2. PRELIMINARIES
An algebra (X,∗,0) is called a BCH-algebra (see [1]) if it satisfies the following assertions. If a set X has a special element 0 and a binary operation ∗ satisfying the conditions:
(∀u∈X)(u∗u=0),
(∀u,v∈X)(u∗v=0,v∗u=0⇒u=v),
(∀u,v,w∈X)((u∗v)∗w=(u∗w)∗v).
Any BCH-algebra X satisfies the following conditions (see [1,4]):
(∀u∈X)u∗0=u,(1)
(∀u∈X)u∗0=0⇒u=0,(2)
(∀u,v∈X)0∗(u∗v)=(0∗u)∗(0∗v),(3)
(∀u∈X)0∗(0∗(0∗u))=0∗u.(4)
(∀u,v∈X)u∗v=0⇒0∗u=0∗v. (5)
A subset S of a BCH-algebra X is called a subalgebra of X if u∗v∈S for all u,v∈S. A subset I of a BCH-algebra X is called a closed ideal of X (see [5]) if it satisfies
(∀u∈X)(u∈I⇒0∗u∈I),(6)
(∀u,v∈X)(u∗v∈I,v∈I⇒u∈I). (7)
Note that every closed ideal is a subalgebra, but the converse is not valid (see [5]).
A subset F of a BCH-algebra X is called a filter of X (see [6]) if it satisfies
(∀u,v∈X)(u∈F,u∗v=0⇒v∈F),(8)
(∀u,v∈X)(u∈F,v∈F⇒u∗(u∗v)∈F,v∗(v∗u)∈F). (9)
A filter F of a BCH-algebra X is said to be closed (see [6]) if 0∗u∈F for all u∈F.
Denote by ℱ(X,[−1,0]) the collection of functions from a set X to [−1,0]. We say that an element of ℱ(X,[−1,0]) is a negative-valued function from X to [−1,0] (briefly, N-function on X.) By an N-structure we mean an ordered pair (X,φ) of X and an N-function φ on X (see [16]).
3. k-FOLDED N-IDEALS/SUBALGEBRAS
In what follows, let k be a natural number unless otherwise specified and [−1,0]k denote the k-Cartesian product of [−1,0], that is,
[−1,0]k=[−1,0]×[−1,0]×⋯×[−1,0]
in which
[−1,0] is repeated
k times.
We give orders ≾ and ⋨ on [−1,0]k as follows:
t̃≾s̃⇔ti≤si, t̃⋨s̃⇔ti≩si,
respectively, for
i=1,2,⋯,k where
t̃:=(t1,t2,⋯,tk)∈[−1,0]k and
s̃:=(s1,s2,⋯,sk)∈[−1,0]k.
We define
Max{t̃,s̃}=(max{t1,s1},max{t2,s2},⋯,max{tk,sk}), Min{t̃,s̃}=(min{t1,s1},min{t2,s2},⋯,min{tk,sk}).
Definition 3.1.
A multi-folded N-structure with finite degree k (briefly, k-folded N-structure) over a universe X is defined to be a pair (f̃,X) where f̃:X→[−1,0]k is a mapping.
For any x∈X, the membership value of x is denoted by
f̃(x)=(ℓ1∘f̃)(x),(ℓ2∘f̃)(x),⋯,(ℓk∘f̃)(x),
where
ℓi:[−1,0]k→[−1,0] is the
i-th projection for
i=1,2,⋯,k, that is,
ℓi(t̃)=ti where
t̃:=(t1,t2,⋯,tk)∈[−1,0]k.
Given a k-folded N-structure (f̃,X) over a universe X, we consider the set
L(f̃;t̃):=x∈X|f̃(x)≾t̃, (10)
that is,
L(f̃;t̃):={x∈X|(ℓi∘f̃)(x)≤ti,i=1,2,⋯,k} =⋂i=1kL(f̃;t̃)i,
which is called a
k-
folded level set of
(f̃,X) related to
t̃, where
L(f̃;t̃)i:={x∈X|(ℓi∘f̃)(x)≤ti}
for
i=1,2,⋯,k.
Definition 3.2.
