The aim of this article is to obtain more general forms than the papers of (Jun et al. (2010); Jun et al. (in press)). The notions of
𝒩-subalgebras of types (∈,qk),(∈,∈∨qk), and (q,∈∨qk) are introduced, and the concepts of qk-support and ∈∨qk-support are also introduced. Several related properties are investigated. Characterizations of 𝒩-subalgebra of type (∈,∈∨qk) are discussed, and conditions for an 𝒩-subalgebra of type (∈,∈∨qk) to be an 𝒩-subalgebra of type (∈,∈) are considered.

1. Introduction

A (crisp) set A 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 generalizations 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 fits 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, Jun et al. [3] defined the notions of 𝒩-subalgebras of types (∈,∈),(∈,q),(∈,∈∨q),(q,∈),(q,q), and (q,∈∨q) and investigated related properties. They also gave conditions for an 𝒩-structure to be an 𝒩-subalgebra of type (q,∈∨q). Jun et al. provided a characterization of an 𝒩-subalgebra of type (∈,∈∨q) (see [3, 4]).

In this paper, we try to have more general form of the papers [3, 4]. We introduce the notions of 𝒩-subalgebras of types (∈,qk),(∈,∈∨qk), and (q,∈∨qk). We also introduce the concepts of qk-support and ∈∨qk-support and investigate several properties. We discuss characterizations of 𝒩-subalgebra of type (∈,∈∨qk). We consider conditions for an 𝒩-subalgebra of type (∈,∈∨qk) to be an 𝒩-subalgebra of type (∈,∈). The important achievement of the study of 𝒩-subalgebras of types (∈,qk),(∈,∈∨qk), and (q,∈∨qk) is that the notions of 𝒩-subalgebras of types (∈,q),(∈,∈∨q), and (q,∈∨q) are a special case of 𝒩-subalgebras of types (∈,qk),(∈,∈∨qk), and (q,∈∨qk), and thus so many results in the papers [3, 4] are corollaries of our results obtained in this paper.

2. Preliminaries

Let K(τ) be the class of all algebras with type τ=(2,0). By a BCI-algebra, we mean a system X:=(X,*,0)∈K(τ) in which the following axioms hold:

((x*y)*(x*z))*(z*y)=0,

(x*(x*y))*y=0,

x*x=0,

x*y=y*x=0⇒x=y,

for all x,y,z∈X. If a BCI-algebra X satisfies 0*x=0 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=0).

In a BCK/BCI-algebra X, the following hold:

(a1) (∀x∈X)(x*0=x),

(a2) (∀x,y,z∈X)((x*y)*z=(x*z)*y),

for all x,y,z∈X.

A nonempty subset S of a BCK/BCI-algebras 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 [5, 6] 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 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,f) of X and an 𝒩-function f on X. In what follows, let X denote a BCK/BCI-algebras and f an 𝒩-function on X unless otherwise specified.

Definition 2.1 (see [<xref ref-type="bibr" rid="B5">1</xref>]).

By a subalgebra 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 any 𝒩-structure (X,f) and t∈[-1,0), the set
(2.4)C(f;t):={x∈X∣f(x)≤t}
is called a closed t-support of (X,f), and the set
(2.5)O(f;t):={x∈X∣f(x)<t}
is called an open t-support of (X,f).

Using the similar method to the transfer principle in fuzzy theory (see [7, 8]), Jun et al. [2] considered transfer principle in 𝒩-structures as follows.

Theorem 2.2 (see [<xref ref-type="bibr" rid="B6">2</xref>]; <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M122"><mml:mrow><mml:mi>𝒩</mml:mi></mml:mrow></mml:math></inline-formula>-transfer principle).

An 𝒩-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 [<xref ref-type="bibr" rid="B5">1</xref>]).

An 𝒩-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. General Form of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M136"><mml:mrow><mml:mi>𝒩</mml:mi></mml:mrow></mml:math></inline-formula>-Subalgebras with Type <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M137"><mml:mo stretchy="false">(</mml:mo><mml:mo>∈</mml:mo><mml:mo>,</mml:mo><mml:mo>∈</mml:mo><mml:mo>∨</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>

In what follows, let t and k denote arbitrary elements of [-1,0) and (-1,0], respectively, unless otherwise specified.

Let (X,f) be an 𝒩-structure in which f is given by
(3.1)f(y)={0ify≠x,tify=x.
In this case, f is denoted by xt, and we call (X,xt) a point 𝒩-structure. For any 𝒩-structure (X,g), we say that a point 𝒩-structure (X,xt) is an 𝒩∈-subset (resp., 𝒩q-subset) of (X,g) if g(x)≤t (resp., g(x)+t+1<0). If a point 𝒩-structure (X,xt) is an 𝒩∈-subset of (X,g) or an 𝒩q-subset of (X,g), we say (X,xt) is an 𝒩∈∨q-subset of (X,g). We say that a point 𝒩-structure (X,xt) is an 𝒩qk-subset of (X,g) if g(x)+t-k+1<0. Clearly, every 𝒩qk-subset with k=0 is an 𝒩q-subset. Note that if k,r∈(-1,0] with k<r, then every 𝒩qk-subset is an 𝒩qr-subset.

Definition 3.1.

An 𝒩-structure (X,f) is called an 𝒩-subalgebra of type

(∈,∈) (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).

(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).

Definition 3.2.

An 𝒩-structure (X,f) is called an 𝒩-subalgebra of type (∈,∈∨qk) (resp., (q,∈∨qk)) if whenever two point 𝒩-structures (X,xt1) and (X,yt2) are 𝒩∈-subsets (resp., 𝒩q-subsets) of (X,f) then the point 𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩∈∨qk-subset of (X,f).

Example 3.3.

Consider a BCI-algebra X={0,a,b,c} with the following Cayley table:
(3.2)*0abc00abcaa0cbbbc0accba0
Let (X,f) be an 𝒩-structure in which f is defined by
(3.3)f=(0abc-0.6-0.7-0.3-0.3).
It is routine to verify that (X,f) is an 𝒩-subalgebra of type (∈,∈∨q-0.2).

Note that if k,r∈(-1,0] with k<r, then every 𝒩-subalgebra of type (∈,∈∨qk) is an 𝒩-subalgebra of type (∈,∈∨qr), but the converse is not true as seen in the following example.

Example 3.4.

The 𝒩-subalgebra (X,f) of type (∈,∈∨q-0.2) in Example 3.3 is not of type (∈,∈∨q-0.4) since (X,a-0.65) and (X,a-0.68) are 𝒩∈-subsets of (X,f), but
(3.4)(X,(a*a)⋁{-0.65,-0.68})
is not an 𝒩∈∨q-0.4-subset of (X,f).

Theorem 3.5.

Every 𝒩-subalgebra of type (∈,∈) is of type (∈,∈∨qk).

Proof.

Straightforward.

Taking k=0 in Theorem 3.5 induces the following corollary.

Corollary 3.6.

Every 𝒩-subalgebra of type (∈,∈) is of type (∈,∈∨q).

The converse of Theorem 3.5 is not true as seen in the following example.

Example 3.7.

Consider the 𝒩-subalgebra (X,f) of type (∈,∈∨q-0.2) which is given in Example 3.3. Then (X,f) is not an 𝒩-subalgebra of type (∈,∈) since (X,a-0.65) and (X,a-0.68) are 𝒩∈-subsets of (X,f), but (X,(a*a)⋁{-0.65,-0.68}) is not an 𝒩∈-subset of (X,f).

Definition 3.8.

An 𝒩-structure (X,f) is called an 𝒩-subalgebra of type (∈,qk) if whenever two point 𝒩-structure (X,xt1) and (X,yt2) are 𝒩∈-subsets of (X,f) then the point 𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩qk-subset of (X,f).

Theorem 3.9.

Every 𝒩-subalgebra of type (∈,qk) is of type (∈,∈∨qk).

Proof.

Straightforward.

Taking k=0 in Theorem 3.9 induces the following corollary.

Corollary 3.10.

Every 𝒩-subalgebra of type (∈,q) is of type (∈,∈∨q).

The converse of Theorem 3.9 is not true as seen in the following example.

Example 3.11.

Consider the 𝒩-subalgebra (X,f) of type (∈,∈∨q-0.2) which is given in Example 3.3. Then (X,a-0.65) and (X,b-0.25) are 𝒩-subsets of (X,f), but
(3.5)(X,(a*b)⋁{-0.65,-0.25})=(X,c-0.2)
is not an 𝒩qk-subset of (X,f) for k=-0.2 since f(c)-0.25-0.2+1>0.

We consider a characterization of an 𝒩-subalgebra of type (∈,∈∨qk).

Theorem 3.12.

An 𝒩-structure (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk) if and only if it satisfies
(3.6)(∀x,y∈X)(f(x*y)≤⋁{f(x),f(y),k-12}).

Proof.

Let (X,f) be an 𝒩-structure of type (∈,∈∨qk). Assume that (3.6) is not valid. Then there exists a,b∈X such that
(3.7)f(a*b)>⋁{f(a),f(b),k-12}.
If ⋁{f(a),f(b)}>(k-1)/2, then f(a*b)>⋁{f(a),f(b)}. Hence
(3.8)f(a*b)>t≥⋁{f(a),f(b)}
for some t∈[-1,0). It follows that point 𝒩-structures (X,at) and (X,bt) are 𝒩∈-subsets of (X,f), but the point 𝒩-structure (X,(a*b)t) is not an 𝒩∈-subset of (X,f). Moreover,
(3.9)f(a*b)+t-k+1>2t-k+1=0,
and so (X,(a*b)t) is not an 𝒩qk-subset of (X,f). Consequently, (X,(a*b)t) is not an 𝒩∈∨qk-subset of (X,f). This is a contradiction. If ⋁{f(a),f(b)}≤(k-1)/2, then f(a)≤(k-1)/2,f(b)≤(k-1)/2 and f(a*b)>(k-1)/2. Thus (X,a(k-1)/2) and (X,b(k-1)/2) are 𝒩∈-subsets of (X,f), but (X,(a*b)(k-1)/2) is not an 𝒩∈-subset of (X,f). Also,
(3.10)f(a*b)+k-12-k+1>k-12+k-12-k+1=0,
that is, (X,(a*b)(k-1)/2) is not an 𝒩qk-subset of (X,f). Hence (X,(a*b)(k-1)/2) is not an 𝒩∈∨qk-subset of (X,f), a contradiction. Therefore (3.6) is valid.

Conversely, suppose that (3.6) 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). Then
(3.11)f(x*y)≤⋁{f(x),f(y),k-12}≤⋁{t1,t2,k-12}.
Assume that t1≥(k-1)/2 or t2≥(k-1)/2. Then f(x*y)≤⋁{t1,t2}, and so (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset of (X,f). Now suppose that t1<(k-1)/2 and t2<(k-1)/2. Then f(x*y)≤(k-1)/2, and thus
(3.12)f(x*y)+⋁{t1,t2}-k+1<k-12+k-12-k+1=0,
that is, (X,(x*y)⋁{t1,t2}) is an 𝒩qk-subset of (X,f). Therefore (X,(x*y)⋁{t1,t2}) is an 𝒩∈∨qk-subset of (X,f) and consequently (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk).

Corollary 3.13 (see [<xref ref-type="bibr" rid="B2">3</xref>]).

An 𝒩-structure (X,f) is an 𝒩-subalgebra of type (∈,∈∨q) if and only if it satisfies
(3.13)(∀x,y∈X)(f(x*y)≤⋁{f(x),f(y),-0.5}).

Proof.

It follows from taking k=0 in Theorem 3.12.

We provide conditions for an 𝒩-structure to be an 𝒩-subalgebra of type (q,∈∨qk).

Theorem 3.14.

Let S be a subalgebra of X and let (X,f) be an 𝒩-structure such that

(∀x∈X)(x∈S⇒f(x)≤(k-1)/2),

(∀x∈X)(x∉S⇒f(x)=0).

Then (X,f) is an 𝒩-subalgebra of type (q,∈∨qk).
Proof.

Let x,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)≤(k-1)/2. If ⋁{t1,t2}<(k-1)/2, then f(x*y)+⋁{t1,t2}-k+1<((k-1)/2)+((k-1)/2)-k+1=0 and so the point 𝒩-structure (X,(x*y)⋁{t1,t2}) is an 𝒩qk-subset of (X,f). If ⋁{t1,t2}≥(k-1)/2, then f(x*y)≤(k-1)/2≤⋁{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∈∨qk-subset of (X,f). This shows that (X,f) is an 𝒩-subalgebra of type (q,∈∨qk).

Taking k=0 in Theorem 3.14, we have the following corollary.

Corollary 3.15 (see [<xref ref-type="bibr" rid="B2">3</xref>]).

Let S be a subalgebra of X and let (X,f) be an 𝒩-structure such that

(∀x∈X)(x∈S⇒f(x)≤-0.5),

(∀x∈X)(x∉S⇒f(x)=0).

Then (X,f) is an 𝒩-subalgebra of type (q,∈∨q).
Theorem 3.16.

Let (X,f) be an 𝒩-subalgebra of type (qk,∈∨qk). If f is not constant on the open 0-support of (X,f), then f(x)≤(k-1)/2 for some x∈X. In particular, f(0)≤(k-1)/2.

Proof.

Assume that f(x)>(k-1)/2 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(0)=t0. Then either t0<tx or t0>tx. For the case t0<tx, choose r<(k-1)/2 such that t0+r-k+1<0<tx+r-k+1. Then the point 𝒩-structure (X,0r) is an 𝒩qk-subset of (X,f). Since (X,x-1) is an 𝒩qk-subset of (X,f). It follows from (a1) that the point 𝒩-structure (X,(x*0)⋁{r,-1})=(X,xr) is an 𝒩∈∨qk-subset of (X,f). But, f(x)>(k-1)/2>r implies that the point 𝒩-structure (X,xr) is not an 𝒩∈-subset of (X,f). Also, f(x)+r-k+1=tx+r-k+1>0 implies that the point 𝒩-structure (X,xr) is not an 𝒩qk-subset of (X,f). This is a contradiction. Assume that t0>tx and take r<(k-1)/2 such that tx+r-k+1<0<t0+r-k+1. Then (X,xr) is an 𝒩qk-subset of (X,f). Since
(3.14)f(x*x)=f(0)=t0>-r+k-1>-k-12+k-1=k-12>r,(X,(x*x)⋁{r,r}) is not an 𝒩∈-subset of (X,f). Since
(3.15)f(x*x)+⋁{r,r}-k+1=f(0)+r-k+1=t0+r-k+1>0,(X,(x*x)⋁{r,r}) is not an 𝒩qk-subset of (X,f). Hence (X,(x*x)⋁{r,r}) is not an 𝒩∈∨qk-subset of (X,f), which is a contradiction. Therefore f(x)≤(k-1)/2 for some x∈X. We now prove that f(0)≤(k-1)/2. Assume that f(0)=t0>(k-1)/2. Note that there exists x∈X such that f(x)=tx≤(k-1)/2 and so tx<t0. Choose t1<t0 such that tx+t1-k+1<0<t0+t1-k+1. Then f(x)+t1-k+1=tx+t1-k+1<0, and thus the point 𝒩-structure (X,xt1) is an 𝒩qk-subset of (X,f). Now we have
(3.16)f(x*x)+⋁{t1,t1}-k+1=f(0)+t1-k+1=t0+t1-k+1>0
and f(x*x)=f(0)=t0>t1=⋁{t1,t1}. Hence (X,(x*x)⋁{t1,t1}) is not an 𝒩∈∨qk-subset of (X,f). This is a contradiction, and therefore f(0)≤(k-1)/2.

Corollary 3.17 (see [<xref ref-type="bibr" rid="B2">3</xref>]).

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(0)≤-0.5.

Theorem 3.18.

An 𝒩-structure (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk) if and only if for every t∈[(k-1)/2,0] the nonempty closed t-support of (X,f) is a subalgebra of X.

Proof.

Assume that (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk) and let t∈[(k-1)/2,0] be such that C(f;t)≠∅. Let x,y∈C(f;t). Then f(x)≤t and f(y)≤t. It follows from Theorem 3.12 that
(3.17)f(x*y)≤⋁{f(x),f(y),k-12}≤⋁{t,k-12}=t
so that x*y∈C(f;t). Therefore C(f;t) is a subalgebra of X.

Conversely, let (X,f) be an 𝒩-structure such that the nonempty closed t-support of (X,f) is a subalgebra of X for all t∈[(k-1)/2,0]. If there exist a,b∈X such that f(a*b)>⋁{f(a),f(b),(k-1)/2}, then we can take s∈[-1,0] such that
(3.18)f(a*b)>s≥⋁{f(a),f(b),k-12}.
Thus a,b∈C(f;s) and s≥(k-1)/2. Since C(f,s) is a subalgebra of X, it follows that a*b∈C(f;s) so that f(a*b)≤s. This is a contradiction, and therefore f(x*y)≤⋁{f(x),f(y),(k-1)/2} for all x,y∈X. Using Theorem 3.12, we conclude that (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk).

Taking k=0 in Theorem 3.18, we have the following corollary.

Corollary 3.19 (see [<xref ref-type="bibr" rid="B3">4</xref>]).

An 𝒩-structure (X,f) is an 𝒩-subalgebra of type (∈,∈∨q) if and only if for every t∈[-0.5,0] the nonempty closed t-support of (X,f) is a subalgebra of X.

Theorem 3.20.

Let S be a subalgebra of X. For any t∈[(k-1)/2,0), there exists an 𝒩-subalgebra (X,f) of type (∈,∈∨qk) for which S is represented by the closed t-support of (X,f).

Proof.

Let (X,f) be an 𝒩-structure in which f is given by
(3.19)f(x)={tifx∈S,0ifx∉S,
for all x∈X where t∈[(k-1)/2,0). Assume that f(a*b)>⋁{f(a),f(b),(k-1)/2} for some a,b∈X. Since the cardinality of the image of f is 2, we have f(a*b)=0 and ⋁{f(a),f(b),(k-1)/2}=t. Since t≥(k-1)/2, it follows that f(a)=t=f(b) so that a,b∈S. Since S is a subalgebra of X, we obtain a*b∈S and so f(a*b)=t<0. This is a contradiction. Therefore f(x*y)≤⋁{f(x),f(y),(k-1)/2} for all x,y∈X. Using Theorem 3.12, we conclude that (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk). Obviously, S is represented by the closed t-support of (X,f).

Corollary 3.21 (see [<xref ref-type="bibr" rid="B3">4</xref>]).

Let S be a subalgebra of X. For any t∈[-0.5,0), there exists an 𝒩-subalgebra (X,f) of type (∈,∈∨q) for which S is represented by the closed t-support of (X,f).

Proof.

It follows from taking k=0 in Theorem 3.20.

Note that every 𝒩-subalgebra of type (∈,∈) is an 𝒩-subalgebra of type (∈,∈∨qk), but the converse is not true in general (see Example 3.7). Now, we give a condition for an 𝒩-subalgebra of type (∈,∈∨qk) to be an 𝒩-subalgebra of type (∈,∈).

Theorem 3.22.

Let (X,f) be an 𝒩-subalgebra of type (∈,∈∨qk) such that f(x)>(k-1)/2 for all x∈X. Then (X,f) is an 𝒩-subalgebra of type (∈,∈).

Proof.

Let x,y∈X and t∈[-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 Theorem 3.12 and the hypothesis that
(3.20)f(x*y)≤⋁{f(x),f(y),k-12}=⋁{f(x),f(y)}≤⋁{t1,t2}
so that (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset of (X,f). Therefore (X,f) is an 𝒩-subalgebra of type (∈,∈).

Corollary 3.23 (see [<xref ref-type="bibr" rid="B3">4</xref>]).

Let (X,f) be an 𝒩-structure of type (∈,∈∨q) such that f(x)>-0.5 for all x∈X. Then (X,f) is an 𝒩-subalgebra of type (∈,∈).

Proof.

It follows from taking k=0 in Theorem 3.22.

Theorem 3.24.

Let {(X,fi)∣i∈Λ} be a family of 𝒩-subalgebras of type (∈,∈∨qk). Then (X,⋃i∈Λfi) is an 𝒩-subalgebra of type (∈,∈∨qk), where ⋃i∈Λfi is an 𝒩-function on X given by (⋃i∈Λfi)(x)=⋁i∈Λfi(x) for all x∈X.

Proof.

Let x,y∈X and t1,t2∈[-1,0) be such that (X,xt1) and (X,yt2) are 𝒩∈-subsets of (X,⋃i∈Λfi). Assume that (X,(x*y)⋁{t1,t2}) is not an 𝒩∈∨qk-subset of (X,⋃i∈Λfi). Then (X,(x*y)⋁{t1,t2}) is neither an 𝒩∈-subset nor an 𝒩qk-subset of (X,⋃i∈Λfi). Hence (⋃i∈Λfi)(x*y)>⋁{t1,t2} and
(3.21)(⋃i∈Λfi)(x*y)+⋁{t1,t2}-k+1≥0,
which imply that
(3.22)(⋃i∈Λfi)(x*y)>k-12.
Let A1:={i∈Λ∣(X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset of(X,fi)} and A2:={i∈Λ∣(X,(x*y)⋁{t1,t2}) is an 𝒩qk-subsetof(X,fi)}∩{j∈Λ∣(X,(x*y)⋁{t1,t2}) is not an 𝒩∈-subsetof(X,fj)}. Then Λ=A1∪A2 and A1∩A2=∅. If A2=∅, then (X,(x*y)⋁{t1,t2}) is an 𝒩∈-subset of (X,fi) for all i∈Λ, that is, fi(x*y)≤⋁{t1,t2} for all i∈Λ. Thus (⋃i∈Λfi)(x*y)≤⋁{t1,t2}. This is a contradiction. Hence A2≠∅, and so for every i∈A2, we have fi(x*y)>⋁{t1,t2} and fi(x*y)+⋁{t1,t2}-k+1<0. It follows that ⋁{t1,t2}<(k-1)/2. Since (X,xt1) is an 𝒩∈-subset of (X,⋃i∈Λfi), we have
(3.23)fi(x)≤(⋃i∈Λfi)(x)≤t1≤⋁{t1,t2}<k-12
for all i∈Λ. Similarly, fi(y)<(k-1)/2 for all i∈Λ. Next suppose that t:=fi(x*y)>(k-1)/2. Taking (k-1)/2<r<t, we know that (X,xr) and (X,yr) are 𝒩∈-subsets of (X,fi), but (X,(x*y)⋁{r,r})=(X,(x*y)r) is not an 𝒩∈∨qk-subset of (X,fi). This contradicts that (X,fi) is an 𝒩-subalgebra of type (∈,∈∨qk). Hence fi(x*y)≤(k-1)/2 for all i∈Λ, and so (⋃i∈Λfi)(x*y)≤(k-1)/2 which contradicts (3.22). Therefore (X,(x*y)⋁{t1,t2}) is an 𝒩∈∨qk-subset of (X,⋃i∈Λfi) and consequently (X,⋃i∈Λfi) is an 𝒩-subalgebra of type (∈,∈∨qk).

For any 𝒩-structure (X,f) and t∈[-1,0), the q-support and the ∈∨q-support of (X,f) related to t are defined to be the sets (see [4])
(3.24)𝒩q(f;t):={x∈X∣(X,xt)isan𝒩q-subsetof(X,f)},(3.25)𝒩∈∨q(f;t):={x∈X∣(X,xt)isan𝒩∈∨q-subsetof(X,f)},
respectively. Note that the ∈∨q-support is the union of the closed support and the q-support, that is,
(3.26)𝒩∈∨q(f;t)=C(f;t)∪𝒩q(f;t),t∈[-1,0).
The qk-support and the ∈∨qk-support of (X,f) related to t are defined to be the sets
(3.27)𝒩qk(f;t):={x∈X∣(X,xt)isan𝒩qk-subsetof(X,f)},(3.28)𝒩∈∨qk(f;t):={x∈X∣(X,xt)isan𝒩∈∨qk-subsetof(X,f)},
respectively. Clearly, 𝒩∈∨qk(f;t)=C(f;t)∪𝒩qk(f;t) for all t∈[-1,0).

Theorem 3.25.

An 𝒩-structure (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk) if and only if the ∈∨qk-support of (X,f) related to t is a subalgebra of X for all t∈[-1,0).

Proof.

Suppose that (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk). Let x,y∈𝒩∈∨qk(f;t) for t∈[-1,0). Then (X,xt) and (X,yt) are 𝒩∈∨qk-subsets of (X,f). Hence f(x)≤t or f(x)+t-k+1<0, and f(y)≤t or f(y)+t-k+1<0. Then we consider the following four cases:

(c1) f(x)≤t and f(y)≤t,

(c2) f(x)≤t and f(y)+t-k+1<0,

(c3) f(x)+t-k+1<0 and f(y)≤t,

(c4) f(x)+t-k+1<0 and f(y)+t-k+1<0.

Combining (3.6) and (c1), we have f(x*y)≤⋁{t,(k-1)/2}. If t≥(k-1)/2, then f(x*y)≤t and so (X,(x*y)t) is an 𝒩∈-subset of (X,f). Hence x*y∈C(f;t)⊆𝒩∈∨qk(f;t). If t<(k-1)/2, then f(x*y)≤(k-1)/2 and so f(x*y)+t-k+1<((k-1)/2)+((k-1)/2)-k+1=0, that is, (X,(x*y)t) is an 𝒩qk-subset of (X,f). Therefore x*y∈𝒩qk(f;t)⊆𝒩∈∨qk(f;t). For the case (c2), assume that t<(k-1)/2. Then
(3.29)f(x*y)≤⋁{f(x),f(y),k-12}≤⋁{t,f(y),k-12}=⋁{f(y),k-12}={f(y)iff(y)>k-12,k-12iff(y)≤k-12,<k-1-t,
and so f(x*y)+t-k+1<0. Thus (X,(x*y)t) is an 𝒩qk-subset of (X,f). If t≥(k-1)/2, then
(3.30)f(x*y)≤⋁{f(x),f(y),k-12}≤⋁{t,f(y),k-12}=⋁{t,f(y)}={f(y)iff(y)>t,tiff(y)≤t,
and thus x*y∈𝒩qk(f;t) or x*y∈C(f;t). Consequently, x*y∈𝒩∈∨qk(f;t). For the case (c3), it is similar to the case (c2). Finally, for the case (c4), if t≥(k-1)/2, then k-1-t≤(k-1)/2≤t. Hence
(3.31)f(x*y)≤⋁{f(x),f(y),k-12}≤⋁{k-1-t,k-12}=k-12≤t,
which implies that x*y∈C(f;t). If t<(k-1)/2, then t<(k-1)/2<k-1-t. Therefore
(3.32)f(x*y)≤⋁{f(x),f(y),k-12}≤⋁{k-1-t,k-12}=k-1-t,
that is, f(x*y)+t-k+1<0, which means that (X,(x*y)t) is an 𝒩qk-subset of (X,f). Consequently, the ∈∨qk-support of (X,f) related to t is a subalgebra of X for all t∈[-1,0).

Conversely, let (X,f) be an 𝒩-structure for which the ∈∨qk-support of (X,f) related to t is a subalgebra of X for all t∈[-1,0). Assume that there exist a,b∈X such that f(a*b)>⋁{f(a),f(b),(k-1)/2}. Then
(3.33)f(a*b)>s≥⋁{f(a),f(b),k-12}
for some s∈[(k-1)/2,0). It follows that a,b∈C(f;s)⊆𝒩∈∨qk(f;s) but a*b∉C(f;s). Also, f(a*b)+s-k+1>2s-k+1≥0, that is, a*b∉𝒩qk(f;s). Thus a*b∉𝒩∈∨qk(f;s) which is a contradiction. Therefore
(3.34)f(x*y)≤⋁{f(x),f(y),k-12}
for all x,y∈X. Using Theorem 3.12, we conclude that (X,f) is an 𝒩-subalgebra of type (∈,∈∨qk).

If we take k=0 in Theorem 3.25, we have the following corollary.

Corollary 3.26 (see [<xref ref-type="bibr" rid="B3">4</xref>]).

An 𝒩-structure (X,f) is an 𝒩-subalgebra of type (∈,∈∨q) if and only if the ∈∨q-support of (X,f) related to t is a subalgebra of X for all t∈[-1,0).

JunY. B.LeeK. J.SongS. Z.𝒩-ideals of BCK/BCI-algebrasJunY. B.ÖztürkM. A.RohE. H.𝒩-structures applied to closed ideals in BCH-algebrasJunY. B.KangM. S.ParkC. H.𝒩-subalgebras in BCK/BCI-algebras based on point 𝒩-structuresJunY. B.LeeK. J.ZhangX. H.𝒩-subalgerbas of type (∈,∈Vq)
based on point 𝒩-structures in BCK/BCI-algebrasCommunications of the Korean Mathematical Society. In pressHuangY. S.MengJ.JunY. B.JunY. B.KondoM.On transfer principle of fuzzy BCK/BCI-algebrasKondoM.DudekW. A.On the transfer principle in fuzzy theory