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
.
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.
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.