# 𝒩-Subalgebras in BCK/BCI-Algebras Based on Point

Authors: Young Bae Jun; Min Su Kang; Chul Hwan Park
Journal: International Journal of Mathematics and Mathematical
                    Sciences
 (2010)

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## Abstract

The notion of𝒩-subalgebras of several types is introduced, and
related properties are investigated. Conditions for an 𝒩-structure to be an 𝒩-subalgebra of type (q,∈⋁q) are provided, and a characterization of an
𝒩-subalgebra of type (∈,∈⋁q) is considered.


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## Body

## 1. Introduction

A (crisp) setA in a universe X can be defined in the form of its characteristic
function μA:X→{0,1} yielding the value 1 for elements belonging to the set
A and the value 0 for elements excluded from the set
A. So far, most of the generalization of the crisp set
have been conducted on the unit interval [0,1] and they are consistent with the asymmetry
observation. In other words, the generalization of the crisp set to fuzzy sets
relied on spreading positive information that fit the crisp point
{1} into the interval [0,1]. Because no negative meaning of information is
suggested, we now feel a need to deal with negative information. To do so, we also
feel a need to supply mathematical tool. To attain such object, Jun et al. [1] introduced a new function which is called
negative-valued function, and constructed 𝒩-structures. They applied 𝒩-structures to BCK/BCI-algebras, and discussed
𝒩-subalgebras and 𝒩-ideals in BCK/BCI-algebras. Jun et al. [2] considered closed ideals in BCH-algebras
based on 𝒩-structures. To obtain more general form of an
𝒩-subalgebra in BCK/BCI-algebras, we define the
notions of 𝒩-subalgebras of types (∈,∈),(∈,q),(∈,∈⋁q),(q,∈),(q,q), and (q,∈⋁q), and investigate related properties. We provide a
characterization of an 𝒩-subalgebra of type (∈,∈⋁q). We give conditions for an 𝒩-structure to be an 𝒩-subalgebra of type (q,∈⋁q).


## 2. Preliminaries

LetK(τ) be the class of all algebras with type
τ=(2,0). By a BCI-algebra we mean a
system X:=(X,*,θ)∈K(τ) in which the following axioms hold: (i)
((x*y)*(x*z))*(z*y)=θ,(ii)
(x*(x*y))*y=θ,(iii)
x*x=θ,(iv)
x*y=y*x=θ⇒x=yfor allx,y,z∈X. If a BCI-algebra X satisfies θ*x=θ for all x∈X, then we say that X is a BCK-algebra. We can define a
partial ordering ⪯ by (2.1)(∀x,y∈X)(x⪯y⇔x*y=θ). In a BCK/BCI-algebra X, the following hold: (a1)
(forallx∈X)(x*θ=x),(a2)
(forallx,y,z∈X)((x*y)*z=(x*z)*y)for allx,y,z∈X.A nonempty subsetS of a BCK/BCI-algebra X is called a subalgebra of
X if x*y∈S for all x,y∈S. For our convenience, the empty set
∅ is regarded as a subalgebra of
X.We refer the reader to the books [3, 4] for further information regarding
BCK/BCI-algebras.For any family{ai∣i∈Λ} of real numbers, we define (2.2)⋁{ai∣i∈Λ}:={max{ai∣i∈Λ},ifΛisfinite,sup{ai∣i∈Λ},otherwise,⋀{ai∣i∈Λ}:={min{ai∣i∈Λ},ifΛisfinite,inf{ai∣i∈Λ},otherwise.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,f) of X and an 𝒩-function f on X. In what follows, let X denote a BCK/BCI-algebra and
f an 𝒩-function on X unless otherwise specified.Definition 2.1 (see [1]).
By asubalgebra of X based on 𝒩-function f (briefly, 𝒩-subalgebra of
X), we mean an 𝒩-structure (X,f) in which f satisfies the following assertion:
(2.3)(∀x,y∈X)(f(x*y)≤⋁{f(x),f(y)}).
For any𝒩-structure (X,f) and t∈[-1,0), the set (2.4)C(f;t):={x∈X∣f(x)≤t} is called a closedt-support of
(X,f), and the set (2.5)O(f;t):={x∈X∣f(x)<t} is called an opent-support of
(X,f).
Using the similar method to the transfer principle in fuzzy theory (see [5, 6]), Jun et al. [2] considered
transfer principle in 𝒩-structures as follows.Theorem 2.2 (𝒩-transfer principle [2]).
An𝒩-structure (X,f) satisfies the property
𝒫¯ if and only if for all
α∈[-1,0],(2.6)C(f;α)≠∅⇒C(f;α)satisfiestheproperty𝒫.Lemma 2.3 (see [1]).
An𝒩-structure (X,f) is an 𝒩-subalgebra of X if and only if every open
t-support of (X,f) is a subalgebra of X for all t∈[-1,0).


## 3. Generalized𝒩-Subalgebras

Let(X,f) be an 𝒩-structure in which f is given by (3.1)f(y)={0,ify≠x,α,ify=x, where α∈[-1,0). In this case, f is denoted by xα and we call (X,xα) a point𝒩-structure. For any
𝒩-structure (X,g), we say that a point 𝒩-structure (X,xα) is an 𝒩∈-subset (resp.,
𝒩q-subset) of
(X,g) if g(x)≤α (resp., g(x)+α+1<0). If a point 𝒩-structure (X,xα) is an 𝒩∈-subset of (X,g) or an 𝒩q-subset of (X,g), we say (X,xα) is an 𝒩∈⋁q-subset of
(X,g).Theorem 3.1.
For any𝒩-structure (X,f), the following are equivalent: (1)
(X,f) is an 𝒩-subalgebra of
X;(2)
for anyx,y∈X and t1,t2∈[-1,0), if two point
𝒩-structures
(X,xt1) and (X,yt2) are 𝒩∈-subsets of
(X,f), then the point
𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset of (X,f).Proof.
(1)⇒ (2). Let x,y∈X and t1,t2∈[-1,0) be such that (X,xt1) and (X,yt2) are 𝒩∈-subsets of (X,f). Then f(x)≤t1 and f(y)≤t2. It follows from (2.3) that (3.2)f(x*y)≤⋁{f(x),f(y)}≤⋁{t1,t2} so that the point 𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset of (X,f).
(2)⇒ (1). For any x,y∈X, note that (X,xf(x)) and (X,yf(y)) are point 𝒩-structures which are 𝒩∈-subsets of (X,f). Using (2), we know that the point
𝒩-structure (X,(x*y)⋁{f(x),f(y)}) is an 𝒩∈-subset of (X,f). Thus f(x*y)≤⋁{f(x),f(y)}, and so (X,f) is an 𝒩-subalgebra of X.Definition 3.2.
An𝒩-structure (X,f) is called an 𝒩-subalgebra of type (i)
(∈,∈) (resp., (∈,q) and (∈,∈⋁q)) if whenever two point
𝒩-structures
(X,xt1) and (X,yt2) are 𝒩∈-subsets of
(X,f) then the point
𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset (resp.,
𝒩q-subset and
𝒩∈⋁q-subset) of
(X,f);(ii)
(q,∈) (resp., (q,q) and (q,∈⋁q)) if whenever two point
𝒩-structures
(X,xt1) and (X,yt2) are 𝒩q-subsets of
(X,f) then the point
𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset (resp.,
𝒩q-subset and
𝒩∈⋁q-subset) of
(X,f).
Note that every𝒩-subalgebra of type (∈,∈) is an 𝒩-subalgebra of X (see Theorem 3.1). Note also that every 𝒩-subalgebra of types (∈,∈) and (∈,q) is an 𝒩-subalgebra of type (∈,∈⋁q).Example 3.3.
LetX={θ,a,b,c,d} be a set with a *-operation table which is given by Table 1. Then (X;*,θ) is a BCK-algebra (see [4]). Consider an 𝒩-structure (X,f) in which f is defined by (3.3)f=(θabcd-0.9-0.8-0.5-0.7-0.3). It is routine to verify that
(X,f) is an 𝒩-subalgebra of types (∈,∈) and (∈,∈⋁q). But it is not of type (q,∈⋁q).Table 1
*-operation.
*
θ
a
b
c
d
θ
θ
θ
θ
θ
θ
a
a
θ
θ
θ
θ
b
b
a
θ
a
θ
c
c
a
a
θ
θ
d
d
b
a
b
θExample 3.4.
LetX={θ,a,b,c} be a BCI-algebra with a
*-operation table which is given by Table 2. Consider an 𝒩-structure (X,f) in which f is defined by (3.4)f=(θabc-0.5-0.8-0.3-0.3). Then (X,f) is an 𝒩-subalgebra of type (∈,∈⋁q). But(1)
(X,f) is not of type
(∈,∈) since two point
𝒩-structures
(X,a-0.7) and (X,a-0.76) are 𝒩∈-subsets of
(X,f), but the point
𝒩-structure (3.5)(X,(a*a)⋁{-0.7,-0.76})=(X,θ-0.7) is not an 𝒩∈-subset of (X,f) since f(θ)=-0.5≰-0.7;(2)
(X,f) is not of type
(q,∈⋁q) since two point
𝒩-structures
(X,a-0.42) and (X,b-0.88) are 𝒩q-subsets of
(X,f), but the point
𝒩-structure (3.6)(X,(a*b)⋁{-0.42,-0.88})=(X,c-0.42) is not an 𝒩∈⋁q-subset of (X,f);(3)
(X,f)is not of type
(∈⋁q,∈⋁q) since two point
𝒩-structures
(X,a-0.6) and (X,c-0.82) are 𝒩∈⋁q-subsets of
(X,f), but the point
𝒩-structure (3.7)(X,(a*c)⋁{-0.6,-0.82})=(X,b-0.6) is not an 𝒩∈⋁q-subset of (X,f).Table 2
*-operation.
*
θ
a
b
c
θ
θ
a
b
c
a
a
θ
c
b
b
b
c
θ
a
c
c
b
a
θExample 3.5.
LetX={θ,a,b,c,d} be a set with a *-operation table which is given by Table 3. Then (X;*,θ) is a BCK-algebra (see [4]). Consider an 𝒩-structure (X,f) in which f is defined by (3.8)f=(θabcd-0.8-0.700-0.6). Then (X,f) is an 𝒩-subalgebra of type (q,∈⋁q).Table 3
*-operation.
*
θ
a
b
c
d
θ
θ
θ
θ
θ
θ
a
a
θ
θ
θ
θ
b
b
b
θ
θ
b
c
c
b
a
θ
b
d
d
d
d
d
θTheorem 3.6.
If(X,f) is an 𝒩-subalgebra of type (∈,∈), then the open 0-support of (X,f) is a subalgebra of X.Proof.
Let(X,f) be an 𝒩-subalgebra of type (∈,∈). If f is zero, that is, f(x)=0 for all x∈X, then O(f;0)=∅ which is a subalgebra of
X. Assume that f is nonzero and let x,y∈O(f;0). Then f(x)<0 and f(y)<0. Suppose that f(x*y)=0. Note that (X,xf(x)) and (X,yf(y)) are point 𝒩-structures which are 𝒩∈-subsets of (X,f). But the point 𝒩-structure (X,(x*y)⋁{f(x),f(y)}) is not an 𝒩∈-subset of (X,f) because f(x*y)=0>⋁{f(x),f(y)}. This is a contradiction, and so
f(x*y)<0, that is, x*y∈O(f;0). Hence O(f;0) is a subalgebra of X.Theorem 3.7.
If(X,f) is an 𝒩-subalgebra of type (∈,q), then the open 0-support of (X,f) is a subalgebra of X.Proof.
Letx,y∈O(f;0). Then f(x)<0 and f(y)<0. If f(x*y)=0, then (3.9)f(x*y)+⋁{f(x),f(y)}+1=⋁{f(x),f(y)}+1≥0. Thus the point 𝒩-structure (X,(x*y)⋁{f(x),f(y)}) is not an 𝒩q-subset of (X,f), which is impossible since
(X,xf(x)) and (X,yf(y)) are point 𝒩-structures which are 𝒩∈-subsets of (X,f). Therefore, f(x*y)<0, that is, x*y∈O(f;0). This shows that the open
0-support of (X,f) is a subalgebra of X.Theorem 3.8.
If(X,f) is an 𝒩-subalgebra of type (q,∈), then the open 0-support of (X,f) is a subalgebra of X.Proof.
Letx,y∈O(f;0). Then f(x)<0 and f(y)<0, which imply that (X,x-1) and (X,y-1) are point 𝒩-structures which are 𝒩q-subsets of (X,f). If f(x*y)=0, then the point 𝒩-structure (X,(x*y)⋁{-1,-1}) is not an 𝒩∈-subset of (X,f), a contradiction. Therefore,
f(x*y)<0, that is, x*y∈O(f;0), and so the open 0-support of (X,f) is a subalgebra of X.Theorem 3.9.
If(X,f) is an 𝒩-subalgebra of type (q,q), then f is constant on the open
0-support of (X,f).Proof.
Assume thatf is not constant on the open
0-support of (X,f). Then there exists y∈O(f;0) such that ty=f(y)≠f(θ)=t0. Then either ty<t0 or ty>t0. Suppose that ty>t0 and choose t1,t2∈[-1,0) such that t2<-1-ty<t1<-1-t0. Then f(0)+t1+1=t0+t1+1<0 and f(y)+t2+1=ty+t2+1<0, and so (X,θt1) and (X,yt2) are point 𝒩-structures which are 𝒩q-subsets of (X,f). Since (3.10)f(y*θ)+⋁{t1,t2}+1=f(y)+t1+1=ty+t1+1>0, the point 𝒩-structure (X,(y*θ)⋁{t1,t2}) is not an 𝒩q-subset of (X,f), which is a contradiction. Next assume that
ty<t0. Then f(y)+(-1-t0)+1=ty-t0<0, and so (X,y-1-t0) is an 𝒩q-subset of (X,f). Note that (3.11)f(y*y)+(-1-t0)+1=f(θ)-t0=t0-t0=0, and thus (X,(y*y)⋁{-1-t0,-1-t0}) is not an 𝒩q-subset of (X,f). This is impossible, and therefore
f is constant on the open
0-support of (X,f).Theorem 3.10.
An𝒩-structure (X,f) is an 𝒩-subalgebra of type (∈,∈⋁q) if and only if it satisfies (3.12)(∀x,y∈X)(f(x*y)≤⋁{f(x),f(y),-0.5}).Proof.
Suppose that(X,f) is an 𝒩-subalgebra of type (∈,∈⋁q). For any x,y∈X, assume that ⋁{f(x),f(y)}>-0.5. If f(a*b)>⋁{f(a),f(b)} for some a,b∈X, then there exists t∈[-1,0) such that f(a*b)>t≥⋁{f(a),f(b)}. Thus, point 𝒩-structures (X,at) and (X,bt) are 𝒩∈-subsets of (X,f), but the point 𝒩-structure (X,(a*b)⋁{t,t}) is not an 𝒩∈⋁q-subset of (X,f), a contradiction. Hence
f(x*y)≤⋁{f(x),f(y)} whenever ⋁{f(x),f(y)}>-0.5 for all x,y∈X. Now suppose that ⋁{f(x),f(y)}≤-0.5. Then point 𝒩-structures (X,x-0.5) and (X,y-0.5) are 𝒩∈-subsets of (X,f), which imply that the point
𝒩-structure (X,(x*y)⋁{-0.5,-0.5}) is an 𝒩∈⋁q-subset of (X,f). Hence f(x*y)≤-0.5. Otherwise, f(x*y)-0.5+1>-0.5-0.5+1=0, that is, (X,(x*y)-0.5) is not an 𝒩q-subset of (X,f). This is a contradiction. Consequently,
f(x*y)≤⋁{f(x),f(y),-0.5} for all x,y∈X.
Conversely, assume that (3.12)
is valid. Let x,y∈X and t1,t2∈[-1,0) be such that two point
𝒩-structures (X,xt1) and (X,yt2) are 𝒩∈-subsets of (X,f). If f(x*y)≤⋁{t1,t2}, then (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset of (X,f). Suppose that f(x*y)>⋁{t1,t2}. Then ⋁{f(x),f(y)}≤-0.5. Otherwise, we have (3.13)f(x*y)≤⋁{f(x),f(y),-0.5}=⋁{f(x),f(y)}≤⋁{t1,t2}, a contradiction. It follows that (3.14)f(x*y)+⋁{t1,t2}+1<2f(x*y)+1≤2⋁{f(x),f(y),-0.5}+1=0 and so (X,(x*y)⋁{t1,t2}) is an 𝒩q-subset of (X,f). Consequently, (X,(x*y)⋁{t1,t2}) is an 𝒩∈⋁q-subset of (X,f), and thus (X,f) is an 𝒩-subalgebra of type (∈,∈⋁q).We provide conditions for an𝒩-structure to be an 𝒩-subalgebra of type (q,∈⋁q).Theorem 3.11.
LetS be a subalgebra of X and let (X,f) be an 𝒩-structure such that (1)
(forallx∈X)(x∈S⇒f(x)≤-0.5),(2)
(forallx∈X)(x∉S⇒f(x)=0).Then (X,f) is an 𝒩-subalgebra of type (q,∈⋁q).Proof.
Letx,y∈X and t1,t2∈[-1,0) be such that two point
𝒩-structures (X,xt1) and (X,yt2) are 𝒩q-subsets of (X,f). Then f(x)+t1+1<0 and f(y)+t2+1<0. Thus x*y∈S because if it is impossible, then
x∉S or y∉S. Thus f(x)=0 or f(y)=0, and so t1<-1 or t2<-1. This is a contradiction. Hence
f(x*y)≤-0.5. If ⋁{t1,t2}<-0.5, then f(x*y)+⋁{t1,t2}+1<0 and thus the point 𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩q-subset of (X,f). If ⋁{t1,t2}≥-0.5, then f(x*y)≤-0.5≤⋁{t1,t2} and so the point 𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset of (X,f). Therefore, the point 𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩∈⋁q-subset of (X,f). This completes the proof.Theorem 3.12.
Let(X,f) be an 𝒩-subalgebra of type (q,∈⋁q). If f is not constant on the open
0-support of (X,f), then f(x)≤-0.5 for some x∈X. In particular, f(θ)≤-0.5.Proof.
Assume thatf(x)>-0.5 for all x∈X. Since f is not constant on the open
0-support of (X,f), there exists x∈O(f;0) such that tx=f(x)≠f(θ)=t0. Then either t0<tx or t0>tx. For the case t0<tx, choose r<-0.5 such that t0+r+1<0<tx+r+1. Then the point 𝒩-structure (X,θr) is an 𝒩q-subset of (X,f). Since (X,x-1) is an 𝒩q-subset of (X,f). It follows from (a1) that the point
𝒩-structure (X,(x*θ)⋁{r,-1})=(X,xr) is an 𝒩∈⋁q-subset of (X,f). But, f(x)>-0.5>r implies that the point
𝒩-structure (X,xr) is not an 𝒩∈-subset of (X,f). Also, f(x)+r+1=tx+r+1>0 implies that the point
𝒩-structure (X,xr) is not an 𝒩q-subset of (X,f). This is a contradiction. Now, if
t0>tx then we can take r<-0.5 such that tx+r+1<0<t0+r+1. Then (X,xr) is an 𝒩q-subset of (X,f), and f(x*x)=f(θ)=t0>r=⋁{r,r} induces that (X,(x*x)⋁{r,r}) is not an 𝒩∈-subset of (X,f). Since (3.15)f(x*x)+⋁{r,r}+1=f(θ)+r+1=t0+r+1>0,(X,(x*x)⋁{r,r}) is not an 𝒩q-subset of (X,f). Hence (X,(x*x)⋁{r,r}) is not an 𝒩∈⋁q-subset of (X,f), which is a contradiction. Therefore
f(x)≤-0.5 for some x∈X. We now prove that f(θ)≤-0.5. Assume that f(θ)=t0>-0.5. Note that there exists
x∈X such that f(x)=tx≤-0.5 and so tx<t0. Choose t1<t0 such that tx+t1+1<0<t0+t1+1. Then f(x)+t1+1=tx+t1+1<0, and thus the point 𝒩-structure (X,xt1) is an 𝒩q-subset of (X,f). Now we have (3.16)f(x*x)+⋁{t1,t1}+1=f(θ)+t1+1=t0+t1+1>0 and f(x*x)=f(θ)=t0>t1=⋁{t1,t1}. Hence (X,(x*x)⋁{t1,t1}) is not an 𝒩∈⋁q-subset of (X,f), a contradiction. Therefore
f(θ)≤-0.5.Corollary 3.13.
If(X,f) is an 𝒩-subalgebra of types (q,∈) or (q,q) in which f is not constant on the open
0-support of (X,f), then f(x)≤-0.5 for some x∈X. In particular, f(θ)≤-0.5.Theorem 3.14.
LetX be a BCK-algebra and let
(X,f) be an 𝒩-subalgebra of type (q,∈⋁q) such that f is not constant on the open
0-support of (X,f). If (3.17)f(θ)=⋀x∈Xf(x), then f(x)≤-0.5 for all x∈O(f;0).Proof.
Assume thatf(x)>-0.5 for all x∈X. Since f is not constant on the open
0-support of (X,f), there exists y∈O(f;0) such that ty=f(y)≠f(θ)=t0. Then ty>t0. Choose t1<-0.5 such that t0+t1+1<0<ty+t1+1. Then (X,θt1) is an 𝒩q-subset of (X,f). Note that the point 𝒩-structure (X,y-1) is an 𝒩q-subset of (X,f). It follows that (X,(y*θ)⋁{-1,t1})=(X,yt1) is an 𝒩∈⋁q-subset of (X,f). But f(y)>-0.5>t1 induces that (X,yt1) is not an 𝒩∈-subset of (X,f), and f(y)+t1+1=ty+t1+1>0 induces that (X,yt1) is not an 𝒩q-subset of (X,f). This is a contradiction, and so
f(x)≤-0.5 for some x∈X. Now, if possible, let t0=f(θ)>-0.5. Then there exists x∈X such that tx=f(x)≤-0.5. Thus tx<t0. Take t1<t0 such that tx+t1+1<0<t0+t1+1. Then two point 𝒩-structures (X,xt1) and (X,θ-1) are 𝒩q-subsets of (X,f), but (X,(θ*x)⋁{-1,t1})=(X,θt1) is not an 𝒩∈⋁q-subset of (X,f), a contradiction. Hence
f(θ)≤-0.5. Finally let tx=f(x)>-0.5 for some x∈O(f;0). Taking t1<0 such that tx+t1>-0.5, then two point 𝒩-structures (X,x-1) and (X,θ-0.5+t1) are 𝒩q-subsets of (X,f). But (3.18)f(x)-0.5+t1+1=tx-0.5+t1+1>-0.5-0.5+1=0 implies that the point 𝒩-structure (X,x-0.5+t1) is not an 𝒩q-subset of (X,f). Hence the point 𝒩-structure (X,(x*θ)⋁{-1,-0.5+t1})=(X,x-0.5+t1) is not an 𝒩∈⋁q-subset of (X,f), a contradiction. Therefore
f(x)≤-0.5 for all x∈O(f;0).



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*Source: 303412-2010-09-26.xml*