Let X be a BCH-algebra. A k-folded N-structure (f̃,X) over X is called a k-folded N-subalgebra of X if it satisfies
(∀x,y∈X)f̃(x∗y)≾Max{f̃(x),f̃(y)}, (11)
that is,
(∀x,y∈X)(ℓi∘f̃)(x∗y)≤max{(ℓi∘f̃)(x),(ℓi∘f̃)(y)} (12)
for
i=1,2,⋯,k.
Example 3.3.
Let X={0,1,2,3,4} be a set with the binary operation “∗” which is given in Table 1.
| ∗ |
0 |
1 |
2 |
3 |
4 |
| 0 |
0 |
0 |
0 |
0 |
4 |
| 1 |
1 |
0 |
0 |
1 |
4 |
| 2 |
2 |
2 |
0 |
0 |
4 |
| 3 |
3 |
3 |
3 |
0 |
4 |
| 4 |
4 |
4 |
4 |
4 |
0 |
Table 1Cayley table for the binary operation “*”.
Then (X,∗,0) is a BCH-algebra (see [5]). Let (f̃,X) be a 3-folded N-structure over X given by
f̃:X→[−1,0]3,x↦−0.82,−0.45,−0.66ifx=0,−0.33,−0.25,−0.44ifx=1,−0.33,−0.25,−0.44ifx=2,−0.33,−0.25,−0.44ifx=3,−0.82,−0.45,−0.66ifx=4.
It is routine to verify that (f̃,X) is a 3-folded N-subalgebra of X.
Proposition 3.4.
Every k-folded N-subalgebra (f̃,X) of X satisfies the following inequality
(∀x∈X)(f̃(0∗x)≾f̃(x)), (13)
that is,
(ℓi∘f̃)(0∗x)≤(ℓi∘f̃)(x) for all x∈X and i=1,2,⋯,k.
Proof.
For any x∈X and i=1,2,⋯,k, we have
(ℓi∘f̃)(x)=max{max{(ℓi∘f̃)(x),(ℓi∘f̃)(x)},(ℓi∘f̃)(x)} =max{(ℓi∘f̃)(x∗x),(ℓi∘f̃)(x)} =max{(ℓi∘f̃)(0),(ℓi∘f̃)(x)} ≥(ℓi∘f̃)(0∗x),
that is,
f̃(0∗x)≾f̃(x) for all
x∈X.
Definition 3.5.
Let X be a BCH-algebra. A k-folded N-structure (f̃,X) over X is called a k-folded N-closed ideal of X if it satisfies
(∀x,y∈X)f̃(0∗x)≾f̃(x)≾Max{f̃(x∗y),f̃(y)}, (14)
that is,
(∀x,y∈X)((ℓi∘f~)(0∗x)≤(ℓi∘f~)(x)≤max(ℓi∘f~)(x∗y),(ℓi∘f~)(y))(15)
for
i=1,2,⋯,k.
Example 3.6.
(1) Consider the BCH-algebra (X,∗,0) in Example 3.3. Let (f̃,X) be a 3-folded N-structure over X given as follows:
f̃:X→[−1,0]3,y↦−14,−0.45,−16ifx=4,−13,−0.88,−25otherwise.
It is easy to check that (f̃,X) is a 3-folded N-closed ideal of X.
(2) Consider a BCH-algebra X={0,1,2,3} with the binary operation “∗” which is given in Table 2.
| ∗ |
0 |
1 |
2 |
3 |
| 0 |
0 |
3 |
0 |
3 |
| 1 |
1 |
0 |
3 |
2 |
| 2 |
2 |
3 |
0 |
1 |
| 3 |
3 |
0 |
3 |
0 |
Table 2Cayley table for the binary operation “*”.
Let (f̃,X) be a 4-folded N-structure over X given by
f̃:X→[−1,0]4,y↦−0.82,−0.45,−0.66,−0.23ify∈{0,3}−0.33,−0.25,−0.44,−0.13ify∈{1,2}.
It is routine to prove that (f̃,X) is a 4-folded N-closed ideal of X.
It is clear that if a k-folded N-structure (f̃,X) over X is a k-folded N-closed ideal or a k-folded N-subalgebra of X, then f̃(0)≾f̃(x) for all x∈X, that is, (ℓi∘f̃)(0)≤(ℓi∘f̃)(x) for all x∈X and i=1,2,⋯,k.
We provide relations between k-folded N-closed ideal and k-folded N-subalgebra.
Theorem 3.7.
Every k-folded N-closed ideal is a k-folded N-subalgebra.
Proof.
Let (f̃,X) be a k-folded N-closed ideal of a BCH-algebra X. For any x,y∈X and i=1,2,⋯,k, we have
(ℓi∘f̃)(x∗y)≤max{(ℓi∘f̃)((x∗y)∗x),(ℓi∘f̃)(x)} =max{(ℓi∘f̃)((x∗x)∗y),(ℓi∘f̃)(x)} =max{(ℓi∘f̃)(0∗y),(ℓi∘f̃)(x)} ≤max{(ℓi∘f̃)(y),(ℓi∘f̃)(x)},
that is,
f̃(x∗y)≾Max{f̃(x),f̃(y)} for all
x,y∈X. Hence
(f̃,X) is a
k-folded
N-subalgebra of
X.
The 3-folded N-subalgebra (f̃,X) in Example 3.3 is not a 3-folded N-closed ideal since
(ℓ2∘f̃)(3)=−0.25>−0.45=max{(ℓ2∘f̃)(3∗4),(ℓ2∘f̃)(4)}.
Hence we know that the converse of Theorem 3.7 is not true in general.
We provide conditions for the converse of Theorem 3.7 to be true.
Theorem 3.8.
If a k-folded N-subalgebra (f̃,X) of X satisfies
(∀x,y∈X)(f̃(x)≾Max{f̃(x∗y),f̃(y)}), (16)
that is,
(ℓi∘f̃)(x)≤max{(ℓi∘f̃)(x∗y),(ℓi∘f̃)(y)} for all x,y∈X and i=1,2,⋯,k,
then (f̃,X) is a k-
folded N-
closed ideal of X.
Proof.
It is straightforward by (13) and (16).
Proposition 3.9.
If a k-folded N-closed ideal (f̃,X) of X satisfies
(∀x∈X)(f̃(x)≾f̃(0∗x)), (17)
that is,
(ℓi∘f̃)(x)≤(ℓi∘f̃)(0∗x) for all x∈X and i=1,2,⋯,k,
then (f̃,X) satisfies the following inequality:
(∀x,y∈X)(f̃(y∗x)≾f̃(x∗y)), (18)
that is,
(ℓi∘f̃)(y∗x)≤(ℓi∘f̃)(x∗y) for all x,y∈X and i=1,2,⋯,k.
Proof.
For any x,y∈X and i=1,2,⋯,k, we have
(ℓi∘f~)(y∗x)≤(ℓi∘f~)(0∗(y∗x))≤max{(ℓi∘f~)((0∗(y∗x))∗(x∗y)),(ℓi∘f~)(x∗y)}=max{(ℓi∘f~)(((0∗y)∗(0∗x))∗(x∗y)),(ℓi∘f~)(x∗y)}=max{(ℓi∘f~)(((0∗y)∗(x∗y))∗(0∗x)),(ℓi∘f~)(x∗y)}=max{(ℓi∘f~)(((0∗(x∗y))∗y)∗(0∗x)),(ℓi∘f~)(x∗y)}=max{(ℓi∘f~)((((0∗x)∗(0∗y))∗(0∗x))∗y),(ℓi∘f~)(x∗y)}=max{(ℓi∘f~)((0∗(0∗y))∗y),(ℓi∘f~)(x∗y)}=max{(ℓi∘f~)(0),(ℓi∘f~)(x∗y)}=(ℓi∘f~)(x∗y)
by (I), (III),
(3),
(15) and
(17). Hence
(18) is valid.
Theorem 3.10.
If a k-folded N-subalgebra (f̃,X) of X satisfies the condition (18), then (f̃,X) is a k-folded N-closed ideal of X.
Proof.
If we put y=0 in (18) and use (1), then f̃(0∗x)≾f̃(x∗0)=f̃(x) for all x∈X. Using (I), (III), (1), (12) and (18), we have
(ℓi∘f̃)(x)=(ℓi∘f̃)(x∗0)≤(ℓi∘f̃)(0∗x) =(ℓi∘f̃)((y∗y)∗x)=(ℓi∘f̃)((y∗x)∗y) ≤max{(ℓi∘f̃)(y∗x),(ℓi∘f̃)(y)} ≤max{(ℓi∘f̃)(x∗y),(ℓi∘f̃)(y)}
for all
x,y∈X and
i=1,2,⋯,k, that is,
f̃(x)≾Max{f̃(x∗y),f̃(y)} for all
x,y∈X. Therefore
(f̃,X) is a
k-folded
N-closed ideal of
X.
Using the notion of k-folded level sets, we consider characterization of k-folded N-closed ideal and k-folded N-subalgebra.
Theorem 3.11.
Given a k-folded N-structure (f̃,X) over a BCH-algebra X, the following are equivalent:
(f̃,X) is a k-folded N-closed ideal (resp. k-folded N-subalgebra) of X.
The k-folded level set L(f̃;t̃) of (f̃,X) is a closed ideal (resp. subalgebra) of X for all t̃∈[−1,0]k with L(f̃;t̃)≠∅.
Proof.
Assume that (f̃,X) is a k-folded N-closed ideal of X and let t̃∈[−1,0]k be such that L(f̃;t̃)≠∅. If x∈L(f̃;t̃), then x∈L(f̃;t̃)i for all i=1,2,⋯,k. Hene (ℓi∘f̃)(0∗x)≤(ℓi∘f̃)(x)≤ti for all i=1,2,⋯,k, and so 0∗x∈⋂i=1kL(f̃;t̃)i=L(f̃;t̃). Let x,y∈X be such that x∗y∈L(f̃;t̃) and y∈L(f̃;t̃). Then x∗y∈L(f̃;t̃)i and y∈L(f̃;t̃)i for all i=1,2,⋯,k. It follows that
(ℓi∘f̃)(x)≤max{(ℓi∘f̃)(x∗y),(ℓi∘f̃)(y)}≤ti.
Hence x∈L(f̃;t̃)i for all i=1,2,⋯,k and thus x∈⋂i=1kL(f̃;t̃)i=L(f̃;t̃). Therefore L(f̃;t̃) is a closed ideal of X. Similarly, we can show that if (f̃,X) is a k-folded N-subalgebra of X, then L(f̃;t̃) is a subalgebra of X.
Conversely, suppose that the k-folded level set L(f̃;t̃) of (f̃,X) is a closed ideal of X for all t̃∈[−1,0]k with L(f̃;t̃)≠∅. If the inequality f̃(0∗a)≾f̃(a) is false for some a∈X, then there exists t̃∈(−1,0)k such that f̃(a)≾t̃⋨f̃(0∗a), and so a∈L(f̃;t̃) and 0∗a∉L(f̃;t̃). This is a contradiction, and thus f̃(0∗x)≾f̃(x) for all x∈X. Now suppose that the inequality f̃(a)≾Max{f̃(a∗b),f̃(b)} is not true for some a,b∈X. Then Max{f̃(a∗b),f̃(b)}≾s̃⋨f̃(a) for some s̃∈[−1,0]k, which implies that a∗b∈L(f̃;s̃) and b∈L(f̃;s̃) but a∈L(f̃;s̃). This is impossible, and hence f̃(x)≾Max{f̃(x∗y),f̃(y)} for all x,y∈X. Therefore (f̃,X) is a k-folded N-closed ideal of X. By the similar way, we can show that if the k-folded level set L(f̃;t̃) of (f̃,X) is a subalgebra of X for all t̃∈[−1,0]k with L(f̃;t̃)≠∅, then (f̃,X) is a k-folded N-subalgebra of X.
Let X be a BCH-algebra. Then the BCA-part X+:={x∈X|0∗x=0} of X is a closed ideal of X, and the medial part Med(X):={x∈X|0∗(0∗x)=x} of X is a subalgebra of X (see [5]). Hence the following theorem is a direct result of Theorem 3.11.
Theorem 3.12.
Let (f̃,X) be a k-folded N-structure over a BCH-algebra X given by
f̃:X→[−1,0]k,x↦t̃ifx∈X+(resp. Med(X))s̃otherwise
where t̃=(t1,t2,⋯,tk)⋨(s1,s2,⋯,sk)=s̃ in [−1,0]k.
Then (f̃,X) is a k-
folded N-
closed ideal (resp.
k-
folded N-
subalgebra) of X.
Theorem 3.13.
If (f̃,X) is a k-folded N-closed ideal of a BCH-algebra X, then the set
Xe:={x∈X|f̃(x)≾f̃(e)} (19)
is a closed ideal of X for all e∈X.
Proof.
Note that Xe=⋂i=1kXei where Xei:={x∈X|(ℓi∘f̃)(x)≤(ℓi∘f̃)(e)}. For any i=1,2,⋯,k, if x∈Xei, then (ℓi∘f̃)(0∗x)≤(ℓi∘f̃)(x)≤(ℓi∘f̃)(e) and so 0∗x∈Xei. Let x,y∈X be such that x∗y∈Xei and y∈Xei. Then (ℓi∘f̃)(x∗y)≤(ℓi∘f̃)(e) and (ℓi∘f̃)(y)≤(ℓi∘f̃)(e). It follows from (15) that
(ℓi∘f̃)(x)≤max{(ℓi∘f̃)(x∗y),(ℓi∘f̃)(y)}≤(ℓi∘f̃)(e).
Hence x∈Xei, which shows that Xei is a closed ideal of X for all i=1,2,⋯,k. Therefore Xe=⋂i=1kXei is a closed ideal of X.
Proposition 3.14.
Given a k-folded N-structure (f̃,X) over a BCH-algebra X, if the set Xei:={x∈X|(ℓi∘f̃)(x)≤(ℓi∘f̃)(e)} is a closed ideal of X for all e∈X and i=1,2,…,k, then (f̃,X) satisfies the following argument:
(∀x,y,z∈X)(Max{f̃(y∗z),f̃(z)}≾f̃(x)⇒f̃(y)≾f̃(x)), (20)
that is,
(ℓi∘f̃)(x)≥max{(ℓi∘f̃)(y∗z),(ℓi∘f̃)(z)} implies (ℓi∘f̃)(x)≥(ℓi∘f̃)(y) for all x,y,z∈X and i=1,2,…,k.
Proof.
Let x,y,z∈X be such that Max{f̃(y∗z),f̃(z)}≾f̃(x), that is,
(ℓi∘f̃)(x)≥max{(ℓi∘f̃)(y∗z),(ℓi∘f̃)(z)}
for
i=1,2,…,k. Then
y∗z∈Xxi and
z∈Xxi. Since
Xxi is a closed ideal of
X, we have
y∈Xxi, and so
(ℓi∘f̃)(x)≥(ℓi∘f̃)(y) for
i=1,2,…,k. Hence the argument
(20) is valid.
Theorem 3.15.
If a k-folded N-structure (f̃,X) over a BCH-algebra X satisfies the conditions (13) and (20), then the set Xe in (19) is a closed ideal of X for all e∈X.
Proof.
If x∈Xe, then f̃(0∗x)≾f̃(x)≾f̃(e) by (13) and so 0∗x∈Xe. Let x,y∈X be such that x∗y∈Xe and y∈Xe. Then f̃(x∗y)≾f̃(e) and f̃(y)≾f̃(e), which imply Max{f̃(x∗y),f̃(y)}≾f̃(e). Using (20), we get f̃(x)≾f̃(e), that is, x∈Xe. Therefore Xe is a closed ideal of X for all e∈X.
4. k-FOLDED N-FILTERS
Definition 4.1.
Let X be a BCH-algebra. A k-folded N-structure (f̃,X) over X is called a k-folded N-filter of X if it satisfies
(∀x,y∈X)x∗y=0⇒f̃(x)≾f̃(y),(21)
(∀x,y∈X)Min{f̃(x),f̃(y)}≾Min{f̃(x∗(x∗y)),f̃(y∗(y∗x))}, (22)
that is,
x∗y=0⇒(ℓi∘f̃)(x)≤(ℓi∘f̃)(y)
and
min{(ℓi∘f~)(x),(ℓi∘f~)(y)}≤min{(ℓi∘f~)(x∗(x∗y)),(ℓi∘f~)(y∗(y∗x))}
for all
x,y∈X and
i=1,2,⋯,k.
A k-folded N-filter of X is said to be closed if f̃(x)≾f̃(0∗x) for all x∈X, that is, (ℓi∘f̃)(x)≤(ℓi∘f̃)(0∗x) for all x∈X and i=1,2,⋯,k.
Example 4.2.
Consider the BCH-algebra X={0,1,2,3,4} in Example 3.3.
(1) Let (f̃,X) be a 5-folded N-structure over X given by
f~:X→[−1,0]5,x↦−0.86,−0.77,−0.55,−0.49,−0.33ifx=4−0.13,−0.22,−0.28,−0.36,−0.05otherwise.
Then (f̃,X) is a 5-folded N-filter of X.
(2) Let (f̃,X) be a 2-folded N-structure over X given by
f̃:X→[−1,0]2,x↦−0.7,−0.6ifx∈{0,1,2,3}−0.2,−0.3ifx=4
Then (f̃,X) is a 2-folded N-filter of X.
Theorem 4.3.
A k-folded N-structure (f̃,X) over a BCH-algebra X is a (closed) k-folded N-filter of X if and only if the k-folded level set L(f̃;t̃) of (f̃,X) is a (closed) filter of X for all t̃∈[−1,0]k with L(f̃;t̃)≠∅.
Theorem 4.4.
Let (f̃,X) be a k-folded N-structure over a BCH-algebra X given as follows:
f̃:X→[−1,0]k,x↦t̃ifx∈B(x0),x0∈Med(X),s̃otherwise
where B(x0) is the branch of X,
i.e.,
B(x0)={x∈X|x0∗x=0},
and t̃=(t1,t2,⋯,tk)⋨(s1,s2,⋯,sk)=s̃ in [−1,0]k.
Then (f̃,X) is a k-
folded N-
filter of X.
Proof.
Using Theorem 4.3, it is sufficient to show that B(x0) is a filter of X for x0∈Med(X). Let x,y∈X be such that x∈B(x0) and x∗y=0. Then 0∗x0=0∗x=0∗y by (5). It follows that 0∗(0∗y)=0∗(0∗x0)=x0. Hence x0∗y=(0∗(0∗y))∗y=0, and so y∈B(x0). Let x,y∈B(x0). Then x0∗x=0 and x0∗y=0, which imply from (5) that 0∗x0=0∗x and 0∗x0=0∗y. It follows from (I), (III), (1) and (3) that
x0∗(y∗(y∗x))=(0∗(0∗x0))∗(y∗(y∗x)) =(0∗(y∗(y∗x)))∗(0∗x0) =((0∗y)∗((0∗y)∗(0∗x)))∗(0∗x0) =((0∗x0)∗((0∗x0)∗(0∗x0)))∗(0∗x0) =0.
Similarly, we have x0∗(x∗(x∗y))=0. Therefore x∗(x∗y)∈B(x0) and y∗(y∗x)∈B(x0). This completes the proof.
Proposition 4.5.
Every closed k-folded N-filter (f̃,X) of a BCH-algebra X satisfies
(∀x,y∈X)(f̃(x∗y)≾f̃(y∗x)), (23)
that is,
(ℓi∘f̃)(x∗y)≤(ℓi∘f̃)(y∗x) for all x,y∈X and i=1,2,⋯,k.
Proof.
Using (I), (III) and (3), we have
(0∗(x∗y))∗(y∗x)=((0∗x)∗(0∗y))∗(y∗x) =((0∗x)∗(y∗x))∗(0∗y) =((0∗(y∗x))∗x)∗(0∗y) =(((0∗y)∗(0∗x))∗x)∗(0∗y) =(((0∗y)∗(0∗x))∗(0∗y))∗x =(((0∗y)∗(0∗y))∗(0∗x))∗x =(0∗(0∗x))∗x =0
for all
x,y∈X. It follows from the closedness of
(f̃,X) and
(21) that
f̃(x∗y)≾f̃((0∗(x∗y)))≾f̃(y∗x) for all
x,y∈X.
