ON n-FOLD FUZZY IMPLICATIVE / COMMUTATIVE IDEALS OF BCK-ALGEBRAS

We consider the fuzzification of the notion of an n-fold implicative ideal, an n-fold (weak) commutative ideal. We give characterizations of an n-fold fuzzy implicative ideal. We establish an extension property for n-fold fuzzy commutative ideals. 2000 Mathematics Subject Classification. 06F35, 03G25, 03E72.

(I2) x * y ∈ I and y ∈ I imply x ∈ I.A nonempty subset I of X is said to be an implicative ideal of X if it satisfies: (I1) 0 ∈ I, (I3) (x * (y * x)) * z ∈ I and z ∈ I imply x ∈ I.A nonempty subset I of X is said to be a commutative ideal of X if it satisfies: (I1) 0 ∈ I, (I4) (x * y) * z ∈ I and z ∈ I imply x * (y * (y * x)) ∈ 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 implicative ideal of X if it satisfies: (F1) µ(0) ≥ µ(x) for all x ∈ X, 3. n-fold fuzzy implicative ideals.For any elements x and y of a BCK-algebra X, x * y n denotes in which y occurs n times.Huang and Chen [1] introduced the concept of n-fold implicative ideals as follows.
We consider the fuzzification of the concept of n-fold implicative ideal.
Notice that the 1-fold fuzzy implicative ideal is a fuzzy implicative ideal.
Theorem 3.3.Every n-fold fuzzy implicative ideal is a fuzzy ideal.
The following example shows that the converse of Theorem 3.3 may not be true.
Example 3.4.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 BCKalgebra (see [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 implicative ideal of X because We give a condition for a fuzzy ideal to be an n-fold fuzzy implicative ideal.
Proof.Necessity is by taking z = 0 in (F5).Suppose that a fuzzy ideal µ satisfies the inequality µ(x) ≥ µ(x * (y * x n )) for all x, y ∈ X.Then Hence µ is an n-fold fuzzy implicative ideal of X.
Lemma 3.8 (see [3,Theorem 3.13]).Let µ be a fuzzy set in X.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].Lemma 3.9 (see [1,Theorem 2.5]).Every n-fold implicative ideal is an n-fold positive implicative ideal.
Using Theorem 3.6 and Lemmas 3.8 and 3.9, we have the following theorem.Theorem 3.10.Every n-fold fuzzy implicative ideal is an n-fold fuzzy positive implicative ideal.

n-fold fuzzy commutative ideals
We consider the fuzzification of n-fold (weak) commutative ideals as follows.
Note that the 1-fold fuzzy commutative ideal is a fuzzy commutative ideal.Putting y = 0 and y = x in (F7) and (F8), respectively, we know that every n-fold fuzzy commutative (or fuzzy weak commutative) ideal is a fuzzy ideal.Theorem 4.3.Let µ be a fuzzy ideal of X.Then (i) µ is an n-fold fuzzy commutative ideal of X if and only if (ii) µ is an n-fold fuzzy weak commutative ideal of X if and only if Proof.The proof is straightforward.
Lemma 4.4 (see [3,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: Theorem 4.5.If µ is both an n-fold fuzzy positive implicative ideal and an n-fold fuzzy weak commutative ideal of X, then it is an n-fold fuzzy implicative ideal of X.
Theorem 4.6 (extension property for n-fold fuzzy commutative ideals).Let µ and ν be fuzzy ideals of X such that µ(0) = ν(0) and µ ⊆ ν, that is, µ(x) ≤ ν(x) for all x ∈ X.If µ is an n-fold fuzzy commutative ideal of X, then so is ν.
and since ν is order reversing, it follows that Hence, by Theorem 4.3(i), ν is an n-fold fuzzy commutative ideal of X.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation