∗ | 0 | a | b | c |
0 | 0 | 0 | 0 | 0 |
a | a | 0 | 0 | a |
b | b | a | 0 | b |
c | c | c | c | 0 |
Citation: Anas Al-Masarwah, Abd Ghafur Ahmad. On (complete) normality of m-pF subalgebras in BCK/BCI-algebras[J]. AIMS Mathematics, 2019, 4(3): 740-750. doi: 10.3934/math.2019.3.740
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Imai and Iséki [14] in 1966 introduced a significant algebraic structure called a BCK-algebra. In the same year, Iséki [15] introduced the notion of a BCI-algebra as a generalization of a BCK-algebra. Today, BCK/BCI-algebras have been extensively studied by several researchers and they have been applied to several fields of mathematics, such as fuzzy set theory, group theory, ring theory, functional analysis, and so on.
The theory of fuzzy sets (FSs), initiated by Zadeh [28] in 1965, has obtained more attention by authors in a wide range of scientific domains, including decision theory, robotics, management sciences and numerous other disciplines. In 1986, Atanassov [11] introduced the notion of intuitionistic fuzzy sets (IFSs) in which there are two functions, membership function and non-membership function. In 1994, Zhang [29] introduced the new notion of bipolar fuzzy sets (BFSs) in which there are two functions, positive membership function and negative membership function. Applications of BFSs and IFSs appear in different areas, including decision-making, optimization problems, and medical diagnosis. In algebraic structures, Xi [27] implemented the idea of FSs into BCK/BCI-algebras and gave the notions of fuzzy subalgebras and ideals, while Lee [16] generalized the Xi's idea and gave the notions of bipolar fuzzy subalgebras and ideals in BCK/BCI-algebras. After that, many researchers used the ideas of fuzzy sets and hybrid models of fuzzy sets and gave several results in various algebraic structures, for instance BCK/BCI-algebras [6,7,8,22,23,25,26], B-algebras [18,24], G-algebras [21] and BG-algebras [19,20]. In several real-life situations, information sometimes comes from m factors (m≥2), that is, multi-attribute data arise which cannot be handled using the existing ideals (e.g., fuzzy ideals, bipolar fuzzy ideals, etc.). For the time being, experts trust that the real world is proceeding to multipolarity. Multi-polar vagueness in information performs a crucial role in different domains of the sciences, such as technology and neurobiology.
In view of this motivation, the notion of m-polar fuzzy (m-pF) sets was initiated by Chen et al. [12] in 2014 which is a generalization of the BFSs. In an m-pF set, the degree of membership of an object ranges over [0,1]m, which depicts m distinct characteristics of the object. Akram et al. [3], for the first time, introduced the new concept of m-pF Lie subalgebras of a Lie algebra, which is a generalization of BF Lie subalgebras. Al-Masarwah and Ahmad [9] defined the idea of m-pF subalgebras and ideals in BCK/BCI-algebras and described several properties of m-pF BCK/BCI-algebras. After that, many authors applied the idea of m-pF sets to other mathematical theories such as groups [13], Lie algebras [2], BCK/BCI-algebras [10], matroid theory [17] and graph Theory [1,4,5].
In this paper, we establish the normalization of m-pF subalgebras in BCK/BCI-algebras. We introduce the concepts of normal m-pF subalgebras, maximal m-pF subalgebras and completely normal m-pF subalgebras in BCK/BCI-algebras. We discuss some properties of normal (resp., maximal, completely normal) m-pF subalgebras. We prove that any non-constant normal m-pF subalgebra which is a maximal element of (NO(X),⊆) takes only the values ˆ0=(0,0,...,0) and ˆ1=(1,1,...,1), and every maximal m-pF subalgebra is completely normal. Moreover, we state an m-pF characteristic subalgebra in BCK/BCI-algebras.
We first recall some elementary aspects which are used to present the paper. In this paper, X always denotes a BCK/BCI-algebra without any specifications.
