒ x∗ ((y ∗(y ∗x)) ∗ (0∗(0∗ (x∗ y)))) ∈I) is satisfied.

Proof. Straightforward.

3.LEVEL 1 OF FUZZIFICATION

We now review some fuzzy logic concepts.

Definition 3.1 ([36]). A fuzzy set µ of a universe X is a function from X into the unit closed interval [0, 1], that is µ: X → [0, 1].

Definition 3.2 ([9]). For a fuzzy setµof a BCH-algebra X and t∈ (0, 1], the crisp set

µ_t={x∈X|µ(x)≥t}

is called the level subset ofµ.

Recall that ([0, 1],∧= min,∨= max, 0, 1) is a complete lattice (chain).

Definition 3.3 ([22]).Let X be a BCH-algebra. A fuzzy setµin X is said to be a fuzzy subalgebra of X if it satisfies, for all x, y∈X,

µ(x∗y)≥µ(x)∧µ(y).

Theorem 3.4. Let µ be a fuzzy set of a BCH-algebra X. Then µ is a fuzzy subalgebra of X if and only if µt = {x ∈ X |µ(x) ≥ t} is a subalgebra of X for all t ∈ (0,1], for our convenience, the empty set φ is regarded as a subalgebra of X.

Proof. Straightforward.

Definition 3.5 ([31]).A fuzzy set µof a BCH-algebra X is called a fuzzy ideal of X if it satisfies (F1) and (F2), where

(F1) µ(0)≥µ(x),

(F2) µ(x)≥µ(x∗y)∧µ(y), for all x, y∈X.

Theorem 3.6. A fuzzy set µ of a BCH-algebra X is a fuzzy ideal if and only if µt6=φis an ideal of X.

Proof. The proof of the following theorem is obvious.

Definition 3.7.A fuzzy setµof a BCH-algebra X is called a commutative fuzzy ideal of X if it satisfies (F1) and (F3), where

(F1) µ(0)≥µ(x),

(F3) µ(x ∗ ((y∗ (y∗ x)) ∗ (0∗(0∗ (x∗ y))))) ≥µ((x ∗y) ∗ z)∧µ(z), for all x, y, z∈ X.

Theorem 3.8. A fuzzy setµof a BCH-algebra Xis a commutative fuzzy ideal of X if and only if µt6=φis a commutative ideal of X.

Proof. Letµbe a commutative fuzzy ideal of X andµ_t6=φfor t ∈(0, 1].

Since µ(0)≥µ(x) ≥tfor all x ∈µt, we get 0 ∈µt. If (x ∗ y) ∗ z ∈ µt and z

∈µ_t, then

µ((x∗ y) ∗z)≥t and µ(z)≥ t.

It follows from (F3) that

µ(x ∗ ((y∗ (y∗ x)) ∗ (0∗(0∗ (x∗ y))))) ≥µ((x ∗y) ∗z)∧ µ(z)

≥t ∧t ≥t.

Hence, x ∗ ((y ∗ (y∗ x)) ∗ (0∗ (0∗ (x ∗ y))))∈µt. This shows that µt

is a commutative ideal of X by (I3).

Conversely, suppose that for each t ∈ (0, 1], µt is either empty or a commutative ideal of X. For any x∈X, setting 06=µ(x) = t, then x∈µ_t. Since µ_t(6=φ) is a commutative ideal of X, we have 0∈µ_tand hence,µ(0)≥t=µ(x).

If µ(x) = 0 then obviously µ(0) ≥0 = µ(x). Thus,µ(0)≥µ(x) for all x∈X.

Now we prove thatµsatisfies (F3). If not, then there exist x₁, y₁, z₁ ∈X such that

µ(x₁∗((y₁∗(y₁∗x₁))∗(0∗(0∗(x₁∗y₁)))))< µ((x₁∗y₁)∗z₁)∧µ(z₁).

Select t ∈(0, 1] such that

µ(x₁∗((y₁∗(y₁∗x₁))∗(0∗(0∗(x₁∗y₁)))))< t≤µ((x₁∗y₁)∗z₁)∧µ(z₁).

Hence, (x₁∗y₁)∗z₁ ∈µ_t and z₁∈µ_t, butx₁∗((y₁∗(y₁∗x₁))∗(0∗(0∗ (x₁∗y₁))))∈µ_t, which is a contradiction. Therefore,

µ(x ∗((y ∗ (y∗ x)) ∗(0∗ (0∗ (x∗ y))))) ≥µ((x∗ y)∗ z)∧ µ(z).

Consequently, µis a commutative fuzzy ideal of X.

Theorem 3.9. Any commutative fuzzy ideal of a BCH-algebra is a fuzzy ideal.

Proof. Letµbe a fuzzy commutative ideal of a BCH-algebra X and let x, z ∈X.

Then

µ(x ∗z)∧ µ(z) =µ((x ∗ 0)∗z)∧ µ(z) (by Proposition 2.1(3))

≤ µ(x ∗ ((0∗ (0∗x)) ∗ (0∗(0∗ (x∗ 0))))) (by F3)

= µ(x ∗ ((0∗ (0∗x)) ∗ (0∗(0∗ x)))) (by Proposition 2.1(3))

= µ(x ∗ 0) (BCH-I)

= µ(x) (by Proposition 2.1(3)) Hence, µis a fuzzy ideal of X.

Remark 3.10. The converse of the above theorem may not be true as shown in the following example.

Example 3.11.Let X = {0,1,2,3,4}in which the operation is defined by

* 0 1 2 3 4

0 0 0 0 0 0

1 1 0 1 0 0

2 2 2 0 0 0

3 3 3 3 0 0

4 4 3 4 3 0

Then (X, ∗, 0) is a BCH-algebra. Let t0, t1, t2 ∈ [0, 1] be such that t₀ > t₁ > t₂. We define a mapµ: X→[0, 1] byµ(0) =t₀,µ(1) =t₁ andµ(2)

=µ(3) = µ(4) =t₂. By simple calculations show that µ is a fuzzy ideal of X.

