# 𝒩-Structures Applied to Closed Ideals in BCH-Algebras

Authors: Young Bae Jun; Mehmet Ali Öztürk; Eun Hwan Roh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

The notions of𝒩-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.


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## Body

## 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 aBCH-algebra we mean an algebra (X,*,0) of type (2,0) satisfying the following axioms:(H1)
x*x=0,(H2)
x*y=0 and y*x=0 imply x=y,(H3)
(x*y)*z=(x*z)*yfor allx,y,z∈X.In a BCH-algebraX, the following conditions are valid (see [1, 4]).(a1)
x*0=x,(a2)
x*0=0 implies x=0,(a3)
0*(x*y)=(0*x)*(0*y),(a4)
0*(0*(0*x))=0*x.A nonempty subsetS 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:(1)
(forallx∈X)(x∈A⇒0*x∈A),(2)
(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(2.1)∨{ai∣i∈Λ}:={max{ai∣i∈Λ}ifΛisfinite,sup{ai∣i∈Λ}otherwise,(2.2)⋀{ai∣i∈Λ}:={min{ai∣i∈Λ}ifΛisfinite,inf{ai∣i∈Λ}otherwise.


## 3.𝒩-Closed Ideals of BCH-Algebras

Denote byℱ(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 any𝒩-structure (X,μ) and t∈[-1,0], the set(3.1)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:(3.2)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(3.3)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],(3.4)C(μ;α)≠∅⇒C(μ;α)satisfiestheproperty𝒫.Definition 3.2.
By an𝒩-subalgebra of X we mean an 𝒩-structure (X,μ) in which μ satisfies:
(3.5)(∀x,y∈X)(μ(x*y)≤∨{μ(x),μ(y)}).Theorem 3.3.
For an𝒩-structure (X,μ), the following are equivalent:  (1)
(X,μ) is an 𝒩-subalgerba of X;(2)
(forallt∈[-1,0])(C(μ;t)∈𝒮(X)∪{∅}).Proof.
It follows from the𝒩-transfer principle.Definition 3.4.
By an𝒩-closed ideal of X we mean an 𝒩-structure (X,μ) in which μ satisfies:
(3.6)(∀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.
Every𝒩-closed ideal is an 𝒩-subalgebra.Proof.
Let(X,μ) be an 𝒩-closed ideal of X. For any x,y∈X, we have
(3.7)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 Theorem3.5 may not be true as seen in the following example.Example 3.6.
Consider a BCH-algebraX={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
(3.8)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)}.Table 1
Cayley table.
*01234000004110014222004333304444440In order to discuss the converse of Theorem3.5 we need to strengthen some conditions. We first consider the following lemma.Lemma 3.7.
Every𝒩-subalgebra (X,μ) of X satisfies the following inequality:
(3.9)(∀x∈X)(μ(x)≥μ(0*x)).Proof.
For anyx∈X, we get
(3.10)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 an𝒩-subalgerba (X,μ) satisfies
(3.11)(∀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 Lemma3.7.Proposition 3.9.
Let(X,μ) be an 𝒩-closed ideal of X that satisfies the following inequality
(3.12)(∀x∈X)(μ(x)≤μ(0*x)).
Then (X,μ) satisfies the inequality
(3.13)(∀x,y∈X)(μ(y*x)≤μ(x*y)).Proof.
Using (3.12), (3.6), (a3), (H1), and (H3), we have
(3.14)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 the𝒩-transfer principle, we have a characterization of an 𝒩-closed ideal.Theorem 3.10.
For an𝒩-structure (X,μ), the following are equivalent:  (1)
(X,μ) is an 𝒩-closed ideal of X.(2)
(forallt∈[-1,0])(C(μ;t)∈ℐ(X)∪{∅}).Consider two subsets ofX as follows:(3.15)D1:={x∈X∣0*x=0},D2:={x∈X∣0*(0*x)=x}.SinceD1 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
(3.16)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
(3.17)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 an𝒩-subalgebra to be an 𝒩-closed ideal.Theorem 3.13.
Let(X,μ) be an 𝒩-subalgebra of X in which μ satisfies
(3.18)(∀x,y∈X)(μ(y*x)≥μ(x*y)).
Then (X,μ) is an 𝒩-closed ideal of X.Proof.
Takingx=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
(3.19)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 any𝒩-structure (X,μ) and any element w∈X, we consider the set(3.20)Xw:={x∈X∣μ(x)≤μ(w)}.
Then Xw is nonempty subset of X.Theorem 3.14.
If an𝒩-structure (X,μ) is an 𝒩-closed ideal of X, then Xw is a closed ideal of X for all w∈X.Proof.
Ifx∈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
(3.21)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:
(3.22)μ(x)≥∨{μ(y*z),μ(z)}⇒μ(x)≥μ(y)
for all x,y,z∈X.Proof.
Letx,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 an𝒩-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 eachw∈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 an𝒩-structure (X,μ), let (X,μ*) be an 𝒩-structure in which μ* is defined by
(3.23)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 anys∈Im(μ*), let sn=s+(1/n) for any n∈ℕ. Let x∈C(μ*;s). Then μ*(x)≤s, and so
(3.24)⋀{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
(3.25)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
(3.26)s∈{t∈[-1,0]∣x∈C(μ;t)}.
Then x∈C(μ;s) and thus x∈〈C(μ;s)〉. It follows that
(3.27)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
(3.28)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.



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