By a BCI-algebra we mean an algebra (X;∗,0) of type (2,0) satisfying the axioms:
(a1) ((x∗y)∗(x∗z))∗(z∗y)=0,
(a2) (x∗(x∗y))∗y=0,
(a3) x∗x=0,
(a4) x∗y=0 and y∗x=0 imply x=y.
for all x,y,z∈X. If a BCI-algebra X satisfies the axiom (a5) 0∗x=0 for all x∈X, then X is called a BCK-algebra. A partial ordering ≤ on X can be defined by x≤y if and only if x∗y=0. Any BCK/BCI-algebra X satisfies the following axioms:
(1) (x∗y)∗z=(x∗z)∗y,
(2) x∗y≤x,
(3) (x∗y)∗z≤(x∗z)∗(y∗z),
(4) x≤y⇒x∗z≤y∗z, z∗y≤z∗x.
for all x,y,z∈X. A non-empty subset I of X is called a subalgebra of X if x∗y∈I for any x,y∈I.
Definition 2.1. [12] A function ˆH is defined from X(≠ϕ) to a m-tuple of real number in [0,1] is said to be an m-pF set, that is, a mapping ˆH:X→[0,1]m. The membership degree of any element x∈X is denoted by
ˆH(x)=(p1∘ˆH(x),p2∘ˆH(x),...,pm∘ˆH(x)) |
where pj∘ˆH:[0,1]m→[0,1] is defined the j-th projection mapping. The smallest and largest values in [0,1]m are ˆ0=(0,0,...,0) and ˆ1=(1,1,...,1), respectively.
By KˆH we denote the set {x∈X∣ˆH(x)=ˆH(0)}. For any m-pF sets ˆH and ˆC in a set X, we define
ˆH⊆ˆC⇔ˆH(x)≤ˆC(x),∀x∈X. |
Definition 2.2. [9] Let ˆH be an m-pF set of X. Then, ˆHˆt={x∈X∣ˆH(x)≥ˆt} is said to be the level cut subset of ˆH for all ˆt∈(0,1]m.
If M is a nonempty subsets of X, then the m-pF characteristic function ˆCM denoted and defined by
ˆCM(x)={ˆ1=(1,1,...,1), if x∈Mˆ0=(0,0,...,0), otherwise. |
Clearly, the m-pF characteristic function of any subset of X is an m-pF subset of X.
In the current section, we present the concepts of normal m-pF subalgebras, maximal m-pF subalgebras and completely normal m-pF subalgebras in X and investigate several fundamental properties.
Definition 3.1. [9] An m-pF set ˆH in X is called an m-pF subalgebra of X if
ˆH(x∗y)≥inf{ˆH(x),ˆH(y)},∀x,y∈X. |
Example 3.1. [9] Consider a BCK-algebra X={0,a,b,c} with the Cayley table which is given in Table 1.
∗ | 0 | a | b | c |
0 | 0 | 0 | 0 | 0 |
a | a | 0 | 0 | a |
b | b | a | 0 | b |
c | c | c | c | 0 |
Let ˆH:X→[0,1]m be an m-pF set in X defined by:
ˆH(x)={(0.8,0.8,...,0.8), if x=0,a,c(0.5,0.5,...,0.5), if x=b. |
By routine computations, we can verify that ˆH is an m-pF subalgebra of X.
Lemma 3.2 ([9]). If ˆH is an m-pF subalgebra of X, then ˆH(0)≥ˆH(x), ∀x∈X.
Theorem 3.3. Let ϕ≠M⊆X and let ˆHM:X→[0,1]m be an m-pF set in X defined by
ˆHM(x)={ˆα=(α1,α2,...,αm),ifx∈Mˆβ=(β1,β2,...,βm),otherwise, |
for all x∈X and ˆα,ˆβ∈[0,1]m with ˆα>ˆβ. Then, ˆHM is an m-pF subalgebra of X if and only if M is a subalgebra of X. Moreover, in this case KˆHM={x∈X∣ˆHM(x)=ˆHM(0)}=M.
Proof. Let ˆH be an m-pF subalgebra of X. Let x,y∈X be such that x,y∈M. Then, we have
ˆHM(x∗y)≥inf{ˆHM(x),ˆHM(y)}={ˆα,ˆα}=ˆα, |
and so x∗y∈M. Hence, M is a subalgebra of X.
Conversely, suppose that M is a subalgebra of X and let x,y∈X. Then, we have the following cases:
Case(1). If x,y∈M, then x∗y∈M. Therefore
ˆHM(x∗y)=ˆα=inf{ˆHM(x)ˆHM(y)}. |
Case(2). If x∉M or y∉M, then
ˆHM(x∗y)≥ˆβ=inf{ˆHM(x),ˆHM(y)}. |
This shows that ˆHM is an m-pF subalgebra of X.
Moreover, we have KˆHM={x∈X∣ˆHM(x)=ˆHM(0)}={x∈X∣ˆHM(x)=ˆα}=M.
Now, we introduce and characterize normal m-pF subalgebras of a BCK/BCI-algebra X.
Definition 3.4. An m-pF subalgebra ˆH of X is said to be normal if there exists x∈X such that ˆH(x)=ˆ1=(1,1,...,1).
Example 3.2. Let X be a BCK-algebra in Example 3.1. Then, an m-pF subalgebra ˆH in X defined by
ˆH(x)={(1,1,...,1), if x=0,a,c(0.7,0.7,...,0.7), if x=b, |
is a normal m-pF subalgebra of X.
We know that if ˆH is a normal m-pF subalgebra of X, then clearly ˆH(0)=ˆ1=(1,1,...,1), and hence ˆH is normal if and only if ˆH(0)=ˆ1=(1,1,...,1).
Theorem 3.5. Given an m-pF subalgebra ˆH of X and let ˆH+ be an m-pF set in X defined by
ˆH+(x)=ˆH(x)+ˆHc(0)),∀x∈X. |
Then, ˆH+ is a normal m-pF subalgebra of X which contains ˆH.
Proof. Let x,y∈X. Then, we have
ˆH+(x∗y)=ˆH(x∗y)+ˆHc(0))≥inf{ˆH(x),ˆH(y)}+ˆHc(x)=inf{ˆH(x)+ˆHc(0),ˆH(y)+ˆHc(0)}=inf{ˆH+(x),ˆH+(y)}. |
Moreover, ˆH+(0)=ˆH(0)+ˆHc(0)=ˆ1. Therefore, ˆH+ is a normal m-pF subalgebra of X. Clearly, ˆH⊆ˆH+. Thus, ˆH+ is a normal m-pF subalgebra of X which contains ˆH.
Corollary 3.6. Let ˆH and ˆH+ be as in Theorem 3.5. If there is x∈X such that ˆH+(x)=ˆ0, then ˆH(x)=ˆ0.
Proof. Since ˆH⊆ˆH+, it is straightforward.
Using Theorem 3.3, we know that for any subalgebra M of X. The m-pF characteristic function ˆCM of M is a normal m-pF subalgebra of X. It is clear that ˆH is normal if and only if ˆH+=ˆH.
Proposition 3.7. If ˆH is an m-pF subalgebra of X, then (ˆH+)+=ˆH+. Moreover, if ˆH is normal, then (ˆH+)+=ˆH.
Proof. Straightforward.
Theorem 3.8. If ˆH and ˆC are m-pF subalgebras of X, such that ˆH⊆ˆC and ˆH(0)=ˆC(0), then KˆH⊆KˆC.
Proof. Let x∈KˆH. Then,
ˆC(x)≥ˆH(x)=ˆH(0)=ˆC(0) |
and so ˆC(x)=ˆC(0), i.e., x∈KˆC. Hence, KˆH⊆KˆC.
Corollary 3.9. If ˆH and ˆC are normal m-pF subalgebras of X such that ˆH⊆ˆC, then KˆH⊆KˆC.
Theorem 3.10. Let ˆH be an m-pF subalgebra of X. If there exists an m-pF subalgebra ˆC of X such that ˆC+⊆ˆH, then ˆH is normal.
Proof. Suppose that there exists an m-pF subalgebra ˆC of X such that ˆC+⊆ˆH. Then, ˆ1=ˆC+(0)≤ˆH(0), and so ˆH(0)=ˆ1. This completes the proof.
Theorem 3.11. Let ψ:[0,1]m→[0,1]m be an increasing function and ˆH be an m-pF set of X. Then, an m-pF set ˆHψ:X→[0,1]m defined by
ˆHψ(x)=ψ(ˆH(x)),∀x∈X |
is an m-pF subalgebra of X if and only if ˆH is an m-pF subalgebra of X. In particular, if ψ(ˆH(0))=ˆ1, then ˆHψ is normal, and if ψ(ˆt)=ˆt for all ˆt∈[0,1]m, then ˆH is contained in ˆHψ.
Proof. Let ˆHψ be an m-pF subalgebra of X. Then, for all x,y∈X, we have
ψ(ˆH(x∗y))=ˆHψ(x∗y)≥inf{ˆHψ(x),ˆHψ(y)}=inf{ψ(ˆH(x)),ψ(ˆH(y))}=ψ(inf{ˆH(x),ˆH(y)}). |
Since ψ is an increasing, it follows that
ˆH(x∗y)≥inf{ˆH(x),ˆH(y)}. |
Hence, ˆH is an m-pF subalgebra of X.
Conversely, if ˆH is an m-pF subalgebra of X, then for all x,y∈X, we have
ˆHψ(x∗y)=ψ(ˆH(x∗y))≥ψ(inf{ˆH(x),ˆH(y)})=inf{ψ(ˆH(x)),ψ(ˆH(y))}=inf{ˆHψ(x),ˆHψ(y)}. |
Hence, ˆHψ is an m-pF subalgebra of X.
Now, if ψ(ˆH(0))=ˆ1=(1,1,...,1), then clearly ˆHψ is normal. Assume that ψ(ˆt)=ˆt for all ˆt∈[0,1]m. Then,
ˆHψ(x)=ψ(ˆH(x))≥ˆH(x) |
for all x∈X, which proves that ˆH is contained in ˆHψ.
Denote by NO(X) the set of all normal m-pF subalgebras of X. Note that NO(X) is a poset under the set inclusion.
Theorem 3.12. Let ˆH∈NO(X) be a non-constant such that it is a maximal element of (NO(X),⊆). Then, ˆH takes only the values ˆ0=(0,0,...,0) and ˆ1=(1,1,...,1).
Proof. Let ˆH be a non-constant maximal element of (NO(X),⊆). Since ˆH is normal, so ˆH(0)=ˆ1. Let x∈X be such that ˆH(x)≠ˆ1. We claim that ˆH(x)=ˆ0. If not, then there exists b∈X such that ˆ0<ˆH(b)<ˆ1. Let ˆC:X→[0,1]m be an m-pF set in X defined by
ˆC(x)=12(ˆH(x)+ˆH(b)). |
for all x∈X. Then, clearly ˆC is well defined, and for all x,y∈X, we have
ˆC(x∗y)=12(ˆH(x∗y)+ˆH(b))≥12(inf{ˆH(x),ˆH(y)}+ˆH(b))=inf{12(ˆH(x)+ˆH(b)),12(ˆH(y)+ˆH(b))}=inf{ˆC(x),ˆC(y)}. |
Hence, ˆC is an m-pF subalgebra of X. It follows from Theorem 3.5 that ˆC+∈NO(X) where ˆC+ is defined by ˆC+(x)=ˆC(x)+ˆCc(0), ∀x∈X. Clearly, ˆC+(x)≥ˆH(x), ∀x∈X. Note that
ˆC+(b)=ˆC(b)+ˆCc(0))=ˆC(b)+ˆ1−ˆC(0)=12(ˆH(b)+ˆH(b))+ˆ1−12(ˆH(0)+ˆH(b))=12(ˆH(b)+ˆ1)>ˆH(b) |
and ˆC+(b)<ˆ1=ˆC+(0). Hence, ˆC+ is a non-constant and ˆH is not a maximal element of NO(X). This is a contradiction. This completes the proof.
Definition 3.13. Let ˆH be an m-pF subalgebra of X. Then, ˆH is said to be maximal if
(ⅰ) ˆH is non-constant.
(ⅱ) ˆH+ is a maximal element of the poset (NO(X),⊆).
Theorem 3.14. A maximal m-pF subalgebra ˆH of X is normal and takes the values ˆ0=(0,0,...,0) and ˆ1=(1,1,...,1).
Proof. Let ˆH be a maximal m-pF subalgebra of X. Then, ˆH+ is a non-constant maximal element of the poset (NO(X),⊆). It follows that from Theorem 3.12 that ˆH+ takes only the values ˆ0 and ˆ1. Note that ˆH+(x)=ˆ1 if and only if ˆH(x)=ˆH(0), and ˆH+(x)=ˆ0 if and only if ˆH(x)=ˆH(0)−ˆ1. By Corollary 3.6, we have ˆH(x)=ˆ0 i.e., ˆH(0)=ˆ1. Hence, ˆH is normal, and clearly ˆH+=ˆH. This completes the proof.
Theorem 3.15. If ˆH is a maximal m-pF subalgebra of X, then ˆHKˆH=ˆH.
Proof. Clearly, ˆHKˆH⊆ˆH and ˆHKˆH takes only the values ˆ0 and ˆ1. Let x∈X. If ˆH(x)=0, then obviously ˆH⊆ˆHKˆH. If ˆH(x)=1, then x∈KˆH, and so ˆHKˆH(x)=ˆ1. This shows that ˆH⊆ˆHKˆH.
Theorem 3.16. For a maximal m-pF subalgebra ˆH of X, KˆH is a maximal subalgebra of X.
Proof. Let KˆH be a proper subalgebra of X because ˆH is non-constant. Let M be a subalgebra of X such that KˆH⊆M. Noticing that for every subalgebras M and N of X, M⊆N if and only if ˆHM⊆ˆHN, then we obtain ˆH=ˆHKˆH⊆ˆHM. Since ˆH and ˆHM are normal and since ˆH=ˆH+ is a maximal element of NO(X), we have that either ˆH=ˆHM or ˆHM=ˆ1, where ˆ1:X→[0,1]m is an m-pF set defined by ˆ1(x)=(1,1,...,1)=ˆ1 for all x∈X. The other case implies that M=X. If ˆH=ˆHM, then KˆH=KˆHM=M by Theorem 3.3. This proves that KˆH is a maximal subalgebra of X. This completes the proof.
Definition 3.17. A normal m-pF subalgebra ˆH of X is said to be completely normal if there exists x∈X such that ˆH(x)=ˆ0. Denote by CN(X) the set of all completely normal m-pF subalgebra of X.
We note that CN(X)⊆NO(X) and the restriction of partial ordering ⊆ of NO(X) gives a partial ordering on CN(X).
Theorem 3.18. A non-constant maximal element of (NO(X),⊆) is also a maximal element of (CN(X),⊆).
Proof. Let ˆH be a non-constant maximal element of (N(X),⊆). By Theorem 3.12, ˆH takes only the values ˆ0 and ˆ1. Now, ˆH(0)=ˆ1 and ˆH(x)=ˆ0 for some x∈X. Thus, ˆH∈CN(X). Suppose there exists ˆC∈CN(X) such that ˆH⊆ˆC. It follows that ˆH⊆ˆC in NO(X). Since ˆH is maximal in (NO(X),⊆) and since ˆC is non-constant, therefore ˆH=ˆC. Hence, ˆH is maximal element of (CN(X),⊆). This completes the proof.
Theorem 3.19. Every maximal m-pF subalgebra of X is completely normal.
Proof. Let ˆH be a maximal m-pF subalgebra of X. Then, by Theorem 3.14, ˆH is normal and ˆH=ˆH+ takes only the values ˆ0 and ˆ1. Since ˆH is a non constant, it follows that ˆH(0)=ˆ1 and ˆH(x)=ˆ0 for some x∈X. Hence, ˆH is completely normal. This completes the proof.
Definition 4.1. For an endomorphism Ψ of X and an m-pF set ˆH in X. We define a new m-pF set ˆH[Ψ]:X→[0,1]m by ˆH[Ψ](x)=ˆH(Ψ(x)) for all x∈X.
Theorem 4.2. If ˆH is an m-pF subalgebra of X, then so is ˆH[Ψ].
Proof. Let x,y∈X. Then,
ˆH[Ψ](x∗y)=ˆH(Ψ(x∗y))=ˆH(Ψ(x)∗Ψ(y))≥inf{ˆH(Ψ(x)),ˆH(Ψ)(y)}=inf{ˆH[Ψ](x),ˆH[Ψ](y)} |
Hence, ˆH[Ψ] is an m-pF subalgebra of X.
Example 4.1. Consider a BCK-algebra X={0,a,b} with the Cayley table which is given in Table 2.
∗ | 0 | a | b |
0 | 0 | 0 | 0 |
a | a | 0 | 0 |
b | b | b | 0 |
Let ˆH:X→[0,1]m be an m-pF set in X defined by:
ˆH(x)={ˆγ=(γ1,γ2,...,γm), if x=0,aˆδ=(δ1,δ2,...,δm), if x=b, |
where ˆγ>ˆδ. By routine computations, we can verify that ˆH is an m-pF subalgebra of X. There are four endomorphisms of X as follows:
Ψ1:0→0,a→0,b→0,Ψ2:0→0,a→0,b→a,Ψ3:0→0,a→0,b→b,Ψ4:0→0,a→a,b→b. |
By Theorem 4.2, we have ˆH[Ψi] for i=1,2,3,4 are m-pF subalgebras.
Definition 4.3. A subalgebra K of X is called characteristic if Ψ(K)=K for all Ψ∈Aut(X), where Aut(X) is the set of all automorphisms of X.
Definition 4.4. An m-pF subalgebra ˆH of X is called an m-pF characteristic if ˆH[Ψ](x)=ˆH(x) for all x∈X and Ψ∈Aut(X).
Example 4.2. In Example 4.1, Ψ4 is an automorphism of X. It is clear that Ψ4(ˆH(x))=ˆH(x) for all x∈X. Therefore, ˆH is characteristic. Also, ˆH[Ψ4](x)=ˆH(Ψ4(x))=ˆH(x) for all x∈X. Hence, ˆH is an m-pF characteristic.
Lemma 4.5. Let ˆH be an m-pF subalgebra of X and let x∈X. Then, ˆH(x)=ˆt if and only if x∈ˆHˆt and x∉ˆHˆs for all ˆs>ˆt.
Proof. Let ˆH be an m-pF subalgebra of X and let x∈X. Suppose ˆH(x)=ˆt, so that x∈ˆHˆt. If possible, let x∈ˆHˆs for ˆs>ˆt. Then, ˆH(x)≥ˆs>ˆt. this contradicts the fact that ˆH(x)=ˆt, concluding that x∉ˆHˆs for all ˆs>ˆt.
Conversely, let x∈ˆHˆt and x∉ˆHˆs for all ˆs>ˆt. Now, let x∈ˆHˆt⇒ˆH(x)≥ˆt, since x∉ˆHˆs for all ˆs>ˆt. Therefore, ˆH(x)=ˆt.
Theorem 4.6. For an m-pF subalgebra ˆH of X, the following are equivalent:
(1) ˆH is an m-pF characteristic.
(2) Each level cut subset ˆHˆt is characteristic subalgebra.
Proof. Suppose ˆH is an m-pF characteristic and let ˆt∈Im(ˆH),Ψ∈Aut(X) and x∈ˆHˆt. Then,
ˆH[Ψ](x)=ˆH(Ψ(x))=ˆH(x)≥ˆt, |
i.e., ˆH(Ψ(x))≥ˆt. Thus, Ψ(x)∈ˆHˆt, i.e., Ψ(ˆHˆt)⊆ˆHˆt. Now, let x∈ˆHˆt and y∈X be such that Ψ(y)=x. Then,
ˆH(y)=ˆH[Ψ](y)=ˆH(Ψ(y))=ˆH(x). |
Hence, y∈ˆHˆt, so that x=Ψ(y)∈Ψ(ˆHˆt). Consequently, ˆHˆt⊆Ψ(ˆHˆt). Therefore, ˆHˆt=Ψ(ˆHˆt) and ˆH is characteristic.
Conversely, suppose that each level cut subset ˆHˆt is characteristic subalgebra and let x∈X,Ψ∈Aut(X) and ˆH(x)=ˆt. Then, by Lemma 4.5, x∈ˆHˆt and x∉ˆHˆs for all ˆs>ˆt. Thus, Ψ(x)∈Ψ(ˆHˆt)=ˆHˆt, so that ˆH(Ψ(x))≥ˆt. Let ˆt1=ˆH[Ψ](x) and suppose ˆt1>ˆt. Then, Ψ(x)∈ˆHˆt1=Ψ(ˆHˆt1), which implies from the injectivity of Ψ that x∈ˆHˆt1, a contradiction. Thus, ˆH(Ψ(x))=ˆH(x). Therefore, ˆH[Ψ] is an m-pF characteristic.
The idea of m-pF algebraic structures plays a significant rule in several fields of applied mathematics, computer sciences and information systems. In [9], we have already introduced the concepts of m-pF subalgebras and ideals of BCK/BCI-algebras and investigated some of their related properties. In this study, as a continuation of [9], we have introduced the concepts of normal m-pF subalgebras, maximal m-pF subalgebras and completely normal m-pF subalgebras in BCK/BCI-algebras and discussed some of their properties. We have proved that any non-constant normal m-pF subalgebra which is a maximal element of (NO(X),⊆) takes only the values ˆ0=(0,0,...,0) and ˆ1=(1,1,...,1), and every maximal m-pF subalgebra is completely normal. Moreover, we have stated an m-pF characteristic subalgebra in BCK/BCI-algebras. In the future, the results of this work can be further expanded to several algebraic structures, for instance UP-algebras, BRK-algebras, KU-algebras, etc.
We declare that we have no conflict of interest.
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0 | 0 | 0 | 0 | 0 |
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0 | 0 | 0 | 0 |
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