But µis not a commutative fuzzy ideal of X, because Put x = 2, y = 3, z = 0 in (F3) we get

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z) µ(2∗((3∗(3∗2))∗(0∗(0∗(2∗3)))))≥µ((2∗3)∗0)∧µ(0)

µ(2∗((3∗3)∗(0∗(0∗0))))≥µ(0∗0)∧µ(0) µ(2∗(0∗(0∗0)))≥µ(0)∧µ(0)

µ(2∗(0∗0))≥µ(0)∧µ(0)

µ(2∗0)≥µ(0)∧µ(0) µ(2)≥µ(0)∧µ(0)

t2 ≥t0∧t0

≥t₀ t₂ t₀.

Theorem 3.12. A fuzzy ideal µ of a BCH-algebra X is a commutative fuzzy ideal of X if and only if it satisfies the condition

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ(x∗y) (i) for all x, y ∈ X.

Proof. Assume that µ is a commutative fuzzy ideal of X. By (F3), we have

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z).

Put z = 0 in above, we get

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗0)∧µ(0)

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ(x∗y)∧µ(0)(by Proposition 2.1(3))

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ(x∗y) (by using condition (F1)) Conversely, suppose µsatisfies (i). As µis a fuzzy ideal, hence

µ(x∗y)≥µ((x∗y)∗z)∧µ(z) (ii)

for all x, y, z ∈X. Combining (ii) and (i), then we obtain

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z).

Hence, µis a commutative fuzzy ideal of X.

4.LEVEL 2.1 OF FUZZIFICATION

In this section, we define (∈,∈ ∨q_k)-fuzzy subalgebra, (∈,∈ ∨q_k)-fuzzy ideal and commutative (∈,∈ ∨q_k)-fuzzy ideal and investigate some of their properties.

Definition 4.1 ([31]).A fuzzy setµ of a BCH-algebra X having the form µ(y) =

t∈(0,1], ify=x

0, ify6=x

is said to be a fuzzy point with support x and value t and is denoted byx_t.

For a fuzzy point x_t and a fuzzy set µ in a set X, Pu and Liu [26] gave meaning to the writting x_tαµ, whereα∈ {∈, q,∈ ∨q,∈ ∧q}.

A fuzzy pointxtis said to belong to (resp., quasi-coincident with) a fuzzy set µ, written asx_t∈µ(resp. x_tqµ) if µ(x) ≥t (resp. µ(x) + t> 1).

To say that x_t ∈ ∨qµ means that x_t ∈µorx_tqµ. x_tαµ if x_tαµ does not hold for α∈ {∈, q,∈ ∨q}.

Let k denote an arbitrary element of [0, 1) unless otherwise specified. For a fuzzy pointx_tand a fuzzy set µof a BCH-algebra X, we say that

(A0) xtqµifµ(x) + t >1, q =q0. (A) xtq_kµ ifµ(x) + t + k>1.

(B) x_t∈ ∨q_kµifx_t∈µorx_tq_kµ.

(C) xtαµ ifxtαµdoes not hold for α∈ {q_k,∈ ∨q_k}.

Definition 4.2. A fuzzy setµof a BCH-algebra X is called an (∈,∈ ∨q_k)- fuzzy subalgebra of X if it satisfies

xt1 ∈µ,yt2 ∈µ⇒ (x∗y)t1∧t2 ∈ ∨q_kµ, for all x, y ∈X and t₁, t₂ ∈ (0, 1].

Theorem 4.3. Let µ be a fuzzy set of a BCH-algebra X. Then the fol- lowing are equivalent:

(D) µ_t6=φ⇒ µ_tBX, for all t∈(^1−k₂ ,1].

(E) µ satisfies the following assertions:

(i) µ(x) ≤µ(0) ∨^1−k₂ ,

(ii) µ((x∗y)∗z)∧µ(z)≤µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨^1−k₂ , for all x, y, z ∈X.

Proof. Suppose that (D) is hold. If there isa∈X such that the condition (i) is not true, that is, there exist a∈X such that

µ(a)> µ(0)∨^1−k₂

then µ(a)∈(^1−k₂ ,1] and a∈µ_µ(a). But µ(0)< µ(a) implies that 0∈/ µ_µ(a), a contradiction. Hence, (i) is hold. Suppose that (ii) is false, i.e.,

w =µ((a∗b)∗c)∧µ(c)> µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))∨^1−k₂ for somea, b, c∈X. Then

w ∈(^1−k₂ ,1] and (a∗b)∗c, c∈µ_w. But

a∗((b∗(b∗a))∗(0∗(0∗(a∗b))))∈/ µ_w

sinceµ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))< w. This is a contradiction, and so (ii) holds.

Conversely, suppose thatµsatisfies conditions (i) and (ii). Let t∈(^1−k₂ ,1]

be such that µ_t6=φ. For any x∈µ_t, we have

µ(0)∨ ^1−k₂ ≥µ(x) ≥t > ^1−k₂

and so

µ(0) =µ(0)∨ ^1−k₂ ≥t.

Hence, 0 ∈µ_t. Let x, y, z ∈X be such that (x∗y)∗z∈µt and z ∈µt. Then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨^1−k₂ ≥µ((x∗y)∗z)∧µ(z)

≥t∧t ≥t > ^1−k₂

and thus,

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) =µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨^1−k₂ ≥t, i.e.,

x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))∈µ_t. Therefore µ_t is a commutative ideal of X.

Setting k= 0 in Theorem 4.3, then we have the following Corollary.

Corollary 4.4. Let µ be a fuzzy set of a BCH-algebra X. Then the following are equivalent:

(F) µ_t6=φ ⇒ µ_tB X, for allt∈(0.5,1].

(G) µ satisfies the following assertions:

(iii) µ(x) ≤µ(0) ∨0.5,

(iv) µ((x∗y)∗z)∧µ(z)≤µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨ 0.5, for all x, y, z ∈X.

