1. 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.
2. Preliminaries
Here we include some elementary aspects of BCI that are necessary for this paper. For more detail, we refer to [4] [6].
An algebra of type (2, 0) is called BCI-algebra if
the following conditions hold:
BCI-1. ;
BCI-2. ;
BCI-3. ;
BCI-4. and .
A binary relation can be defined by
BCI-5. .
Then is a partially ordered set with least element 0.
The following properties also hold in any BCI-algebra [7] [8]:
1) ;
2) and ;
3) and ;
4) ;
5) ;
6) ; let be a BCI-algebra.
Definition 2.1. A fuzzy subset of a BCK/BCI-algebra X is a function .
Definition 2.2. (C. Lele, C. Wu, P. Weke, T. Mamadou, and C.E. Njock [9] ). Let be the family of all fuzzy sets in X. For and , is a fuzzy point if
We denote by the set of all fuzzy points on X, and we define a binary operation on as follows
It is easy to verify , the following conditions hold:
BCI-1’. ;
BCI-2’. ;
BCI-3’. ;
BCK-5’. .
Remark 2.3. (C. Lele, C. Wu, P. Weke, T. Mamadou, and C.E. Njock [9] ). The condition BCI-4 is not true . So the partial order cannot be extended to .
We can also establish the following conditions :
1’) ;
2’) and ;
3’) and
;
4’) ;
5’) ;
6’) .
We recall that if A is a fuzzy subset of a BCK/BCI algebra X, then we have the following:
. (1)
, and (2)
We also have , and one can easily check that it is a BCK-algebra.
Definition 2.4 (Isèki [10] ). A nonempty subset of BCK/BCI-algebra X is called an ideal of X if it satisfies
1) ;
2) .
Definition 2.5. A nonempty subset I of BCI-algebra X is P-ideal if it satisfies:
1) ;
2)
Definition 2.6 (Xi [11] ). A fuzzy subset A of a BCK/BCI algebra X is a fuzzy ideal if
1) ;
2) .
Definition 2.7 (Xi [11] ). A fuzzy subset A of a BCI-algebra X is called a fuzzy P-ideal of X if.
1) ;
2)
Definition 2.8 [12]. is a weak ideal of if
1) ;
2) . Such that and , we have
.
Theorem 2.9 [13]. Suppose that A is a fuzzy subset of a BCK-algebra X, then the following conditions are equivalent:
1) A is a fuzzy ideal;
2) ;
3) , the t-level subset in an ideal when ;
4) is a weak ideal.
3. Fuzzy n-Fold P-Ideals in BCI-Algebras
Throughout this paper is the set of fuzzy points on BCI-algebra X and (where the set of all the natural numbers).
Let us denote by .
Moreover, by (where y and 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 :
1) ;
2) ,
.
Definition 3.2. A fuzzy subset A of X is called a fuzzy n-fold P-ideal of X if it satisfies :
1) ;
2) ,
.
Definition 3.3. is P-weak ideal of if
1) ;
2) ,
.
Definition 3.4. is an n-fold P-weak ideal of if
1) ;
2) ,
.
Example 3.5. Let with defined by Table 1.
By simple computations, one can prove that is BCI-algebra. Define by , where .
Table 1. Example 3.5.
One can easily check that for any .
Is a fuzzy n-fold P-ideal.
Remark 3.6. is a 1-fold P-weak ideal of a BCK-algebra if is P-weak ideal of .
Theorem 3.7. If A is a fuzzy subset of X, then A is a fuzzy n-fold P-ideal if is an n-fold P-weak ideal.
Proof.
- Let , it is easy to prove that ;
- Let and
and .
Since A is a fuzzy n-fold P-ideal, we have
Therefore .
- Let , it is easy to prove that ;
- Let and let and , then and .
Since is P-weak ideal, we have
Thus . □
Proposition 3.8. An n-fold P-weak ideal is a weak ideal.
Proof. let and , since n-fold P-weak ideal, we have
Thus is a weak ideal.
Corollary 3.9. A fuzzy n-fold P-ideal is a fuzzy ideal.
Theorem 3.10. Let be a family of n-fold P-weak ideals and be a family of fuzzy-fold P-ideals. Then: 1) is an n-fold P-weak ideal.
2) is an n-fold P-weak ideal.
3) is a fuzzy n-fold P-ideal.
4) is a fuzzy n-fold P-ideal.
Proof. 1) , then , so, , i.e. . For every , if and , then
and , thus
So . Thus is an n-fold P-weak ideals.
2) , then , such that , so, , i.e. . For every , if and , then such that
and , thus .
So . Thus is an n-fold P-weak ideals.
3) Follows from 1) and Theorem 3.7.
4) Follows from 2) and Theorem 3.7.
4. 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 4.1. A nonempty subset I of X is called an n-fold weak P-ideal of X if it satisfies
1) ;
2) .
