ON n-FOLD FUZZY POSITIVE IMPLICATIVE IDEALS OF BCK-ALGEBRAS

We consider the fuzzification of the notion of an n-fold positive implicative ideal. We give characterizations of an n-fold fuzzy positive implicative ideal. We establish the extension property for n-fold fuzzy positive implicative ideals, and state a characterization of PIn-Noetherian BCK-algebras. Finally we study the normalization of n-fold fuzzy positive implicative ideals. 2000 Mathematics Subject Classification. 06F35, 03G25, 03E72.

(P4) (x * z) * (y * z) ≤ x * y, (P5) x ≤ y implies x * z ≤ y * z and z * y ≤ z * x.Throughout this paper X will always mean a BCK-algebra unless otherwise specified.A nonempty subset I of X is called an ideal of X if it satisfies (I1) 0 ∈ I, (I2) x * y ∈ I and y ∈ I imply x ∈ I.A nonempty subset I of X is said to be a positive implicative ideal if it satisfies (I1) 0 ∈ I, (I3) (x * y) * z ∈ I and y * z ∈ I imply x * z ∈ I.
Theorem 2.1 (see [3,Theorem 3]).A nonempty subset I of X is a positive implicative ideal of X if and only if it satisfies (I1) 0 ∈ I, (I4) ((x * y) * y) * z ∈ I and z ∈ I imply x * y ∈ I.
We now review some fuzzy logic concepts.A fuzzy set in a set X is a function µ : X → [0, 1].For a fuzzy set µ in X and t ∈ [0, 1] define U(µ; t) to be the set U(µ; t) := {x ∈ X | µ(x) ≥ t}.
A fuzzy set µ in X is said to be a fuzzy ideal of X if (F1) µ(0) ≥ µ(x) for all x ∈ X, (F2) µ(x) ≥ min{µ(x * y), µ(y)} for all x, y ∈ X.Note that every fuzzy ideal µ of X is order reversing, that is, if x ≤ y then µ(x) ≥ µ(y).
A fuzzy set µ in X is called a fuzzy positive implicative ideal of X if it satisfies (F1) µ(0) ≥ µ(x) for all x ∈ X, (F3) µ(x * z) ≥ min{µ((x * y) * z), µ(y * z)} for all x, y, z ∈ X. (2.1) 3. n-fold fuzzy positive implicative ideals.For any elements x and y of a BCKalgebra, x * y n denotes in which y occurs n times.Using Theorem 2.1, Huang and Chen [1] introduced the concept of an n-fold positive implicative ideal as follows.
Definition 3.1.A subset A of X is called an n-fold positive implicative ideal of X if (I1) 0 ∈ A, (I5) x * y n ∈ A whenever (x * y n+1 ) * z ∈ A and z ∈ A for every x, y, z ∈ X.
We try to fuzzify the concept of n-fold positive implicative ideal.
Definition 3.2.Let n be a positive integer.A fuzzy set µ in X is called an n-fold fuzzy positive implicative ideal of X if (F1) µ(0) ≥ µ(x) for all x ∈ X, Notice that the 1-fold fuzzy positive implicative ideal is a fuzzy positive implicative ideal.
Example 3.3.Let X = {0,a,b} be a BCK-algebra with the following Cayley table : Then µ is an n-fold fuzzy positive implicative ideal of X for every natural number n. Proposition 3.4.Every n-fold fuzzy positive implicative ideal is a fuzzy ideal for every natural number n.
Proof.Let µ be an n-fold fuzzy positive implicative ideal of X.Then Hence µ is a fuzzy ideal of X.
The following example shows that the converse of Proposition 3.4 may not be true.
Example 3.5.Let X = N ∪ {0}, where N is the set of natural numbers, in which the operation * is defined by x * y = max{0,x − y} for all x, y ∈ X.Then X is a BCK-algebra [1,Example 1.3].Let µ be a fuzzy set in X given by µ(0) = t 0 > t 1 = µ(x) for all x( = 0) ∈ X.Then µ is a fuzzy ideal of X.But µ is not a 2-fold fuzzy positive implicative ideal of X because µ(5 * 2 2 ) = µ(1) = t 1 and µ((5 * 2 3 ) * 0) = µ(0) = t 0 , and so Let X be an n-fold positive implicative BCK-algebra and let µ be a fuzzy ideal of X.For any x, y, z ∈ X we have Hence µ is an n-fold fuzzy positive implicative ideal of X. Combining this and Proposition 3.4, we have the following theorem.
Theorem 3.6.In an n-fold positive implicative BCK-algebra, the notion of n-fold fuzzy positive implicative ideals and fuzzy ideals coincide.Proposition 3.7.Let µ be a fuzzy ideal of X.Then µ is an n-fold fuzzy positive implicative ideal of X if and only if it satisfies the inequality µ(x * y n ) ≥ µ(x * y n+1 ) for all x, y ∈ X.
Proof.Suppose that µ is an n-fold fuzzy positive implicative ideal of X and let x, y ∈ X.Then µ x * y n ≥ min µ x * y n+1 * 0 ,µ(0) Conversely, let µ be a fuzzy ideal of X satisfying the inequality Hence µ is an n-fold fuzzy positive implicative ideal of X.
Corollary 3.8.Every n-fold fuzzy positive implicative ideal µ of X satisfies the inequality µ(x * y n ) ≥ µ(x * y n+k ) for all x, y ∈ X and k ∈ N.
Proof.Using Proposition 3.7, the proof is straightforward by induction.Lemma 3.9.Let A be a nonempty subset of X and let µ be a fuzzy set in X defined by where Then µ is a fuzzy ideal of X if and only if A is an ideal of X.
Proof.Let A be an ideal of X.Since 0 ∈ A, therefore µ(0) = t 1 ≥ µ(x) for all x ∈ X. Suppose that (F2) does not hold.Then there exist a, b ∈ X such that µ(a) = t 2 and min{µ(a * b), µ(b)} = t 1 .Thus µ(a * b) = t 1 = µ(b), and so a * b ∈ A and b ∈ A. It follows from (I2) that a ∈ A so that µ(a) = t 1 .This is a contradiction.Suppose that µ is a fuzzy ideal of X.Since µ(0) ≥ µ(x) for all x ∈ X, we have µ(0) = t 1 and hence 0 ∈ A. Let x, y ∈ X be such that x * y ∈ A and y ∈ A. Using (F2), we get µ(x) ≥ min{µ(x * y), µ(y)} = t 1 and so µ(x) = t 1 , that is, x ∈ A. Consequently, A is an ideal of X. Proposition 3.10.Let A be a nonempty subset of X, n a positive integer, and µ a fuzzy set in X defined as follows: where Then µ is an n-fold fuzzy positive implicative ideal of X if and only if A is an n-fold positive implicative ideal of X.
Proof.Assume that µ is an n-fold fuzzy positive implicative ideal of X.Then µ is a fuzzy ideal of X.It follows from Lemma 3.9 that A is an ideal of X.Let x, y ∈ X be such that x * y n+1 ∈ A. Using Proposition 3.7, we get µ(x * y n ) ≥ µ(x * y n+1 ) = t 1 and so µ(x * y n ) = t 1 , that is, x * y n ∈ A. Hence by [1, Theorem 1.5], we conclude that A is an n-fold positive implicative ideal of X. Conversely, suppose that A is an n-fold positive implicative ideal of X.Then A is an ideal of X (see [1,Proposition 1.2]).It follows from Lemma 3.9 that µ is a fuzzy ideal of X.For any x, y ∈ X, either x * y n ∈ A or x * y n ∉ A. The former induces µ(x * y n ) = t 1 ≥ µ(x * y n+1 ).In the latter, we know that x * y n+1 ∉ A by [1, Theorem 1.5].Hence µ(x * y n ) = t 2 = µ(x * y n+1 ).From Proposition 3.7 it follows that µ is an n-fold fuzzy positive implicative ideal of X. Proposition 3.11.A fuzzy set µ in X is an n-fold fuzzy positive implicative ideal of X if and only if it satisfies Proof.Suppose that µ is an n-fold fuzzy positive implicative ideal of X and let x, y, z ∈ X.Then µ is a fuzzy ideal of X (see Proposition 3.4), and so µ is order reversing.It follows from (P3), (P4), and (P5) that (3.10) Using (F2) and Corollary 3.8, we get which proves (F5).Conversely, assume that µ satisfies conditions (F1) and (F5).Taking z = 0 in (F5) and using (P1), we conclude that Hence µ is a fuzzy ideal of X. Putting z = y in (F5) and applying (III), (IV), and (F1), we have By Proposition 3.7, we know that µ is an n-fold fuzzy positive implicative ideal of X.
