ON SENSIBLE FUZZY IDEALS OF BCK-ALGEBRAS WITH RESPECT TO A t-CONORM

We introduce the notion of sensible fuzzy ideals of BCK-algebras with respect to a t-conorm and investigate some of their properties. We give the conditions for a sensible fuzzy subalgebra with respect to a t-conorm to be a sensible fuzzy ideal with respect to a t-conorm. Some properties of the direct product and S-product of fuzzy ideals of BCK-algebras with respect to a t-conorm are also discussed.


Introduction
Imai and Iséki [3] introduced the class of logical algebras: BCK-algebras.This notion is originated from two different ways: one of the motivations is based on set theory, another motivation is from classical and nonclassical propositional calculus.
The notion of fuzzy sets was first introduced by Zadeh [8].On the other hand, Schweizer and Sklar [5,6] introduced the notions of triangular norm (t-norm) and triangular conorm (t-conorm).Triangular norm (t-norm) and triangular conorm (t-conorm) are the most general families of binary operations that satisfy the requirement of the conjunction and disjunction operators, respectively.Thus, the t-norm generalizes the conjunctive (AND) operator and the t-conorm generalizes the disjunctive (OR) operator.In application, t-norm T and t-conorm S are two functions that map the unit square into the unit interval.Jun and Kim [4] introduced the notion of imaginable fuzzy ideals of BCK-algebras with respect to a t-norm.Cho et al. [1] have recently introduced the notion of sensible fuzzy subalgebras of BCK-algebras with respect to s-norm and studied some of their properties.In this paper, we introduce the notion of sensible fuzzy ideals of BCK-algebras with respect to a t-conorm and investigate some of their properties.We give conditions for a sensible fuzzy subalgebra with respect to a t-conorm to be a sensible fuzzy ideal with respect to a t-conorm.Some properties of the direct product and S-product of fuzzy ideals of BCK-algebras with respect to a t-conorm are also obtained.
2 Sensible fuzzy ideals of BCK-algebras with t-conorms

Preliminaries
In this section, we review some definitions and results that will be used in the sequel.
Replacing 0 by 1 in condition S 1 , we obtain the concept of t-norm T. Definition 2.3.Given a t-norm T and a t-conorm S, T and S are dual (with respect to the negation ) if and only if (T(x, y)) = S(x , y ).Proposition 2.4.Conjunctive (AND) operator is a t-norm T and disjunctive (OR) operator is its dual t-conorm S. Proposition 2.5 [5].For a t-conorm T, the following statement holds:

Fuzzy ideals with respect to a t-conorm
In what follows, let X denote a BCK-algebra unless otherwise specified.
Example 3.2.Let X = {0, a,b,1} be a BCK-algebra with the following Cayley table . By routine calculations, it is easy to check that μ is a sensible fuzzy ideal of X with respect to S m .Proposition 3.3.Let S be a t-conorm.Then every sensible fuzzy ideal of X with respect to S is an antifuzzy ideal of X.
Proof.The proof is obtained dually by using the notion of t-conorm S instead of t-norm T in [4].
The converse of Proposition 3.3 is not true in general as seen in the following example.
Theorem 3.5.Let S be a t-conorm and μ a nonempty fuzzy set of X.Then μ is fuzzy ideal of X with respect to S if and only if each nonempty level subset L(μ;α) of μ is an ideal of X.
is a family of fuzzy ideals of a BCK-algebra X with respect to S, then i∈I μ(x i ) is a fuzzy ideal of X with respect to S.
Proof.Let {μ i | i ∈ I} be a family of fuzzy ideals of X with respect to S. It is easy to see that μ i (0) ≤ μ i (x) for all x ∈ X.For x, y ∈ X, we have Hence i∈I is a fuzzy ideal of X with respect to S.
Proposition 3.8.Every sensible fuzzy ideal of X with respect to S is order preserving.
Proposition 3.9.Let μ be a sensible fuzzy ideal of X with respect to S. If the inequality x * y ≤ z holds in X, then μ(x) ≤ S(μ(y),μ(z)) for all x, y,z ∈ X.
Definition 3.10 [1].A fuzzy set μ is called a fuzzy subalgebra of X with respect to a t-conorm S if μ(x * y) ≤ S(μ(x),μ(y)) for all x, y ∈ X.
Theorem 3.11.Let S be a t-conorm.Then every sensible fuzzy ideal of X with respect to S is a sensible fuzzy subalgebra of X with respect to S. Proof.Straightforward.
The converse of Theorem 3.11 is not true in general as seen in the following example.Remark 3.13.In Example 3.12, we observe that a sensible fuzzy subalgebra with respect to S is not a sensible fuzzy ideal with respect to S. So, a question arises: under what condition(s) a sensible fuzzy subalgebra with respect to S is a sensible fuzzy ideal with respect to S? We answer this question in the following theorems without proofs.
Theorem 3.14.Let S be a t-conorm.A sensible fuzzy subalgebra μ of X with respect to S is a sensible fuzzy ideal of X with respect to S if and only if for all x, y,z ∈ X, the inequality x * y ≤ z implies that μ(x) ≤ S(μ(y),μ(z)).
Theorem 3.15.Let S be a t-conorm and let X be a BCK-algebra in which the equality x = (x * y) * y holds for all distinct elements x and y of X.Then every sensible fuzzy subalgebra of X with respect to S is a sensible fuzzy ideal of X with respect to S.
Definition 3.16.Let f : X → Y be a mapping, where X and Y are nonempty sets, and μ is fuzzy set of Y .The preimage of μ under f written μ f is a fuzzy set of X defined by μ f (x) = μ( f (x)) for all x ∈ X.
Theorem 3.17.Let f : X → Y be a homomorphism of BCK-algebras.If μ is a fuzzy ideal of Y with respect to S, then μ f is a fuzzy ideal of X with respect to S.
Proof.For any x ∈ X, we have Then we have Hence μ f is a fuzzy ideal of X with respect to S.
Theorem 3.18.Let f : X → Y be an epimorphism of BCK-algebras.If μ f is a fuzzy ideal of X with respect to S, then μ is a fuzzy ideal of Y with respect to S.
, where 0 = f (0).Let x, y ∈ Y .Then there exist a,b ∈ X such that 6 Sensible fuzzy ideals of BCK-algebras with t-conorms Hence μ is a fuzzy ideal of Y with respect to S.
Definition 3.19.Let f be a mapping defined on X.If ν is a fuzzy set in f (X), then the fuzzy set μ = ν • f in X (i.e., the fuzzy set defined by μ(x Theorem 3.20.Let S be a t-conorm and let f : X → Y be an epimorphism of BCK-algebras, ν sensible fuzzy ideal of Y with respect to S and μ, the preimage of ν under f .Then μ is a sensible fuzzy ideal of X with respect to S.
Proof.The proof is obtained dually by using the notion of t-conorm S instead of t-norm T in [4].
Theorem 3.21.Let μ be a fuzzy set in X and Im(μ) = {α 0 ,α 1 ,...,α k }, where α i < α j whenever i > j.Let {A n | n = 0,1,...,k} be a family of ideals of X with respect to a t-conorm S such that Then μ is a fuzzy ideal of X with respect to S.
Proof.Since 0 ∈ A 0 , we have μ(0) = α 0 ≤ μ(x) for all x ∈ X.Let x, y ∈ X.Then we discuss the following cases: if x * y ∈ A * n and y ∈ A * n , then x ∈ A n because A n is an ideal of X.Thus μ(x) ≤ α n = S μ(x * y),μ(y) . ( If x * y / ∈ A * n and y / ∈ A * n , then the following four cases arise: But, in either case, we know that μ(x) ≤ S μ(x * y),μ(y) . (3.7) M. Akram and J. Zhan 7 If x * y / ∈ A * n and y ∈ A * n , then by similar process, we have μ(x) ≤ S μ(x * y),μ(y) .(3.9)This completes the proof.
Definition 3.22 [9].A BCK-algebra X is said to satisfy the ascending (resp., descending) chain condition (ACC (resp., DCC)) if for every ascending (resp., descending) sequence ) of ideals of X there exists a natural number n such that A n = A k for all n k.If X satisfies DCC, X is an Artin BCK-algebras.
Theorem 3.23.Let S be a t-conorm.If μ is a fuzzy ideal of X, with respect to S, having finite image, then X is an Artin BCK-algebra.
Proof.Suppose that there exists a strictly descending chain A 0 ⊃ A 1 ⊃ A 2 ⊃ ••• of fuzzy ideals of X which does not terminate at finite step.Define a fuzzy set μ in X by where A 0 = X.We prove that μ is a fuzzy ideal of X with respect to S. Clearly, μ(0) ≤ μ(x) for all x ∈ X.Let x, y ∈ X. Assume that x * y ∈ A n \ A n+1 and y ∈ A k \ A k+1 for n = 0,1,2,...; k = 0,1,2,.... Without loss of generality, we may assume that n ≤ k.Then obviously y ∈ A n , and so x ∈ A n because A n is a fuzzy ideal of X. Hence Consequently, we conclude that μ is a fuzzy ideal of X with respect to S and μ has infinite number of different values.This is a contradiction, and the proof is complete.
Theorem 3.24.Let S be a t-conorm.The following statements are equivalent: (i) every ascending chain of ideals of X with respect to S terminates at finite step, (ii) the set of values of any fuzzy ideal with respect to S is a well-ordered subset of [0, 1].
8 Sensible fuzzy ideals of BCK-algebras with t-conorms Proof.Let μ be a fuzzy ideal of X with respect to S. Suppose that the set of values of μ is not a well-ordered subset of [0,1].Then there exists a strictly increasing sequence {α n } such that μ is a strictly ascending chain of ideals of X which is not terminating.This is a contradiction.
Conversely, suppose that there exists a strictly ascending chain of ideals of X with respect to S which does not terminate at finite step.Define a fuzzy set μ in X by where G = n∈N G n .Since 0 ∈ G n for all n = 0,1,..., therefore, μ(0) ≤ μ(x) for all x ∈ X. Hence μ is an ideal of X with respect to t-conorm S. Since the chain (*) is not terminating, μ has strictly descending sequence of values.This contradicts that the value of any set of fuzzy ideal with respect to S is well ordered.This ends the proof.Theorem 3.26.A fuzzy set μ of a BCK-algebra X is a T-fuzzy ideal of X if and only if its complement μ c is an S-fuzzy ideal of X.