Definition 4.2. A fuzzy subset A of X is called a fuzzy n-fold weak P-ideal of X if it satisfies
1) ;
2) .
Definition 4.3. is a weak P-weak ideal of if
1) ;
2)
.
Definition 4.4. is an n-fold a weak P-weak ideal of if
1) ;
2) ,
.
Example 4.5. Let in which is given by Table 2.
Table 2. Example 4.5.
Then is a BCI-algebra. Let and let us define a fuzzy subset by
It is easy to check that for any
Is an n-fold weak P-weak ideal.
Remark 4.6. is a 1-fold weak P-weak ideal of a BCK-algebra X if is a weak P-weak ideal.
Theorem 4.7. [13] If A is a fuzzy subset of X, then A is a fuzzy n-fold weak P-ideal if is an n-fold weak P-weak ideal.
Proof.
- Let obviously ;
- Let and , then and .
Since A is a fuzzy n-fold weak P-ideal, we have
Therefore .
- Let , it is easy to prove that ;
- Let and .
Then and .
Since is n-fold weak P-weak ideal, we have
Hence .
Proposition 4.8. An n-fold weak P-weak ideal is a weak ideal.
Proof. Let and and .
Since is n-fold weak P-weak ideal, we have .
Corollary 4.9. A fuzzy n-fold weak P-ideal is a fuzzy ideal.
Theorem 4.10. Let be a family of n-fold weak P-weak ideals and be a family of fuzzy n-fold weak P-ideals. then 1) is an n-fold weak P-weak ideal.
2) is an n-fold weak P-weak ideal.
3) is a fuzzy n-fold weak P-ideal.
4) is a fuzzy n-fold weak P-ideal.
Proof. 1) , then , so, , i.e. . For every , if
and , then
and , thus
So . Thus is an n-fold weak P-weak ideal.
2) , then , such that , so, , i.e. . For every , if and , then such that
and , thus .
So . Thus is an n-fold weak P-weak ideal.
3) Follows from 1) and Theorem 4.7.
4) Follows from 2) and Theorem4.7.
5. 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”);
Begin
If then
go to (1.);
EndIf
If then
go to (1.);
EndIf
Stop:=false;
;
While and not (Stop) do
;
While and not (Stop) do
;
While and not (Stop) do
If and then
If
Stop:=true;
EndIf
EndIf
Endwhile
Endwhile
Endwhile
If Stop then
Output (“I is aP-ideal of X”)
Else
(1.) Output (“I is not aP-ideal of X”)
EndIf
End
Algorithm for n-fold P-Ideals of BCI-Algebra
Input(X: BCI-algebra, : binary operation, I: a subset of X);
Output(“I is n-fold P-ideal of X or not”);
Begin
If then
go to (1.);
EndIf
If then
go to (1.);
EndIf
Stop:=false;
;
While and not (Stop) do
;
While and not (Stop) do
;
While and not (Stop) do
If and then
If
Stop:=true;
EndIf
EndIf
Endwhile
Endwhile
Endwhile
If Stop then
Output (“I is ann-fold P-ideal of X”)
Else
(1.) Output (“I is not ann-fold P-ideal of X”)
EndIf
End
Algorithm for Fuzzy P-Ideals of BCI-Algebra
Input(X: BCI-algebra, : binary operation, A: the fuzzy subset of X);
Output(“A is a fuzzy P-ideal of X or not”);
Begin
Stop:=false;
;
While and not (Stop) do
If then
Stop:=true;
EndIf
;
While and not (Stop) do
;
While and not (Stop) do
If then
Stop:=true;
EndIf
Endwhile
Endwhile
Endwhile
If Stop then
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”);
Begin
Stop:=false;
;
While and not (Stop) do
If then
Stop:=true;
EndIf
;
While and not (Stop) do
;
While and not (Stop) do
If
Stop:=true;
EndIf
Endwhile
Endwhile
Endwhile
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, );
Output(“I is ann-fold weak P-ideal of X or not”);
Begin
If then
go to (1.);
EndIf
If then
go to (1.);
EndIf
Stop:=false;
;
While and not (Stop) do
;
While and not (Stop) do
;
While and not (Stop) do
If and then
If
Stop:=true;
EndIf
EndIf
Endwhile
Endwhile
Endwhile
If Stop then
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”);
Begin
Stop:=false;
;
While and not (Stop) do
If then
Stop:=true;
EndIf
;
While and not (Stop) do
;
While and not (Stop) do
If then
Stop=true;
EndIf
Endwhile
Endwhile
Endwhile
If Stop then
Output (“A is not a fuzzy n-foldweakP-ideal of X”)
Else
Output (“A is a fuzzy n-foldweakP-ideal of X”)
EndIf
End
6. 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.
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