The notions of 𝒩-subalgebras and 𝒩-closed ideals in BCH-algebras are introduced, and the relation between 𝒩-subalgebras
and 𝒩-closed ideals is considered.
Characterizations of 𝒩-subalgebras and 𝒩-closed ideals are provided. Using special subsets, 𝒩-subalgebras and
𝒩-closed ideals are constructed.
A condition for an 𝒩-subalgebra to be an
𝒩-closed ideal is discussed. Given an 𝒩-structure,
the greatest 𝒩-closed ideal which is contained in the 𝒩-structure is established.
1. Introduction
In [1, 2], Hu and Li introduced the notion of BCH-algebras which are a generalization of BCK/BCI-algebras. Ahmad [3] classified BCH-algebras, and decompositions of BCH-algebras are considered by Dudek and Thomys [4]. Jun et al. [5] discussed the notion of 𝒩-structures and applied it to BCK/BCI-algebras. In [6], Chaudhry et al. studied closed ideals and filters in BCH-algebras. In this paper, we apply the 𝒩-structures to the closed ideal theory in BCH-algebras. We introduced the notion of 𝒩-subalgebras and 𝒩-closed ideals in BCH-algebras, and investigate the relation between 𝒩-subalgebras and 𝒩-closed ideals. We provide characterizations of 𝒩-subalgebras and 𝒩-closed ideals. Using special subsets, we construct 𝒩-subalgebras and 𝒩-closed ideals. We provide a condition for an 𝒩-subalgebra to be an 𝒩-closed ideal. Given an 𝒩-structure (X,μ), we make the greatest 𝒩-closed ideal which is contained in (X,μ).
2. Preliminaries
By a BCH-algebra we mean an algebra (X,*,0) of type (2,0) satisfying the following axioms:
x*x=0,
x*y=0 and y*x=0 imply x=y,
(x*y)*z=(x*z)*y
for all x,y,z∈X.
In a BCH-algebra X, the following conditions are valid (see [1, 4]).
x*0=x,
x*0=0 implies x=0,
0*(x*y)=(0*x)*(0*y),
0*(0*(0*x))=0*x.
A nonempty subset S of a BCH-algebra X is called a subalgerba of X if x*y∈S for all x,y∈S. A nonempty subset A of a BCH-algebra X is called a closed ideal of X (see [7]) if it satisfies:
(forallx∈X)(x∈A⇒0*x∈A),
(forally∈X)(forallx∈A)(y*x∈A⇒y∈A).
Note that every closed ideal is a subalgebra, but the converse is not true (see [7]). Since every closed ideal is a subalgebra, we know that any closed ideal contains the element 0. Denote by 𝒮(X) and ℐ(X) the set of all subalgebras and closed ideals of X, respectively.
For any family {ai∣i∈Λ} of real numbers, we define
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,μ) of X and an 𝒩-function μ on X. In what follows, let X denote a BCH-algebra and μ an 𝒩-function on X unless otherwise specified.
For any 𝒩-structure (X,μ) and t∈[-1,0], the set
C(μ;t):={x∈X∣μ(x)≤t}
is called a closed (μ,t)-cut of (X,μ).
Using the similar method to the transfer principle in fuzzy theory (see [8, 9]), we can consider transfer principle in 𝒩-structures. Let A be a subset of X and satisfy the following property 𝒫 expressed by a first-order formula:
𝒫:t1(x,…,y)∈A,…,tn(x,…,y)∈At(x,…,y)∈A,
where t1(x,…,y),…,tn(x,…,y) and t(x,…,y) are terms of X constructed by variables x,…,y. We note that the subset A satisfies the property 𝒫 if, for all elements a,…,b∈X,t(a,…,b)∈A whenever t1(a,…,b),…,cn(a,…,b)∈A. For the subset A we define an 𝒩-structure (X,μA) which satisfies the following property
𝒫̅:μA(t(x,…,y))≤∨{μA(t1(x,…,y)),…μA(tn(x,…,y))}.
We establish a statement without proof, and we call it 𝒩-transfer principle in 𝒩-structures.
Theorem 3.1.
(𝒩-transfer principle) An 𝒩-structure (X,μ) satisfies the property 𝒫̅ if and only if for all α∈[-1,0],C(μ;α)≠∅⇒C(μ;α)satisfiestheproperty𝒫.
Definition 3.2.
By an 𝒩-subalgebra of X we mean an 𝒩-structure (X,μ) in which μ satisfies:
(∀x,y∈X)(μ(x*y)≤∨{μ(x),μ(y)}).
Theorem 3.3.
For an 𝒩-structure (X,μ), the following are equivalent:
(X,μ) is an 𝒩-subalgerba of X;
(forallt∈[-1,0])(C(μ;t)∈𝒮(X)∪{∅}).
Proof.
It follows from the 𝒩-transfer principle.
Definition 3.4.
By an 𝒩-closed ideal of X we mean an 𝒩-structure (X,μ) in which μ satisfies:
(∀x,y∈X)(μ(0*x)≤μ(x)≤∨{μ(x*y),μ(y)}).