Definition 4.5. A fuzzy setµof a BCH-algebra X is called an (∈,∈ ∨q_k)- fuzzy ideal of X if it satisfies (H) and (I), where

(H) x_t∈µ⇒ 0_t∈ ∨q_kµ,

(I) (x∗y)_t1 ∈µ,y_t2 ∈µ⇒ x_t1∧t₂ ∈ ∨q_kµ, for all x, y∈X and t, t1, t2 ∈(0,1].

Example 4.6.Let X = {0,1,2,3,4}in which is defined by

* 0 1 2 3 4

0 0 0 0 0 0

1 1 0 1 0 1

2 2 2 0 2 0

3 3 1 3 0 3

4 4 4 2 4 0

Then (X, ∗, 0) is a BCH-algebra. Define µ : X → [0, 1] by µ(0) = 0.4, µ(1) =µ(3) = 0.7 andµ(2) =µ(4) = 0.2. By simple calculations show thatµ is an (∈,∈ ∨q_k)-fuzzy ideal of X fork= 0.4 andµ is not fuzzy ideal of X.

Definition 4.7.A fuzzy setµof a BCH-algebra X is called a commutative (∈,∈ ∨q_k)-fuzzy ideal of X if it satisfies (H) and (J), where

(H) xt∈µ⇒ 0t∈ ∨q_kµ,

(J) ((x∗y)∗z)_t1 ∈µ, z_t2 ∈µ⇒(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))_t1∧t2 ∈

∨q_kµ,

for all x, y, z ∈X andt, t1, t2∈ (0, 1].

Theorem4.8. Every commutative(∈,∈ ∨q_k)-fuzzy ideal of aBCH-algebra X is an (∈,∈ ∨q_k)-fuzzy ideal.

Proof. Let µbe a commutative (∈,∈ ∨q_k)-fuzzy ideal of X. Then for all t₁, t₂ ∈(0, 1] and for all x, y, z∈ X, we have

((x∗y)∗z)t1 ∈µ, zt2 ∈µ⇒(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))t1∧t₂ ∈ ∨q_kµ.

Putting y = 0 in above, we get

((x∗0)∗z)_t1 ∈µ, z_t2 ∈µ⇒ (x∗((0∗(0∗x))∗(0∗(0∗(x∗0)))))_t1∧t₂ ∈ ∨q_kµ (x∗z)t1 ∈ µ, zt2 ∈ µ ⇒ (x∗((0∗(0∗x))∗(0∗(0∗x))))t1∧t2 ∈ ∨q_kµ (by Proposition 2.1(3))

(x∗z)t1 ∈µ, zt2 ∈µ⇒ (x∗0)t1∧t₂ ∈ ∨q_kµ (BCH-I)

(x∗z)t1 ∈µ, zt2 ∈µ⇒ xt1∧t2 ∈ ∨q_kµ(by Proposition 2.1(3))

This means that µ satisfies the condition (I). This implies that µ is an (∈,∈ ∨q_k)-fuzzy ideal of X.

A commutative (∈,∈ ∨q_k)-fuzzy ideal of a BCH-algebra X with k= 0 is called a commutative (∈,∈ ∨q)-fuzzy ideal of X.

Theorem 4.9. Let X be a BCH-algebra. A fuzzy setµof Xis a commu- tative (∈,∈ ∨q_k)-fuzzy ideal of X if and only if it satisfies (K) and (L), where (K) µ(0)≥µ(x)∧^1−k₂ ,

(L) µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧^1−k₂ , for all x, y, z ∈X.

Proof. Assume that µ is a commutative (∈,∈ ∨q_k)-fuzzy ideal of X. Let x ∈ X and suppose that µ(x) < ^1−k₂ . If µ(0) < µ(x), then µ(0) < t ≤ µ(x) for some t ∈ (0, ^1−k₂ ). It follows that x_t ∈ µ but 0_t ∈/ µ. Since µ(0) +t <

2t <1−k, we get 0tq_kµ. Therefore 0t∈ ∨q_kµ, which is a contradiction. Hence, µ(0)≥µ(x).

Now ifµ(x)≥ ^1−k₂ , thenx^1−k

2

∈µand so 0^1−k

2

∈ ∨q_kµ. This implies that

µ(0)≥ ^1−k₂

or

µ(0) + ^1−k₂ >1−k.

Hence,

µ(0)≥ ^1−k₂ . Otherwise,

µ(0) + ^1−k₂ < ^1−k₂ +^1−k₂ = 1−k, a contradiction. Consequently,

µ(0)≥µ(x)∧^1−k₂

for all x ∈X. Let x, y, z ∈X and suppose that µ((x∗y)∗z)∧µ(z)< ^1−k₂ . Now we have to show that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z).

If not, then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))< t≤µ((x∗y)∗z)∧µ(z) for somet∈(0, ^1−k₂ ). It follows that

((x∗y)∗z)_t∈µ andz_t∈µ, but (x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))_t∈/ µ and

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) +t <2t <1−k, i.e.,

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))_tq_kµ.

This is a contradiction. Thus,

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z) whenever

µ((x∗y)∗z)∧µ(z)< ^1−k₂ . If µ((x∗y)∗z)∧µ(z)≥ ^1−k₂ , then

((x∗y)∗z)^1−k

2

∈µ and z^1−k

2

∈µ.

Since µis a commutative (∈,∈ ∨q_k)-fuzzy ideal, it follows that (x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))^1−k

2

∈µ.

So that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥ ^1−k₂ or

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) + ^1−k₂ >1−k.

If µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))< ^1−k₂ , then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) + ^1−k₂ < ^1−k₂ +^1−k₂ = 1 −k a contradiction. Therefore

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥ ^1−k₂ . Consequently,

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧ ^1−k₂ for all x, y, z∈ X.

Conversely, suppose that (K) and (L) are hold. Let x∈X and t ∈(0, 1]

be such that x_t∈µ. Then µ(x)≥t. Supposeµ(0)< t. Ifµ(x)< ^1−k₂ , then µ(0)≥µ(x)∧^1−k₂ = µ(x) ≥t,

a contradiction. Hence, µ(x)≥ ^1−k₂ . This implies that µ(0) +t >2µ(0)≥2(µ(x)∧ ^1−k₂ ) = 1−k.

Thus, 0_t∈ ∨q_kµ. Letx, y, z∈X and t₁, t₂ ∈(0,1] be such that ((x∗y)∗ z)_t1 ∈µ and z_t2 ∈µ. Then

µ((x∗y)∗z)≥t₁ and µ(z)≥t₂. Suppose that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))< t₁∧t₂. If µ((x∗y)∗z)∧µ(z)< ^1−k₂ , then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧ ^1−k₂

=µ((x∗y)∗z)∧µ(z) ≥t₁∧t₂. This is not possible, and so

µ((x∗y)∗z)∧µ(z)≥ ^1−k₂ . It follows that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) +t₁∧t₂

>2µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))

≥2(µ((x∗y)∗z)∧µ(z)∧^1−k₂ ) = 1−k.

So that

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))_t1∧t2 ∈ ∨q_kµ.

Hence, µis a commutative (∈,∈ ∨q_k)-fuzzy ideal of X.

Setting k= 0 in Theorem 4.9, then we have the following Corollary.

Corollary 4.10. Let X be a BCH-algebra. A fuzzy set µ of X is a commutative (∈,∈ ∨q)-fuzzy ideal of X if and only if it satisfies (M) and (N), where

(M) µ(0)≥µ(x)∧0.5,

(N) µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧0.5, for all x, y, z ∈X.

Theorem4.11. A (∈,∈ ∨q_k)-fuzzy idealµof a BCH-algebra Xis a com- mutative (∈,∈ ∨q_k)-fuzzy ideal of X if and only if it satisfies the condition

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ(x∗y)∧ ^1−k₂ (iii) for all x, y ∈ X.

Proof. Assume that µ is a commutative (∈,∈ ∨q_k)-fuzzy ideal of X. By (L), we have

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧^1−k₂ .

Put z = 0 in above, we get

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗0)∧µ(0)∧ ^1−k₂

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ(x∗y)∧µ(0)∧^1−k₂ (by Proposition 2.1(3)) µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ(x∗y)∧^1−k₂ (by using condition (K)) Conversely suppose µ satisfies (iii). As µ is a (∈,∈ ∨q_k)-fuzzy ideal, hence,

µ(x∗y)≥µ((x∗y)∗z)∧µ(z)∧^1−k₂ (iv) for all x, y, z ∈X. Combining (iv) and (iii), then we obtain

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧^1−k₂ ∧^1−k₂

≥µ((x∗y)∗z)∧µ(z)∧^1−k₂

Hence, µis a commutative (∈,∈ ∨q_k)-fuzzy ideal of X.

Setting k = 0 in Theorem 4.11, then we have the following Corollary.

Corollary 4.12. A (∈,∈ ∨q)-fuzzy ideal µ of a BCH-algebra X is a commutative (∈,∈ ∨q)-fuzzy ideal of X if and only if it satisfies the condition

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ(x∗y)∧0.5 for all x, y ∈ X.

Clearly, every commutative fuzzy ideal is a commutative (∈,∈ ∨q_k)-fuzzy ideal.

Here we give a condition for a commutative (∈,∈ ∨q_k)-fuzzy ideal to be a commutative fuzzy ideal.

Theorem4.13. Let µbe a commutative(∈,∈ ∨q_k)-fuzzy ideal of a BCH- algebra X. If µ(0)< ^1−k₂ , then µis a commutative fuzzy ideal of X.

Proof. Suppose thatµ(0)< ^1−k₂ . Thenµ(x)< ^1−k₂ and soµ(x)≤µ(0)<

1−k

2 for all x ∈ X by Theorem 4.9(K). It follows from Theorem 4.9(L) that µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧^1−k₂

=µ((x∗y)∗z)∧µ(z)

Hence, µis a commutative fuzzy ideal of X.

Corollary 4.14. Let µ be a commutative (∈,∈ ∨q)-fuzzy ideal of a BCH-algebra X. If µ(0)<0.5, then µ is a commutative fuzzy ideal of X.

Proof. It follows from Theorem 4.13 by letting k= 0.

Theorem 4.15. Let X be a BCH-algebra. If 0 ≤k < r <1, then every commutative (∈,∈ ∨q_k)-fuzzy ideal is a commutative (∈,∈ ∨q_r)-fuzzy ideal.

Proof. Straightforward.

Theorem 4.16. For a fuzzy setµ of a BCH-algebra X, the following are equivalent:

(O) µ is a commutative(∈,∈ ∨q_k)-fuzzy ideal of X.

(P) µ_t6=φ⇒ µ_tBX, for all t∈(0,^1−k₂ ].

We say that µ_t is an (∈ ∨q_k)-level ideal ofµ in X.

Proof. Suppose thatµ is a commutative (∈,∈ ∨q_k)-fuzzy ideal of X and let t∈(0,^1−k₂ ] be such that µt6=φ. By Theorem 4.9(K), we have

µ(0)≥µ(x)∧^1−k₂

for any x∈µt. It follows that

µ(0)≥t∧^1−k₂ = t.

So that 0∈µ_t. Let x, y, z∈X be such that (x∗y)∗z∈µ_tandz∈µ_t for t∈(0,^1−k₂ ]. Thenµ((x∗y)∗z)≥tand µ(z)≥t. By Theorem 4.9(L) implies that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧ ^1−k₂

≥t∧t∧ ^1−k₂ ≥t∧^1−k₂ = t.