Corollary 4.6.
Every closed k-folded N-filter (f̃,X) of a BCH-algebra X satisfies
(∀x,y∈X)(f̃(x)≾f̃(y∗(y∗x))), (24)
that is,
(ℓi∘f̃)(x)≤(ℓi∘f̃)(y∗(y∗x)) for all
x,y∈X and
i=1,2,⋯,k.
Proof.
Using the closedness of (f̃,X), (I), (III) and (23), we get
f̃(x)≾f̃(0∗x)=f̃((y∗y)∗x)=f̃((y∗x)∗y)≾f̃(y∗(y∗x))
for all
x,y∈X.
Corollary 4.7.
Every closed k-folded N-filter (f̃,X) of a BCH-algebra X satisfies
(∀x,y∈X)(x∗y=0⇒f̃(y)≾f̃(x)), (25)
that is,
(ℓi∘f̃)(y)≤(ℓi∘f̃)(x) for all
x,y∈X with
x∗y=0 and
i=1,2,⋯,k.
Proof.
Let x,y∈X be such that x∗y=0. Then 0∗x=0∗y by (5). It follows from (1) and (23) that
f̃(y)=f̃(y∗0)≾f̃(0∗y)=f̃(0∗x)≾f̃(x∗0)=f̃(x).
This completes the proof.
Theorem 4.8.
Given a k-folded N-structure (f̃,X) over a BCH-algebra X, the following are equivalent.
(f̃,X) is a k-folded N-closed ideal of X satisfying the condition (23).
The k-folded level set L(f̃;t̃) of (f̃,X) is a closed ideal of X for all t̃∈[−1,0]k with L(f̃;t̃)≠∅ which satisfies the following condition:
(∀x,y∈X)y∗x∈L(f̃;t̃)⇒x∗y∈L(f̃;t̃).(26)
Proof.
Recall that (f̃,X) is a k-folded N-closed ideal of X if and only if the k-folded level set L(f̃;t̃) of (f̃,X) is a closed ideal of X for all t̃∈[−1,0]k with L(f̃;t̃)≠∅ (see Theorem 3.11). Assume that the condition (23) is valid and let y∗x∈L(f̃;t̃) for all x,y∈X. Then f̃(x∗y)≾f̃(y∗x)≾t̃, and so x∗y∈L(f̃;t̃). Now suppose the condition (26) is true and let a,b∈X be such that f̃(a∗b)≾̸f̃(b∗a). Then f̃(b∗a)≾s̃⋨f̃(a∗b) for some s̃=(s1,s2,⋯,sk)∈(−1,0]k. Hence b∗a∈L(f̃;s̃) but a∗b∉L(f̃;s̃) which is a contradiction. Therefore the condition (23) is valid.
Theorem 4.9.
If every k-folded N-closed ideal (f̃,X) of a BCH-algebra X satisfies the condition (23), then it is a closed k-folded N-filter of X.
Proof.
Let (f̃,X) be a k-folded N-closed ideal of X satisfying the condition (23). Then L(f̃;t̃) is a closed ideal of X which satisfies (26) (see Theorem 4.8). Let x,y∈X be such that x∈L(f̃;t̃) and x∗y=0. Then x∗y=0∈L(f̃;t̃) which implies from (26) that y∗x∈L(f̃;t̃). Hence 0∗x∈L(f̃;t̃) and y∈L(f̃;t̃) by (6) and (7). It is clear that if x,y∈L(f̃;t̃), then x∗(x∗y)∈L(f̃;t̃) and y∗(y∗x)∈L(f̃;t̃) since L(f̃;t̃) is a subalgebra of X. This shows that L(f̃;t̃) is a closed filter of X. Therefore (f̃,X) is a closed k-folded N-filter of X by Theorem 4.3.
, Kul Hur2, Young Bae Jun3