It is clear that if (X,μ) is an 𝒩-closed ideal or an 𝒩-subalgebra, then μ(0)≤μ(x) for all x∈X.
Theorem 3.5.
Every 𝒩-closed ideal is an 𝒩-subalgebra.
Proof.
Let (X,μ) be an 𝒩-closed ideal of X. For any x,y∈X, we have
μ(x*y)≤∨{μ((x*y)*x),μ(x)}=∨{μ((x*x)*y),μ(x)}=∨{μ(0*y),μ(x)}≤∨{μ(x),μ(y)}.
Hence (X,μ) is an 𝒩-subalgebra of X.
The converse of Theorem 3.5 may not be true as seen in the following example.
Example 3.6.
Consider a BCH-algebra X={0,1,2,3,4} with the Cayley table which is given in Table 1 (see [7]). Let (X,μ) be an 𝒩-structure in which μ is given by
μ=(01234-0.8-0.3-0.3-0.3-0.8).
It is easy to check that (X,μ) is an 𝒩-subalgebra of Xbut it is not an 𝒩-closed ideal of X since μ(3)=-0.3>-0.8=∨{μ(3*4),μ(4)}.
Cayley table.
*
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
In order to discuss the converse of Theorem 3.5 we need to strengthen some conditions. We first consider the following lemma.
Lemma 3.7.
Every 𝒩-subalgebra (X,μ) of X satisfies the following inequality:
(∀x∈X)(μ(x)≥μ(0*x)).
Proof.
For any x∈X, we get
μ(0*x)≤∨{μ(0),μ(x)}=∨{μ(x*x),μ(x)}=∨{∨{μ(x),μ(x)},μ(x)}=μ(x),
which is the desired result.
Theorem 3.8.
If an 𝒩-subalgerba (X,μ) satisfies
(∀x,y∈X)(μ(y)≤∨{μ(y*x),μ(x)}),
then (X,μ) is an 𝒩-closed ideal of X.
Proof.
It is straightforward by Lemma 3.7.
Proposition 3.9.
Let (X,μ) be an 𝒩-closed ideal of X that satisfies the following inequality
(∀x∈X)(μ(x)≤μ(0*x)).
Then (X,μ) satisfies the inequality
(∀x,y∈X)(μ(y*x)≤μ(x*y)).
Proof.
Using (3.12), (3.6), (a3), (H1), and (H3), we have
μ(y*x)≤μ(0*(y*x))≤∨{μ((0*(y*x))*(x*y)),μ(x*y)}=∨{μ(((0*y)*(0*x))*(x*y)),μ(x*y)}=∨{μ(((0*y)*(x*y))*(0*x)),μ(x*y)}=∨{μ(((0*(x*y))*y)*(0*x)),μ(x*y)}=∨{μ((((0*x)*(0*y))*(0*x))*y),μ(x*y)}=∨{μ((0*(0*y))*y),μ(x*y)}=∨{μ(0),μ(x*y)}=μ(x*y)
for all x,y∈X.
Using the 𝒩-transfer principle, we have a characterization of an 𝒩-closed ideal.
Theorem 3.10.
For an 𝒩-structure (X,μ), the following are equivalent:
(X,μ) is an 𝒩-closed ideal of X.
(forallt∈[-1,0])(C(μ;t)∈ℐ(X)∪{∅}).
Consider two subsets of X as follows:
D1:={x∈X∣0*x=0},D2:={x∈X∣0*(0*x)=x}.
Since D1 and D2 are a closed ideal and a subalgebra, respectively, the following theorems are direct results of the 𝒩-transfer principle.
Theorem 3.11.
Let (X,μ) be an 𝒩-structure in which μ is given by
μ(x)={αifx∈D1,βotherwise
for all x∈X where α<β. Then (X,μ) is an 𝒩-closed ideal of X.
Theorem 3.12.
Let (X,μ) be an 𝒩-structure in which μ is given by
μ(x)={αifx∈D2,βotherwise
for all x∈X where α<β. Then (X,μ) is an 𝒩-subalgebra of X.
We provide a condition for an 𝒩-subalgebra to be an 𝒩-closed ideal.
Theorem 3.13.
Let (X,μ) be an 𝒩-subalgebra of X in which μ satisfies
(∀x,y∈X)(μ(y*x)≥μ(x*y)).
Then (X,μ) is an 𝒩-closed ideal of X.
Proof.
Taking x=0 in (3.18) induces μ(0*y)≤μ(y*0)=μ(y) for all y∈X. Using (a1), (3.18), (H1), (H3), and (3.5), we have
μ(y)=μ(y*0)≤μ(0*y)=μ((x*x)*y)=μ((x*y)*x)≤∨{μ(x*y),μ(x)}≤∨{μ(y*x),μ(x)}
for all x,y∈X. Therefore (X,μ) is an 𝒩-closed ideal of X.
For any 𝒩-structure (X,μ) and any element w∈X, we consider the set
Xw:={x∈X∣μ(x)≤μ(w)}.
Then Xw is nonempty subset of X.
Theorem 3.14.
If an 𝒩-structure (X,μ) is an 𝒩-closed ideal of X, then Xw is a closed ideal of X for all w∈X.
Proof.
If x∈Xw, then μ(x)≤μ(w) which implies from (3.6) that μ(0*x)≤μ(x)≤μ(w). Thus 0*x∈Xw. Let x,y∈X be such that y∈Xw and x*y∈Xw. Then μ(y)≤μ(w) and μ(x*y)≤μ(w). Using (3.6), we have
μ(x)≤∨{μ(x*y),μ(y)}≤μ(w),i.e.,x∈Xw.
Therefore Xw is a closed ideal of X.
Proposition 3.15.
Let (X,μ) be an 𝒩-structure such that Xw is a closed ideal of X for all w∈X. Then (X,μ) satisfies the following assertion:
μ(x)≥∨{μ(y*z),μ(z)}⇒μ(x)≥μ(y)
for all x,y,z∈X.
Proof.
Let x,y,z∈X be such that μ(x)≥∨{μ(y*z),μ(z)}. Then y*z∈Xx and z∈Xx. Since Xx is a closed ideal of X, it follows that y∈Xx so that μ(y)≤μ(x). This completes the proof.
Theorem 3.16.
If an 𝒩-structure (X,μ) satisfies (3.22) and μ(0*x)≤μ(x) for all x∈X, then Xw is a closed ideal of X for all w∈X.
Proof.
For each w∈X, let x,y∈X be such that x*y∈Xw and y∈Xw. Then μ(x*y)≤μ(w) and μ(y)≤μ(w), which imply that ∨{μ(x*y),μ(y)}≤μ(w). It follows from (3.22) that μ(x)≤μ(w) so that x∈Xw. If x∈Xw, then μ(0*x)≤μ(x)≤μ(w) by assumption. Hence 0*x∈Xw. Therefore Xw is a closed ideal of X.
Theorem 3.17.
Given an 𝒩-structure (X,μ), let (X,μ*) be an 𝒩-structure in which μ* is defined by
μ*(x)=⋀{t∈[-1,0]∣x∈〈C(μ;t)〉}
for all x∈X. Then (X,μ*) is the greatest 𝒩-closed ideal of X such that (X,μ*)⊆(X,μ), where 〈C(μ;t)〉 is a closed ideal of X generated by C(μ;t).
Proof.
For any s∈Im(μ*), let sn=s+(1/n) for any n∈ℕ. Let x∈C(μ*;s). Then μ*(x)≤s, and so
⋀{t∈[-1,0]∣x∈〈C(μ;t)〉}≤s<s+1n=sn
for all n∈ℕ. Hence there exists t*∈{t∈[-1,0]∣x∈〈C(μ;t)〉} such that t*<sn. Thus C(μ;t*)⊆C(μ;sn), and so x∈〈C(μ;t*)〉⊆〈C(μ;sn)〉 for all n∈ℕ. Consequently x∈⋂n∈ℕ〈C(μ;sn)〉,. On the other hand, if x∈⋂n∈ℕ〈C(μ;sn)〉,, then sn∈{t∈[-1,0]∣x∈〈C(μ;t)〉} for any n∈ℕ. Therefore
μ*(x)=⋀{t∈[-1,0]∣x∈〈C(μ;t)〉}≤sn=s+1n
for all n∈ℕ. Since n is arbitrary, it follows that μ*(x)≤s so that x∈C(μ*;s). Thus C(μ*;s)=⋂n∈ℕ〈C(μ;sn)〉, which is a closed ideal of X. Using Theorem 3.10, we conclude that (X,μ*) is an 𝒩-closed ideal of X. For any x∈X, let
s∈{t∈[-1,0]∣x∈C(μ;t)}.
Then x∈C(μ;s) and thus x∈〈C(μ;s)〉. It follows that
s∈{t∈[-1,0]∣x∈〈C(μ;t)〉}
so that {t∈[-1,0]∣x∈C(μ;t)}⊆{t∈[-1,0]∣x∈〈C(μ;t)〉}. Hence
μ(x)=⋀{t∈[-1,0]∣x∈C(μ;t)}≥⋀{t∈[-1,0]∣x∈〈C(μ;t)〉}=μ*(x),
and so (X,μ*)⊆(X,μ). Finally, let (X,ν) be an 𝒩-closed ideal of X such that (X,ν)⊆(X,μ). Let x∈X. If μ*(x)=0, then clearly ν(x)≤μ*(x). Assume that μ*(x)=s≠0. Then x∈C(μ*;s)=⋂n∈ℕ〈C(μ;sn)〉, and so x∈〈C(μ;sn)〉 for all n∈ℕ. It follows that ν(x)≤μ(x)≤sn=s+(1/n) for all n∈ℕ so that ν(x)≤s=μ*(x) since n is arbitrary. This shows that (X,ν)⊆(X,μ*). This completes the proof.
Acknowledgments
The authors wish to thank the anonymous reviewers for their valuable suggestions. The first author was supported by the fund of sabbatical year program (2009), Gyeongsang National University.
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