Fuzzy Foldness of P-Ideals in BCI-Algebras

This paper aims to introduce new notions of (fuzzy) n-fold P-ideals and (fuzzy) n-fold weak P-ideals in BCI-algebras, and investigate several properties of the foldness theory of P-ideals in BCI-algebras. Finally, we construct a computer-program for studying the foldness theory of P-ideals in BCI-algebras.


Introduction
The study of BCK/BCI-algebras was initiated by Iséki [1] as a generalization of the concept of set-theoretic difference and propositional calculus. Since then, a great deal of theorems has been produced on the theory of BCK/BCI-algebras. In (1965), Zadeh [2] was introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. In 1991, Xi [3] defined fuzzy subsets in BCK/BCI-algebras.
Huang and Chen [4] introduced the notions of n-fold implicative ideal and n-fold (weak) commutative ideals. Y. B. Jun [5] discussed the fuzzification of n-fold positive implicative, commutative, and implicative ideal of BCK-algebras.
In this paper, we redefined a P-ideal of BCI-algebras and studied the foldness theory of fuzzy P-ideals, P-weak ideals, fuzzy weak P-ideals, and weak P-weak ideals in BCI-algebras. This theory can be considered as a natural generalization of P-ideals. Indeed, given any BCI-algebras X, we use the concept of fuzzy point to characterize n-fold P-ideals in X. Finally, we construct some algorithms for studying the foldness theory of P-ideals in BCI-algebras.

Preliminaries
Here we include some elementary aspects of BCI that are necessary for this paper. For more detail, we refer to [4] [6].
Then ( ) , X ≤ is a partially ordered set with least element 0. The following properties also hold in any BCI-algebra [7] [8]: the set of all fuzzy points on X, and we define a binary operation on X  as follows x y x y λ µ λ µ * = * It is easy to verify , , x y z X λ µ α ∀ ∈  , the following conditions hold: We recall that if A is a fuzzy subset of a BCK/BCI algebra X, then we have the following: We also have , , , Definition 2.7 (Xi [11]). A fuzzy subset A of a BCI-algebra X is called a fuzzy P-ideal of X if.

1)
Theorem 2.9 [13]. Suppose that A is a fuzzy subset of a BCK-algebra X, then the following conditions are equivalent:

Fuzzy n-Fold P-Ideals in BCI-Algebras
Throughout this paper X  is the set of fuzzy points on BCI-algebra X and n ∈  (where  the set of all the natural numbers). Let us denote (where y and y µ occurs respectively n times) with , , , Definition 3.1. A nonempty subset I of a BCI-algebra X is an n-fold P-ideal of X if it satisfies : , , , , Table 1.
By simple computations, one can prove that ( )  One can easily check that for any 3 n ≥ .
Is a fuzzy n-fold P-ideal.
Theorem 3.7. If A is a fuzzy subset of X, then A is a fuzzy n-fold P-ideal if A  is an n-fold P-weak ideal.
Since A is a fuzzy n-fold P-ideal, we have 8. An n-fold P-weak ideal is a weak ideal. Proof. , Thus A  is a weak ideal. Corollary 3.9. A fuzzy n-fold P-ideal is a fuzzy ideal.
is an n-fold P-weak ideal.

3)
 is a fuzzy n-fold P-ideal.

4)
i i I A ∈  is a fuzzy n-fold P-ideal.

Proof. 1)
Im  is an n-fold P-weak ideals.

Fuzzy-Fold Weak P-Ideals in BCI-Algebras
In this section, we define and give some characterizations of (fuzzy) n-fold weak P-weak ideals in BCI-algebras.
, , Definition 4.4. A  is an n-fold a weak P-weak ideal of X  if 1)
Is an n-fold weak P-weak ideal.
Remark 4.6. A  is a 1-fold weak P-weak ideal of a BCK-algebra X if A  is a weak P-weak ideal.
[13] If A is a fuzzy subset of X, then A is a fuzzy n-fold weak P-ideal if A  is an n-fold weak P-weak ideal. Proof Since A is a fuzzy n-fold weak P-ideal, we have ( ) Since A  is n-fold weak P-weak ideal, we have Proposition 4.8. An n-fold weak P-weak ideal is a weak ideal.
Since A  is n-fold weak P-weak ideal, we have x A λ µ ∈  . Corollary 4.9. A fuzzy n-fold weak P-ideal is a fuzzy ideal.  is an n-fold weak P-weak ideal.

2)
i i I A ∈   is an n-fold weak P-weak ideal.

3)
 is a fuzzy n-fold weak P-ideal.

4)
i i I A ∈  is a fuzzy n-fold weak P-ideal. Journal of Applied Mathematics and Physics

Proof. 1)
Im  is an n-fold weak P-weak ideal.

Algorithms
Here we give some algorithms for studding the structure of the foldness of (fuzzy) P-ideals In BCI-algebras Algorithm for AP-Ideals of BCI-Algebra Input(X: BCI-algebra, * : binary operation, I: the subset of X); Output("I is aP-ideal of X or not");  Output("A is a fuzzy P-ideal of X or not"); Output ("A is not a fuzzyP-ideal of X") Else Output ("A is a fuzzyP-ideal of X") EndIf End Algorithm for Fuzzy n-fold P-Ideals of BCI-Algebra Input(X: BCI-algebra, * : binary operation, A: the fuzzy subset of X); Output("A is a fuzzy n-fold P-ideal of X or not");

If Stop then
Output ("A is not a fuzzy n-fold P-ideal of X") Else Output ("A is a fuzzy n-fold P-ideal of X") EndIf End Algorithm for N-Fold weak P-Ideals of BCI-Algebra Input(X:BCI-algebra, I: subset of X, n ∈  ); Output("I is ann-fold weak P-ideal of X or not"); Output ("I is ann-fold weak P-ideal of X") Else (1.) Output ("I is not ann-fold weak P-ideal of X") EndIf End Algorithm for Fuzzy n-Fold weak P-Ideals of BCI-Algebra Input(X: BCI-algebra, * : binary operation, A fuzzy subset of X); Output("A is a fuzzy n-fold weak P-ideal of X or not");

Conclusions and Future Research
In this paper, we introduce new notions of (fuzzy) n-fold P-ideals, and (fuzzy) n-fold weak P-ideals in BCI-algebras. Then we studied relationships between different type of n-fold P-ideals and investigate several properties of the foldness theory of P-ideals in BCI-algebras. Finally, we construct some algorithms for studying the foldness theory of P-ideals in BCI-algebras.
In our future study of foldness ideals in BCK/BCI algebras, maybe the following topics should be considered: 1) Developing the properties of foldness of implicative ideals of BCK/BCI algebras.
2) Finding useful results on other structures of the foldness theory of ideals of BCK/BCI algebras.
3) Constructing the related logical properties of such structures. 4) One may also apply this concept to study some applications in many fields like decision making, knowledge base systems, medical diagnosis, data analysis and graph theory.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.