Thus, (x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈µt, and soµtis a commutative ideal of X. Conversely, letµbe a fuzzy set of X such thatµtis non-empty and is a commutative ideal of X for all t∈(0,^1−k₂ ]. If there existsa∈X such that

µ(0)< µ(a)∧ ^1−k₂ , then

µ(0)< t₀ ≤µ(a)∧^1−k₂

for somet0 ∈(0,^1−k₂ ], and so 0∈/µt0. This is a contradiction. Therefore µ(0)≥µ(x)∧^1−k₂

for all x ∈X. Suppose that there exist a, b, c∈ X such that

µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))< µ((a∗b)∗c)∧µ(c)∧^1−k₂ . Then

µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))< t_w≤µ((a∗b)∗c)∧µ(c)∧^1−k₂ . for sometw ∈(0,^1−k₂ ]. It follows that

(a∗b)∗c∈µ_tw and c∈µ_tw, buta∗((b∗(b∗a))∗(0∗(0∗(a∗b))))∈/ µ_tw.

Which is not possible, and thus,

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧ ^1−k₂ for all x, y, z ∈ X. By Theorem 4.9, we conclude that µ is a commutative (∈,∈ ∨q_k)-fuzzy ideal of X.

Letting k= 0 in Theorem 4.16 induces the following Corollary.

Corollary 4.17. For a fuzzy set µ of a BCH-algebra X, the following are equivalent:

(Q) µ is a commutative(∈,∈ ∨q)-fuzzy ideal of X.

(R) µt6=φ ⇒ µtBX, for all t∈(0,0.5].

For a fuzzy point xt and a fuzzy set µof a BCH-algebra X, we say that (S) xtqµifµ(x) +t >1,

(T) x_tq_kµifµ(x) +t+k >1.

Denote by Q(µ;t) (resp. Q(µ;t)) the set {x ∈ X|x_tqµ} (resp. {x ∈ X|x_tqµ}), and

Q^k(µ;t) ={x∈X|x_tq_kµ}

[µ]^k_t ={x∈X|x_t∈ ∨q_kµ}

Q^k(µ;t) ={x∈X|x_tq_kµ}

[µ]^k_t ={x∈X|x_t∈ ∨q

kµ}

Obviously,

[µ]^k_t =µt∪Q^k(µ;t) and

[µ]^k_t =µ_t∪Q^k(µ;t).

Theorem 4.18. If µ is a commutative (∈,∈ ∨q_k)-fuzzy ideal of a BCH- algebra X, then Q^k(µ;t)6=φ⇒ Q^k(µ;t)BX, for allt∈(^1−k₂ ,1].

Proof. Assume t∈(^1−k₂ ,1] be such that Q^k(µ;t) 6=φ. Then there exists x0 ∈Q^k(µ;t), and soµ(x0) +t≥1−k. By Theorem 4.9(K), we have

µ(0)≥µ(x0)∧ ^1−k₂ ≥(1−k−t)∧^1−k₂ = 1−k−t i.e., 0tq_kµ. Hence, 0∈Q^k(µ;t). Let x, y, z∈X be such that

(x∗y)∗z∈Q^k(µ;t) and z∈Q^k(µ;t).

Then

µ((x∗y)∗z) +t≥1−kand µ(z) +t≥1−k.

From Theorem 4.9(L) it follows that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧ ^1−k₂

≥(1−k−t)∧(1−k−t)∧^1−k₂ ≥(1−k−t)∧ ^1−k₂ = 1 −k−t.

So that

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))tq_kµ, i.e.,

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈Q^k(µ;t).

Hence, Q^k(µ;t) is a commutative ideal of X.

Corollary 4.19. If µ is a commutative(∈,∈ ∨q)-fuzzy ideal of a BCH- algebra X, then

Q(µ;t)6=φ⇒ Q(µ;t)BX, for all t ∈(0.5,1].

Corollary 4.20. Let µ be a commutative (∈,∈ ∨q_k)-fuzzy ideal of a BCH-algebra X. If k < r <1, then

Q^r(µ;t)6=φ⇒ Q^r(µ;t)BX, for all t∈(^1−r₂ ,1].

Proof. It is straightforward by Theorems 4.15 and 4.18.

Theorem 4.21. For a fuzzy setµ of a BCH-algebra X, the following are equivalent:

(U) µis a commutative (∈,∈ ∨q_k)-fuzzy ideal of X.

(V) [µ]^k_t 6=φ ⇒ [µ]^k_t BX, for all t ∈(0,1].

We call [µ]^k_t a commutative (∈ ∨q

k)-level ideal of µ.

Proof. Suppose thatµ is a commutative (∈,∈ ∨q_k)-fuzzy ideal of X and let t∈(0, 1] such that [µ]^k_t 6=φ. Then there existsa∈[µ]^k_t 6=φ, and soa∈µt

ora∈Q^k(µ;t),i.e.,

µ(x)≥tor µ(x) +t≥1−k.

By Theorem 4.9(K), we get

µ(0)≥µ(a)∧^1−k₂ (1) Here we consider two cases:

Case 1: µ(a)≤ ^1−k₂ and Case 2: µ(a)> ^1−k₂ .

Case 1: We have µ(0) ≥ µ(a) by (1). Thus, if µ(a) ≥ t, then µ(0) ≥ t and so

0∈µt⊆[µ]^k_t. If µ(a) +t≥1−k, then

µ(0) +t≥µ(a) +t= 1 −k.

This implies that 0_tq_kµ,i.e.,

0∈Q^k(µ;t)⊆[µ]^k_t. Combining the Case 2 and (1) induces

µ(0)≥ ^1−k₂ .

If t≤ ^1−k₂ , then µ(0)≥tand hence, 0∈µ_t⊆[µ]^k_t. Case 2: If t > ^1−k₂ , then

µ(0) +t > ^1−k₂ +^1−k₂ = 1 −k.

This implies that

0∈Q^k(µ;t)⊆Q^k(µ;t)⊆[µ]^k_t.

Therefore [µ]^k_t satisfies the condition (I1). Let x, y, z ∈X be such that (x∗y)∗z∈[µ]^k_t and z∈[µ]^k_t.

Then

(x∗y)∗z∈µ_t or ((x∗y)∗z)_tq_kµ, and

z∈µt orztq_kµ, that is,

µ((x∗y)∗z)≥tor µ((x∗y)∗z) +t≥1−k,

and

µ(z)≥tor µ(z) +t≥1−k.

We consider the following cases:

(a) µ((x∗y)∗z)≥t andµ(z)≥t,

(b) µ((x∗y)∗z)≥t andµ(z) +t≥1−k, (c) µ((x∗y)∗z) +t≥1−k and µ(z)≥t, (d) µ((x∗y)∗z) +t≥1−k and µ(z) +t≥1−k.

Since µis a commutative (∈,∈ ∨q_k)-fuzzy ideal of X, we have

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧^1−k₂ (2) by Theorem 4.9(L). By using (a) and (2), we get

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧^1−k₂ µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥t∧t∧^1−k₂

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥t∧^1−k₂ .

If t≤ ^1−k₂ , then µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥t,i.e., (x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈µ_t⊆[µ]^k_t.

If t > ^1−k₂ , then µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥ ^1−k₂ and so µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) +t > ^1−k₂ +^1−k₂ = 1−k.

Hence,

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈Q^k(µ;t)⊆Q^k(µ;t)⊆[µ]^k_t. Case (b) and (2) imply that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧ ^1−k₂ µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥t∧(1−k−t)∧^1−k₂ . If t≤ ^1−k₂ , then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥t∧(1−k−t) =t and so

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈µ_t⊆[µ]^k_t. If t > ^1−k₂ , then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥(1−k−t)∧ ^1−k₂ = 1−k−t and thus,

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈Q^k(µ;t)⊆[µ]^k_t.

Similarly, we have (x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈[µ]^k_t from the case (c) and (2).

Finally, case (d) and (2) induces

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧^1−k₂ µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥(1−k−t)∧(1−k−t)∧^1−k₂ µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥(1−k−t)∧^1−k₂ .

If t≤ ^1−k₂ , then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥ ^1−k₂ ≥t.

Hence,

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈µt⊆[µ]^k_t. If t > ^1−k₂ , then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥1−k−t.

This implies that

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈Q^k(µ;t)⊆[µ]^k_t.

Consequently, [µ]^k_t is a commutative ideal of X. Conversely, assume that (V) is hold. If there existsa∈X such that

µ(0)< µ(a)∧ ^1−k₂ , then

µ(0)< t₀ ≤µ(a)∧^1−k₂

for some t0 ∈ (0,^1−k₂ ]. It follows that a ∈ µt0 ⊆ [µ]^k_t0 but 0 ∈/ µt0. Also, we have

µ(0) +t₀ <2t₀ ≤1−k, and so 0_t0q

kµ,i.e., 0∈/ Q^k(µ;t). Therefore 0∈/ [µ]^k_t0, a contradiction. Hence, µ(0)≥µ(x)∧^1−k₂

for all x ∈X. Suppose that there exist a, b, c∈ X such that

µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))< µ((a∗b)∗c)∧µ(c)∧^1−k₂ . Then

µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))< t_w ≤µ((a∗b)∗c)∧µ(c)∧ ^1−k₂

for some t_w ∈(0,^1−k₂ ]. It follows that (a∗b)∗c, c∈µ_w0 ⊆[µ]^k_tw from (I3) that (a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))∈[µ]^k_tw.

Thus,

µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))≥tw

or

µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b))))) +tw ≥1−k, a contradiction. Therefore

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥µ((x∗y)∗z)∧µ(z)∧ ^1−k₂ for all x, y, z ∈ X. From Theorem 4.9, we conclude that µ is a commutative (∈,∈ ∨q_k)-fuzzy ideal of X.

Corollary 4.22. For a fuzzy set µ of a BCH-algebra X, the following are equivalent:

(W) µ is a commutative(∈,∈ ∨q)-fuzzy ideal of X, (Y) [µ]t6=φ⇒ [µ]tBX, for all t∈(0,1],

where [µ]t={x∈X|x_t∈ ∨qµ}=µt∪Q(µ;t).

A fuzzy set µ of a BCH-algebra X is said to be proper if Im(µ) has at least two elements. Two fuzzy sets are said to be equivalent if they have the same family of level subsets. Otherwise, they are said to be non-equivalent.

Theorem4.23. Let µbe a commutative(∈,∈ ∨q_k)-fuzzy ideal of a BCH- algebra X such that

] {µ(x)|µ(x)< ^1−k₂ } ≥2, where ] mean cardinal.

Then there exist two proper non-equivalent commutative (∈,∈ ∨q_k)-fuzzy ideals of X such that µ can be expressed as the union of them.

Proof. Suppose that

{µ(x)|µ(x)< ^1−k₂ }={t₁, t₂, ..., t_n},

wheret₁ > t₂ > ... > t_nand n≥2. Then the chain of commutative∈ ∨q

k-level ideal of µis

[µ]^k1−k 2

⊆[µ]^k_t1 ⊆[µ]^k_t2 ⊆...⊆[µ]^k_tn = X.

Let τ and σ be fuzzy sets of X defined by

τ(x) =











t₁ ifx∈[µ]^k_t1, t2 ifx∈[µ]^k_t2\[µ]^k_t1, ...

tn ifx∈[µ]^k_tn\[µ]^k_tn−1, and

σ(x) =



















µ(x) ifx∈[µ]^k1−k 2

, k ifx∈[µ]^k_t2\[µ]^k1−k

2

, t₃ ifx∈[µ]^k_t3\[µ]^k_t2, ...

t_n ifx∈[µ]^k_tn\[µ]^k_tn−1,

respectively, where t3 < k < t2. Then τ and σ are commutative (∈,∈ ∨q_k)- fuzzy ideals of X, and τ, σ≤µ. The chains of commutative ∈ ∨q

k-level ideals of τ and σ are, respectively, given by

[µ]^k_t1 ⊆[µ]^k_t2 ⊆...⊆[µ]^k_tn and

[µ]^k1−k 2

⊆[µ]^k_t2 ⊆...⊆[µ]^k_tn.

Therefore τ and σ are non-equivalent and clearlyµ=τ ∪σ.

Theorem 4.24. Let {µ_i|i ∈ Λ} be a family of commutative (∈,∈ ∨q_k)- fuzzy ideals of a BCH-algebra X. Thenµ=∩i∈Λµiis a commutative(∈,∈ ∨q_k)- fuzzy ideal of X.

Proof. Let x ∈ X and t ∈ (0, 1] be such that xt ∈ µ. Suppose that 0_t∈ ∨q_kµ. Then

µ(0)< t and µ(x) +t≤1−k.

This implies that

µ(0)< ^1−k₂ . (3) Let

Ω1 ={i∈Λ|µ_i(0)≥t}

and

Ω₂={i∈Λ|0_tq_kµ_i and µ_i(0)< t}.

Then

Λ = Ω1∪Ω2

and

Ω₁∩Ω₂ =φ.

If Ω₂ =φ, thenµ_i(0)≥tfor alli∈Λ, and soµ(0)≥tis a contradiction.

Hence, Ω2 6=φ, and so µi(0) +t > 1−k and µi(0) < t for every i ∈ Ω2. It follows thatt > ^1−k₂ so that

µ_i(x)≥µ(x)≥t > ^1−k₂ for all i∈Λ. Now, assume that

t₀ =µ_i(x)< ^1−k₂

for somei∈Λ. Lett⁰₀ ∈(0,^1−k₂ ) be such thatt₀ < t⁰₀. Then µi(x)> ^1−k₂ > t⁰₀,

i.e.,x_t⁰

0 ∈µi. But µi(0) =t0 < t₀⁰ and µi(0) +t₀⁰ <1−k, that is,x_t⁰

0∈ ∨q_kµi. Which is a contradiction, and soµ_i(x)≥ ^1−k₂ for all i∈Λ. Thus,µ(x)≥ ^1−k₂ . This is not possible. Therefore 0t∈ ∨q_kµ. Letx, y, z∈X and t1, t2 ∈(0,1] be such that ((x∗y)∗z)t1 ∈µand zt2 ∈µ. Suppose that

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))t1∧t2∈ ∨q_kµ.

Then

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))< t₁∧t₂ and

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) +t₁∧t₂≤1−k.

It follows that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))< ^1−k₂ . Let

Ω3 ={i∈Λ|µ_i(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥t1∧t2} and

Ω₄ ={i∈Λ|(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))_t1∧t2q_kµ_i and µ_i(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))< t₁∧t₂}.

Then

Λ = Ω₃∪Ω₄ and

Ω3∩Ω4 =φ.

If Ω₄=φ, then

µ_i(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥t₁∧t₂ for all i∈Λ, and so

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥t1∧t2. This is a contradiction. Hence, Ω46=φ, and thus,

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))_t1∧t₂q_kµ_i, i.e.,

µ_i(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) +t₁∧t₂ >1−k, and

µi(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))< t1∧t2. It follows that

t₁∧t₂ > ^1−k₂ . So that

µi((x∗y)∗z)≥µ((x∗y)∗z) ≥t1 ≥t1∧t2 > ^1−k₂ for all i∈Λ. Similarly, we have

µ_i(z)> ^1−k₂ for all i∈Λ. Now, suppose that

t=µ_i(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))< ^1−k₂ for somei∈Λ. Lett⁰ ∈(0,^1−k₂ ) be such thatt < t⁰. Then

µ_i((x∗y)∗z)> ^1−k₂ > t⁰ and

µi(z)> ^1−k₂ > t⁰, that is,

((x∗y)∗z)_t⁰ ∈µi and z_t⁰ ∈µi. But

µi(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) =t < t⁰ and

µ_i(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) +t⁰ <1−k, that is,

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))_t⁰∈ ∨q_kµ_i. Which is a contradiction, and hence,

µ_i(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥ ^1−k₂ for all i∈Λ. Therefore

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))≥ ^1−k₂ . This does not hold. Consequently,

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))t1∧t2 ∈ ∨q_kµ.

Hence, µis a commutative (∈,∈ ∨q_k)-fuzzy ideal of X.

Setting k= 0 in Theorem 4.24, we have the following Corollary.

Corollary 4.25. Let {µ_i|i∈Λ} be a family of commutative (∈,∈ ∨q)- fuzzy ideals of a BCH-algebra X. Thenµ=∩_i∈Λµi is a commutative(∈,∈ ∨q)- fuzzy ideal of X.

Consider the number t∈(^1−k₂ ,1] for which µ_t is a commutative ideal of X, we consider a new kind of a commutative fuzzy ideal as follows.

5.LEVEL 2.2 OF FUZZIFICATION

In this section, we introduce the concepts of (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal and commutative (¯∈,∈ ∨¯ q¯k)-fuzzy ideal in BCH-algebras and investigate some of their properties.

Definition 5.1. A fuzzy setµof a BCH-algebra X is called an (¯∈,∈ ∨¯ q¯_k)- fuzzy ideal of X if it satisfies (Z) and (A1), where

(Z) 0_t∈µ¯ ⇒x_t∈ ∨¯ q¯_kµ,

(A1) xt∧r∈µ¯ ⇒ (x∗y)_t∈ ∨¯ q¯_kµory_r∈ ∨¯ q¯_kµ, for all x, y∈X and t, r∈ (0, 1].

Definition 5.2.A fuzzy setµof a BCH-algebra X is called a commutative (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal of X if it satisfies (Z) and (B1), where

(Z) 0t∈µ¯ ⇒xt∈ ∨¯ q¯kµ,

(B1) (x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))t∧r∈µ¯ ⇒ ((x∗y)∗z)_t∈ ∨¯ q¯_kµ orz_r∈ ∨¯ q¯_kµ,

for all x, y, z∈ X andt, r∈(0, 1].

Proposition 5.3. Every commutative (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal of a BCH- algebra X is an (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal.

Proof. Letµ be a commutative (¯∈,∈ ∨¯ q¯k)-fuzzy ideal of X. Then for all t, r∈(0, 1] and for allx, y, z ∈X, we have

(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))t∧r∈µ¯ ⇒ ((x∗y)∗z)_t∈ ∨¯ q¯_kµorz_r∈ ∨¯ q¯_kµ Putting y= 0 in above, we get

(x∗((0∗(0∗x))∗(0∗(0∗(x∗0)))))t∧r∈µ¯ ⇒((x∗0)∗z)_t∈ ∨¯ q¯_kµorz_r∈ ∨¯ q¯_kµ (x∗((0∗(0∗x))∗(0∗(0∗x))))t∧r∈µ¯ ⇒ (x∗z)t∈ ∨¯ q¯_kµ or zr∈ ∨¯ q¯_kµ (by Proposition 2.1(3))

(x∗0)t∧r∈µ¯ ⇒(x∗z)t∈ ∨¯ q¯kµorzr∈ ∨¯ q¯kµ (BCH-I) xt∧r∈µ¯ ⇒(x∗z)_t∈ ∨¯ q¯_kµorz_r∈ ∨¯ q¯_kµ(by Proposition 2.1(3))

This means that µsatisfies the condition (A1). This implies thatµ is an (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal of X.

A commutative (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal of a BCH-algebra X with k= 0 is called a commutative (¯∈,∈ ∨¯ q)-fuzzy ideal of X.¯

Theorem 5.4. A (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal of a BCH-algebra X is a com- mutative (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal of X if and only if it satisfies the condition

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨^1−k₂ ≥µ(x∗y) for all x, y ∈ X.

Proof. The proof is similar to the proof of Theorem 4.11.

Setting k= 0 in Theorem 5.4, we have the following Corollary.

Corollary 5.5. A (¯∈,∈ ∨¯ q)-fuzzy ideal of a¯ BCH-algebra X is a com- mutative (¯∈,∈ ∨¯ q)-fuzzy ideal of¯ X if and only if it satisfies the condition

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨0.5≥µ(x∗y) for all x, y∈ X.

Corollary 5.6. Every commutative fuzzy ideal is a commutative(¯∈,∈ ∨¯

¯

q_k)-fuzzy ideal.

Let µ be a commutative (¯∈,∈ ∨¯ q¯_k)-fuzzy ideal of a BCH-algebra X.

Assume that there existsa∈X such that µ(a)> µ(0)∨ ^1−k₂ . Then

µ(a)≥t > µ(0)∨^1−k₂

for some t∈(^1−k₂ ,1]. It follows that 0_t∈µ,¯ a_t ∈µ and µ(0) +t ≥2t >1−k, i.e.,a_tq_kµ. Which is a contradiction, and so the following inequality holds.

(C1) µ(a)≤µ(0)∨^1−k₂ for allx∈ X.

Assume that

µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))∨^1−k₂ < µ((a∗b)∗c)∧µ(c) for somea, b, c∈X. Then there existst∈(^1−k₂ ,1] such that

µ(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))∨^1−k₂ < t≤µ((a∗b)∗c)∧µ(c).

Thus,

(a∗((b∗(b∗a))∗(0∗(0∗(a∗b)))))_t∈µ.¯ Fromt≤µ((a∗b)∗c)∧µ(c), we have

((a∗b)∗c)t∈µ,ct∈µ,µ((a∗b)∗c) +t≥2t >1−k, i.e.,

((a∗b)∗c)_tq_kµ, and µ(c) +t≥2t >1−k,

i.e., ctqkµ. Which is not possible, and hence, we know that µ satisfies the following assertion:

(D1) µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨^1−k₂ ≥µ((x∗y)∗z)∧µ(z), for all x, y, z∈ X.

Suppose µ is a fuzzy set of a BCH-algebra X satisfying (C1) and (D1).

Lett∈(^1−k₂ ,1] be such that µt6=φ. Then there exists a∈µt, and so

1−k

2 < t≤µ(a)≤µ(0)∨^1−k₂ =µ(0)

by (C1). Hence, 0∈µt. Letx, y, z∈X be such that (x∗y)∗z∈µtandz∈µt. Then

µ((x∗y)∗z)≥t > ^1−k₂ and µ(z)≥t > ^1−k₂ . By using (D1), we get

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨^1−k₂ ≥µ((x∗y)∗z)∧µ(z)

≥t∧t≥t > ^1−k₂ . This implies that

µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y))))) =µ(x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∨^1−k₂ ≥t.

Thus, (x∗((y∗(y∗x))∗(0∗(0∗(x∗y)))))∈µt. Consequently, µtBX.

Thus, we conclude that if a fuzzy setµof X satisfies conditions (C1) and (D1), then the following assertion holds.

(E1) µt6=φ⇒ µtBX, for all t∈(^1−k₂ ,1].

Let µbe a fuzzy set of a BCH-algebra X satisfying (E1). Let x∈X and t ∈(0, 1] be such that xt∈ ∨¯ q¯_kµ. Then xt∈µand xtq_kµ. Hence,x∈µt,i.e., µ_t6=φ, and soµ_tBX by (E1). Thus, 0∈µ_t, and thus, µ(0)≥t,i.e., 0_t∈µ.

This shows that condition (Z) is held. Letx, y, z ∈X and t1, t2∈(0,1] be such that

((x∗y)∗z)t1∈ ∨¯ q¯kµand z∈ ∨¯ q¯kµ.

Then

((x∗y)∗z)t1 ∈µ,zt2 ∈µ, ((x∗y)∗z)t1q_kµand zt2q_kµ.

This implies that