Now we give a condition for a fuzzy ideal to be an n-fold fuzzy positive implicative ideal.
Theorem 3.12.A fuzzy set µ in X is an n-fold fuzzy positive implicative ideal of X if and only if µ is a fuzzy ideal of X in which the following inequality holds: Proof.Assume that µ is an n-fold fuzzy positive implicative ideal of X.By Proposition 3.4, it follows that µ is a fuzzy ideal of X.
and so µ((a * b) * z n ) = µ(0).Using (F5) we obtain which is condition (F6).Conversely, let µ be a fuzzy ideal of X satisfying condition (F6).It is sufficient to show that µ satisfies condition (F5).For any x, y, z ∈ X we have which is precisely (F5).Hence µ is an n-fold fuzzy positive implicative ideal of X.
Theorem 3.13.Let µ be a fuzzy set in X and let n be a positive integer.Then µ is an n-fold fuzzy positive implicative ideal of X if and only if the nonempty level set U(µ; t) of µ is an n-fold positive implicative ideal of X for every t ∈ [0, 1].
Theorem 3.14.If µ is an n-fold fuzzy positive implicative ideal of X, then the set is an n-fold positive implicative ideal of X.
Proof.Let µ be an n-fold fuzzy positive implicative ideal of X. Clearly 0 ∈ X µ .Let x, y, z ∈ X be such that (x * y n+1 ) * z ∈ X µ and z ∈ X µ .Then It follows from (F1) that µ(x * y n ) = µ(0) so that x * y n ∈ X µ .Hence X µ is an n-fold positive implicative ideal of X.
Proof.Using Proposition 3.7, it is sufficient to show that ν satisfies the inequality ν(x * y n ) ≥ ν(x * y n+1 ) for all x, y ∈ X.Let x, y ∈ X.Then Since ν is a fuzzy ideal, it follows from (F1) and (F2) that This completes the proof.

PI n -Noetherian BCK-algebras
Definition 4.1.A BCK-algebra X is said to satisfy the PI n -ascending (resp., PI ndescending) chain condition (briefly, PI n -ACC (resp., PI n -DCC)) if for every ascending (resp., descending) sequence fold positive implicative ideals of X there exists a natural number r such that A r = A k for all r ≥ k.If X satisfies the PI n -ACC, we say that X is a PI n -Noetherian BCK-algebra.
for all x ∈ X, where A 0 stands for X.Then µ is an n-fold fuzzy positive implicative ideal of X.
Proof.Clearly µ(0) ≥ µ(x) for all x ∈ X.Let x, y, z ∈ X. Suppose that for k = 0, 1, 2,... ; r = 0, 1, 2,.... Without loss of generality, we may assume that k ≤ r .Then obviously z ∈ A k .Since A k is an n-fold positive implicative ideal, it follows that x * y n ∈ A k so that Then z ∈ A j \ A j+1 for some j ∈ N. Hence x * y n ∈ A j , and thus Consequently, µ is an n-fold fuzzy positive implicative ideal of X.
Theorem 4.2 tells that if every n-fold fuzzy positive implicative ideal of X has a finite number of values, then X satisfies the PI n -DCC.Now we consider the converse of Theorem 4.2.
Theorem 4.3.Let X be a BCK-algebra satisfying PI n -DCC and let µ be an n-fold fuzzy positive implicative ideal of X.If a sequence of elements of Im(µ) is strictly increasing, then µ has a finite number of values.
Proof.Let {t k } be a strictly increasing sequence of elements of Im(µ).Hence 0 ≤ , and so we obtain a strictly descending sequence of n-fold positive implicative ideals of X which is not terminating.This contradicts the assumption that X satisfies the PI n -DCC.Consequently, µ has a finite number of values.
(i) X is a PI n -Noetherian BCK-algebra.(ii) The set of values of any n-fold fuzzy positive implicative ideal of X is a wellordered subset of [0, 1].

Proof. (i)⇒(ii).
Let µ be an n-fold fuzzy positive implicative ideal of X. Assume that the set of values of µ is not a well-ordered subset of [0, 1].Then there exists a strictly decreasing sequence is a strictly ascending chain of n-fold positive implicative ideals of X, where U(µ; r ) = {x ∈ X | µ(x) ≥ t r } for every r = 1, 2,....This contradicts the assumption that X is PI n -Noetherian.(ii)⇒(i).Assume that condition (i) is satisfied and X is not PI n -Noetherian.Then there exists a strictly ascending chain of n-fold positive implicative ideals of X.Let A = ∪ k∈N A k .Then A is an n-fold positive implicative ideal of X. Define a fuzzy set ν in X by (4.10) We claim that ν is an n-fold fuzzy positive implicative ideal of X.Since 0 ∈ A k for all k = 1, 2,..., we have ν(0 Similarly for the case Thus ν is an n-fold fuzzy positive implicative ideal of X.Since the chain (4.9) is not terminating, ν has a strictly descending sequence of values.This contradicts the assumption that the value set of any n-fold fuzzy positive implicative ideal is well ordered.Therefore X is PI n -Noetherian.This completes the proof.
We note that a set is well ordered if and only if it does not contain any infinite descending sequence.
where {t k } is a strictly descending sequence in (0, 1).Then a BCK-algebra X is PI n -Noetherian if and only if for each nfold fuzzy positive implicative ideal µ of X, Im(µ) ⊆ S implies that there exists a natural number k such that Im(µ) ⊆ {t 1 ,t 2 ,...,t k }∪{0}.
Proof.Assume that X is a PI n -Noetherian BCK-algebra and let µ be an n-fold fuzzy positive implicative ideal of X.Then by Theorem 4.4 we know that Im(µ) is a well-ordered subset of [0, 1] and so the condition is necessary.
Conversely, suppose that the condition is satisfied.Assume that X is not PI n -Noetherian.Then there exists a strictly ascending chain of n-fold positive implicative ideals Define a fuzzy set µ in X by Consequently, µ is an n-fold fuzzy positive implicative ideal of X.This contradicts our assumption.

Normalizations of n-fold fuzzy positive implicative ideals
Definition 5.1.An n-fold fuzzy positive implicative ideal µ of X is said to be normal if there exists x ∈ X such that µ(x) = 1.
Note that if µ is a normal n-fold fuzzy positive implicative ideal of X, then clearly µ(0) = 1, and hence µ is normal if and only if µ(0) = 1.
Proposition 5.3.Given an n-fold fuzzy positive implicative ideal µ of X let µ + be a fuzzy set in X defined by µ + (x) = µ(x) + 1 − µ(0) for all x ∈ X.Then µ + is a normal n-fold fuzzy positive implicative ideal of X which contains µ.
Definition 5.11.An n-fold fuzzy positive implicative ideal µ of X is said to be fuzzy maximal if µ is nonconstant and µ + is a maximal element of the poset (ᏺ(X), ⊆).
For any positive implicative ideal I of X let µ I be a fuzzy set in X defined by (5.6) Theorem 5.12.Let µ be an n-fold fuzzy positive implicative ideal of X.If µ is fuzzy maximal, then (i) µ is normal, (ii) µ takes only the values 0 and 1, (ii) µ Xµ = µ, (iv) X µ is a maximal n-fold positive implicative ideal of X.
(iv) Since µ is nonconstant, X µ is a proper n-fold positive implicative ideal of X.Let J be an n-fold positive implicative ideal of X containing X µ .Then µ = µ Xµ ⊆ µ J .Since µ and µ J are normal n-fold fuzzy positive implicative ideals of X and since µ = µ + is a maximal element of ᏺ(X), we have that either µ = µ J or µ J = 1 where 1 : X → [0, 1] is a fuzzy set defined by 1(x) = 1 for all x ∈ X.The later case implies that J = X.If µ = µ J , then X µ = X µ J = J.This shows that X µ is a maximal n-fold positive implicative ideal of X.This completes the proof.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: