# Hybrid Structures Applied to Subalgebras of BCH-Algebras

Authors: G. Muhiuddin; D. Al-Kadi; W.A. Khan; C. Jana
Journal: Security and Communication Networks
 (2021)

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## Abstract

The algebraic structures have many applications in coding theory, cryptography, and security networks. In this paper, the notion of hybrid subalgebras ofBCH-algebras is introduced and related properties are investigated. Moreover, some characterizations of hybrid subalgebras of BCH-algebras are given. Furthermore, we state and prove some theorems in hybrid subalgebras of BCH-algebras. The homomorphic images and inverse images of fuzzy BCH-subalgebras are studied and discussed.


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## Body

## 1. Introduction

The notions ofBCK/BCI-algebras were initiated by Imai and Iséki in 1966. A number of research papers have been produced on the theory of BCK/BCI-algebras. Hu and Li [1, 2] introduced the notion of a BCH-algebra as a generalization of BCK/BCI-algebras and subsequently gave examples of proper BCH-algebras and studied some properties. Certain other properties of BCH-algebras have been studied by Ahmad [3], Dudek and Thomys [4], Chaudhry [5], Roh et al. [6, 7], Chaudhry et al. [8], and Dar et al. [9], and Smarandache structure has been applied to BCH-algebra [10].Fuzzy sets, which were introduced in the 1960s by Zadeh [11], have been developed considerably by many research studies. Molodtsov introduced the concept of soft set [12] and pointed out several directions for its applications (for more details, see [12–15]). This concept was applied to BCH-algebras introducing soft BCH-algebras which were studied in [16]. Moreover, the fuzzy set theoretical approach to BCH-algebras was extensively investigated by many researchers on different aspects. For example, fuzzy n-fold ideals [17], fuzzy closed ideals and fuzzy filters [18], filters based on bipolar-valued fuzzy sets [19], and cubic subalgebras [20].Jun et al. [21] combined the concepts of fuzzy sets and soft sets, introduced the notion of hybrid structure in a set of parameters over an initial universe set, and investigated several properties. They also introduced the concepts of hybrid linear space, hybrid subalgebra, and hybrid field. Moreover, hybrid structure applications have been studied in semigroups (see [22–25] and references there in), and recently, hybrid ideals of BCK = BCI-algebras were studied in [26–29]. For more important terminologies, the readers are referred to [30–36].In the present paper, we present an application of fuzzy set theory to an algebraic structure called, BCH-algebra. As we know it algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, and the like. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. The objective of this study is to introduce the concept of hybrid subalgebras of BCH-algebras. The notion of hybrid subalgebras of BCH-algebras is defined, and related properties are investigated. This paper is organized as follows: in Section2, we recall some definitions related to the subject. In Section 3, the concepts and operations of hybrid subalgebras of BCH-algebras are introduced and their properties are discussed in detail. Furthermore, some properties of hybrid subalgebras of BCH-algebras under homomorphisms are explored.


## 2. Preliminaries

This section begins with the following definitions and properties that will be needed in the sequel.An algebraL,∗,0 of type 2,0 is called a BCH-algebra [1] if it satisfies the following axioms, for all q,l,n∈L:(1)
q∗q=0,(2)
q∗l=0andl∗q=0implyq=l,(3)
q∗l∗n=q∗n∗l.AnyBCH-algebra L satisfies the following axioms:(i)
q∗0=q,(ii)
q∗q∗l∗l=0,(iii)
0∗q∗l=0∗q∗0∗l,(iv)
0∗0∗0∗q=0∗q,(v)
q≤l implies 0∗q=0∗l,for all q,l,n∈L [8].A nonempty subsetS of a BCH-algebra L is called a subalgebra of L if q∗l∈S, for all q,l∈S.We now review some fuzzy logic concepts as follows.LetL be the collection of objects denoted generally by q. Then, a fuzzy set [11] A in L is defined as A=q,μA∞q:q∈L where μAq is called the membership degree of q in A and 0≤μAq≤1.Furthermore, we collect some basic notions and results on hybrid structures due to Jun et al. [21]. Let I be the unit interval, L be a set of parameters, and ℙU be the power set of an initial universe set U.Definition 1 (see [21]).
A hybrid structure inL over U is a mapping:(1)h˜η≔h˜;η:L⟶ℙU×I;q↦h˜q;ηq,where h˜:L⟶ℙU and η:L⟶I are mappings.Definition 2 (see [21]).
For hybrid structuresh˜η and g˜μ in L over U, the hybrid intersection denoted by h˜η⋒g˜μ is a hybrid structure:(2)h˜η⋒g˜μ:L⟶ℙU×I,q↦h˜∩˜g˜q,η∨μq,where(3)h˜∩˜g˜:L⟶ℙU,q↦h˜q∩g˜q,η∨μ:L⟶I,q↦∨ηq,ηq.Definition 3 (see [21]).
LetL be a BCK/BCI-algebra. For a hybrid structure h˜η in L over U, h˜η is said to be a hybrid subalgebra of L if the following statements are valid:(4)∀q,l∈Lh˜q∗l⊇h˜q∩h˜l,ηq∗l≤∨ηq,ηl.Lemma 1 (see [21]).
Every hybrid subalgebrah˜η of a BCK/BCI-algebra L over U satisfies(5)∀q∈Lh˜0⊇h˜q,η0≤ηq.


## 3. Hybrid Subalgebras of BCH-Algebras

In this section, we obtain our main results. Throughout our discussion,L will denote a BCH-algebra unless otherwise mentioned.Definition 4.
LetL be a BCH-algebra. A hybrid structure h˜η=h˜;η in L over U is called a hybrid subalgebra of L over U if the following assertions are valid:(6)∀q,l∈Lh˜q∗l⊇h˜q∩h˜l,ηq∗l≤∨ηq,ηl.
Let us illustrate this definition using the following example.Example 1.
Let the initial universe be the setU=u1,u2,u3,u4,u5 and L = 0,l1,l2,l3,l4 be a BCH-algebra with the Cayley table (Table 1).
Leth˜η=h˜;η be a hybrid structure in L over U which is given in Table 2.
It can be easily verified thathη˜ is a hybrid subalgebra of L over U.Table 1
Cayley table of the binary operation∗.
∗0l1l2l3l4000l4l3l2l1l10l4l3l2l2l2l20l4l3l3l3l3l20l4l4l4l4l3l20Table 2
Table representation of the hybrid structurehη˜.
Lh˜η0U0.3l1u1,u2,u3,u40.3l2u3,u40.9l3u3,u40.9l4u3,u40.9Proposition 1.
Every hybrid subalgebrah˜η=h˜;η in L over U satisfies the following assertions:(7)∀q∈Lh˜q⊆h˜0,ηq≥η0.Proof.
For allq∈L, we have h˜0=h˜q∗q⊇h˜q∩h˜q=h˜q and η0=ηq∗q≤∨ηq,ηq=ηq.Proposition 2.
Leth˜η=h˜;η be a hybrid subalgebra in L over U. Then, the following assertions are equivalent:(1)
∀q,l∈Lh˜q∗l⊇h˜l,ηq∗l≤ηq.(2)
∀q∈Lh˜0=h˜q,η0=ηq.Proof.
If we takel=0 in (1), then h˜q⊇h˜0 and ηq≤η0 for all q∈L. Combining this and Proposition 1, we have h˜0=h˜q,η0=ηq for all q∈L.
Conversely, assume that (2) is valid. Then,(8)h˜l=h˜0∩h˜l=h˜q∩h˜l⊆h˜q∗l,ηl=∨η0,ηl=∨ηq,ηl≥ηq∗l,for all q,l∈L.Proposition 3.
Leth˜η=h˜;η be a hybrid subalgebra in L over U. Then, for all q∈L, h˜q⊆h˜0∗q and ηq≥η0∗q.Proof.
Letq∈L. Then,(9)h˜0∗q⊇h˜0∩h˜q=h˜q∗q∩h˜q⊇h˜q∩h˜q∩h˜q=h˜q,η0∗q≤∨η0,ηq=∨ηq∗q,ηq≤∨∨ηq,ηq,ηq=ηq.
This completes the proof.For any hybrid structureh˜η=h˜;η in L over U, we consider two level sets:(10)hη˜α≔q∈L|α⊆h˜q,hη˜t≔q∈L|ηq≤t,where α∈ℙU and t∈I.Theorem 1.
LetL be a BCH-algebra. For a hybrid structure h˜η=h˜;η in L over U, the following are equivalent:(1)
h˜η is a hybrid subalgebra of L over U.(2)
For anyα∈ℙU and t∈I, the nonempty sets hη˜α and hη˜t are subalgebras of L.Proof.
(1)⟹(2). Suppose that h˜η is a hybrid subalgebra of L. Let q,l∈hη˜α. Then, α⊆h˜q and α⊆h˜l. It follows that α⊆h˜q∩h˜l⊆h˜q∗l and so q∗l∈hη˜α. Hence, hη˜α is a subalgebra of L. Also, let q,l∈hη˜t. Then, ηq≤t and ηl≤t. It follows that ηq∗l≤∨ηq,ηl≤t and so q∗l∈hη˜t. Hence, hη˜t is a subalgebras of L.(2)⟹(1). Let for any α∈ℙU and t∈I, the nonempty sets hη˜α and hη˜t are subalgebras of L. For contradiction, let q0,l0∈L such that h˜q0∗l0⊂h˜q0∩h˜l0. Let h˜q0=β1,h˜l0=β2 and h˜q0∗l0=α. Then, α⊂β1∩β2. Let us consider α1|h˜q0∗l0⊂α1⊂h˜q0∩h˜l0. We get that h˜q0∗l0=α⊂α1⊂β1∩β2, and so q0∗l0∉hη˜α which is a contradiction. Thus, h˜q∗l⊇h˜q∩h˜l for all q,l∈L. Also, let q0,l0∈L such that ηq0∗l0>∨ηq0,ηl0. Let ηq0=η1,ηl0=η2 and ηq0∗l0=t. Then, t>∨η1,η2. Let us consider t1|ηq0∗l0>t1>∨ηq0,ηl0. We get that t>t1>∨η1,η2. Hence, ∨η1,η2<t1<t=ηq0∗l0, and so q0∗l0∉hη˜t which is a contradiction. Thus, ηq∗l≤∨ηq,ηl for all q,l∈L. Hence, h˜η is a hybrid subalgebra of L.Next, we defineHh˜=q∈L|h˜q=h˜0 and Hη=q∈L|ηq=η0. These two sets are also subalgebras of a BCH-algebra L over U.Proposition 4.
Leth˜η=h˜;η be a hybrid subalgebra in L over U. Then, the sets Hh˜ and Hη are subalgebras of L over U.Proof.
Letq,l∈Hh˜. Then, h˜q=h˜0=h˜l and so h˜q∗l⊇h˜q∩h˜l=h˜0. By using Proposition 1, we know that h˜q∗l=h˜0. Consequently, q∗l∈Hh˜.
Letq,l∈Iη. Then, ηq=η0=ηl and so ηq∗l≤∨ηq,ηl=η0. Again by Proposition 1, we know that ηq∗l=η0 or equivalently q∗l∈Hη.
Hence, the setsHh˜ and Hη are subalgebras of L over U.Proposition 5.
Lethη˜=h˜;η be a hybrid structure in L over U where h˜:L⟶ℙU and η:L⟶I are mappings given by(11)q↦α1,if0∗0∗q=q,α2,otherwise,such thatα1⊂α2∈ℙU.q↦t1,if0∗0∗q=q,t2,otherwise,such thatt1<t2∈I,for q∈L. Then, hη˜ is a hybrid subalgebra of L.Proof.
Letq,l∈L. If 0∗0∗q=q and 0∗0∗l=l, then h˜q=h˜l=α1,ηq=ηl=t1. Since, q∗l∗0∗0∗q∗l=0 (using condition (3), property (iv), and property (ii)) and 0∗0∗q∗l∗q∗l=0 (using property (iii) and condition (1)). This implies that q∗l=0∗0∗q∗l, by condition (2). Thus, h˜q∗l=α1=h˜q∩h˜l and ηq∗l=t1=∨ηq,ηl. If 0∗0∗q≠q or 0∗0∗l≠l, then h˜q=α2 or h˜l=α2 and ηq=t2 or ηl=t2. Then, h˜q∗l⊇α2=h˜q∩h˜l and ηq∗l≤t2=∨ηq,ηl. Thus, hη˜ is a hybrid subalgebra.Proposition 6.
Let∅≠Q⊆L and let hη˜=h˜;η be a hybrid structure of L over U where h˜:L⟶ℙU and η:L⟶I are mappings given by(12)q↦α1,ifq∗m=0∗m∗0∗q,α2,otherwise,such thatα1⊂α2∈ℙU,q↦t1,ifq∗m=0∗m∗0∗q,t2,otherwise,such thatt1<t2∈I,for q∈L, m∈Q. Then, hη˜ is a hybrid subalgebra of L.Proof.
Letq,l∈L. If there exists m∈Q such that q∗m=0∗m∗0∗q and l∗m=0∗m∗0∗l, then h˜q=h˜l=α1,ηq=ηl=t1. Using condition (3), property (ii), and property (iii), we have q∗l∗m=0∗m∗0∗q∗l. If there exists m∈Q such that either q∗m≠0∗m∗0∗q or l∗m≠0∗m∗0∗l, then h˜q=α2 or h˜l=α2 and ηq=t2 or ηl=t2. It follows that h˜q∗l⊇α2=h˜q∩h˜l and ηq∗l≤t2=∨ηq,ηl. Therefore, hη˜ is a hybrid subalgebra of L.Proposition 7.
Letg˜μ=g˜;μ be a nonempty subset in L over U and h˜η=h˜;η be a hybrid structure of L over U defined by(13)h˜q≔α,ifq∈g˜μ,β,otherwise,ηq≔γ,ifq∈g˜μ,δ,otherwise,for all α,β∈ℙU and γ,δ∈0,1. Then, h˜η is a hybrid subalgebra of L over U if and only if g˜μ is a subalgebra of L over U. Moreover, Hh˜=g˜μ=Hη.Proof.
Leth˜η be a hybrid subalgebra of a BCH-algebra L over U. Let q,l∈L such that q,l∈g˜μ. Then, we have h˜q∗l⊇h˜q∩h˜l=α∩α=α and ηq∗l≤∨ηq,ηl=∨γ,γ. Hence, we have proved that q∗l∈g˜μ. Thus, g˜μ is indeed a subalgebra of L.
Conversely, suppose thatg˜μ is a subalgebra of L. Let q,l∈L. Consider the following two cases:Case (i): ifq,l∈g˜μ, then q∗l∈g˜μ. Thus, h˜q∗l=α=h˜q∩h˜l and ηq∗l=γ=∨ηq,ηl.Case (ii): ifq∉g˜μ or l∉g˜μ, then h˜q∗l⊇β=h˜q∩h˜l and ηq∗l≤δ=∨ηq,ηl.
Hence,h˜η is a hybrid subalgebra of L.
Also,Hh˜=q∈L:h˜q=h˜0=q∈L:h˜q=α=g˜μ and Hη=q∈L:ηq=η0=q∈L:ηq=γ=g˜μ.Proposition 8.
Leth˜η=h˜;η be a hybrid subalgebra in L over U. Then, the set Ω≔q∗l∈L|h˜q∗l∩α≠∅,ηq∗l≤t is a subalgebra in L over U, for ∅≠α∈ℙU, t∈I.Proof.
Letq,l∈L such that q,l∈Ω. Thus, h˜q∩α≠∅, h˜l∩α≠∅ and ηq≤t, ηl≤t. It follows that h˜q∩α∩h˜l∩α=h˜q∩h˜l∩α⊆h˜q∗l∩α≠∅ and ηq∗l≤∨ηq,ηl≤t. Thus, q∗l∈Ω and so Ω is a subalgebra in L.Proposition 9.
Leth˜η=h˜;η be a hybrid structure in L over U. Then, h˜η is a hybrid subalgebra of L over U if and only if for α∈ℙU, t∈I, and the sets h˜ηα≔q∗l∈L|h˜q∗l⊇α and h˜ηt≔q∗l∈L|ηq∗l≤t are subalgebras in L.Proof.
“⇒”. Let h˜η=h˜;η be a hybrid subalgebra in a BCH-algebra L over U and consider the sets h˜ηα≔q∈L|h˜q⊇α and h˜ηt≔q∈L|ηq≤t. Now, let q,l∈L such that q,l∈h˜ηα and q,l∈h˜ηt. Thus, h˜q⊇α,h˜l⊇α and ηq≤t,ηl≤t. Then, from (3), we have h˜q∗l⊇h˜q∩h˜l⊇α and ηq∗l≤∨ηq,ηl≤t. That is, h˜q∗l⊇α and ηq∗l≤t and so q∗l∈h˜ηα and q∗l∈h˜ηt. Hence, h˜ηα and h˜ηt are subalgebras in L.
“⇐”. For α∈ℙU, t∈I, let h˜ηα≔q∗l∈L|h˜q∗l⊇α and h˜ηt≔q∗l∈L|ηq∗l≤t be subalgebras in L. Let q∈L such that h˜q=α and ηq=t. Suppose for contradiction that(14)h˜q∗l⊂h˜q∩h˜l,ηq∗l>∨ηq,ηl.
Takeα and t such that(15)h˜q∗l⊂α⊂h˜q∩h˜l,ηq∗l>t>∨ηq,ηl.
This means that bothh˜ηα and h˜ηt are not subalgebras which contradicts the assumption. Thus, h˜q∗l⊇h˜q∩h˜landηq∗l≤∨ηq,ηl. Hence, h˜η is a hybrid subalgebra of L.For any hybrid structureh˜η in L over U, let hη∗˜≔h∗˜;η∗ be a hybrid structure in L over U defined by(16)h∗˜:L⟶ℙU,q↦h˜q,ifq∈hη˜α,β,otherwise,η∗:L⟶I,q↦ηq,ifq∈hη˜s,t,otherwise,where α,β∈ℙU and s,t∈I with β⊊h˜q and t>ηq.Proposition 10.
LetL be a BCH-algebra. If h˜η is a hybrid subalgebra in L over U, then so is hη∗˜.Proof.
Assume thath˜η is a hybrid subalgebra of a BCH-algebra L over U. Then, h˜ηα and h˜ηt are subalgebras of L for all α∈ℙU and t∈I provided that they are nonempty by Proposition 9. Let q,l∈L. If q,l∈h˜ηα, then q∗l∈h˜ηα. Thus,(17)h∗˜q∗l=h˜q∗l⊇h˜q∩h˜l=h∗˜q∩h∗˜l.
Ifq∉hη˜α or l∉hη˜α, then h∗˜q=β or h∗˜l=β. Hence,(18)h∗˜q∗l⊇β=h∗˜q∩h∗˜l.
Now, ifq,l∈h˜ηs, then q∗l∈h˜ηs. Thus,(19)η∗q∗l=ηq∗l≤∨ηq,ηl=∨η∗q,η∗l.
Ifq∉hη˜s or l∉hη˜s, then η∗q=t or η∗l=t. Hence,(20)η∗q∗l≤t=∨η∗q,η∗l.
Therefore,hη∗˜ is a hybrid subalgebra of L over U.
The converse of Proposition10 may not true in general.Example 2.
LetL = 0,l1,l2,l3 be a BCH-algebra with the Cayley table (Table 3) and U=u1,u2,u3,u4,u5,u6,u7,u8,u9 be an initial universe set.
Leth˜η=h˜;η be a hybrid structure in L over U which is given in Table 4.Let h˜ηα=hη˜s=0,l1, where α=u1,u3,u5,u7,u9 and s=0.6. Define the hybrid structure hη∗˜≔h∗˜;η∗ by Table 5.
It can be easily verified thathη∗˜ is a hybrid subalgebra of L. Moreover, hη˜ is not hybrid subalgebra of L as h˜l1∗l2=h˜l3=u7⊇u5=h˜l1∩h˜l2=u1,u3,u5,u7,u9∩u2,u5,u8.Table 3
Cayley table of the binary operation∗.
∗0l1l2l300l1l2l3l1l10l3l2l2l2l30l1l3l3l2l10Table 4
Table representation of the hybrid structurehη˜.
Lh˜η0U0.5l1u1,u3,u5,u7,u90.6l2u2,u5,u80.8l3u70.8Table 5
Table representation of the hybrid structurehη∗˜.
Lh∗˜η∗0U0.5l1u1,u3,u5,u7,u90.6l2∅0.9l3∅0.9Proposition 11.
Ifh˜η=h˜;η and g˜μ=g˜;μ are two hybrid subalgebras in L over U, then the hybrid intersection h˜η⋒g˜μ is also a hybrid subalgebra of L.Proof.
Letq,l∈L. Then,(21)h˜∩˜g˜q∗l=h˜q∗l∩g˜q∗l⊇h˜q∩h˜l∩g˜q∩g˜l=h˜q∩g˜q∩h˜l∩g˜l=h˜∩˜g˜q∩h˜∩˜g˜l,η∨μq∗l=∨ηq∗l,μq∗l≤∨∨ηq,ηl,∨μq,μl=∨∨ηq,μq,∨ηq,μl=∨η∨μq,η∨μl.
Consequently,h˜η⋒g˜μ is a hybrid subalgebra of L.Definition 5.
Leth˜η=h˜;η be a hybrid structure of a BCH-algebra L over U. Then, the “power-m” operation on a hybrid structure of a BCH-algebra L over U is defined as follows:(22)h˜ηm≔h˜m;ηm,where m is any nonnegative integer.Proposition 12.
Ifh˜η=h˜;η is a hybrid subalgebra in L over U, then h˜ηm is a hybrid subalgebra in L over U.Proof.
Leth˜η=h˜;η be a hybrid subalgebra in a BCH-algebra L over U and let q∈L. Then,(23)h˜m0=h˜0m=h˜q∗qm⊇h˜q∩h˜qm=h˜qm∩h˜qm=h˜mq∩h˜mq=h˜mq,ηm0=η0m=ηq∗qm≤∨ηq,ηqm=∨ηqm,ηqm=∨ηmq,ηmq=ηmq.Let q,l∈L. Then,(24)h˜mq∗l=h˜q∗lm⊇h˜q∩h˜lm=h˜qm∩h˜lm=h˜mq∩h˜mlηmq∗l=ηq∗lm≤∨ηq,ηlm=∨ηqm,ηlm=∨ηmq,ηml.
Hence,h˜ηm is a hybrid subalgebra of L.Definition 6.
Leth˜η=h˜;η be a hybrid structure of a BCH-algebra L over U. Then, the “ρ-multiply” operation on a hybrid structure of a BCH-algebra L over U is defined as(25)ρh˜η≔ρh˜,ρη,where ρ is any nonnegative integer.Proposition 13.
Ifh˜η=h˜;η is a hybrid subalgebra in L over U, then ρh˜η is a hybrid subalgebra of L over U.Proof.
Leth˜η=h˜;η be a hybrid subalgebra of a BCH-algebra L over U and let ρ be any nonnegative integer. Then, for any q∈L, we have(26)ρh˜0=ρh˜q∗q⊇ρh˜q∩h˜q=ρh˜q∩ρh˜q=ρh˜q,ρη0=ρηq∗q≤ρ∨ηq,ηq=∨ρηq,ρηq=ρηq.
Letq,l∈L. Then,(27)ρh˜q∗l⊇ρh˜q∩h˜l=ρh˜q∩ρh˜l,ρηq∗l≤ρ∨ηq,ηl=∨ρηq,ρηl.
Hence,ρh˜η is a hybrid subalgebra of L over U.Letφ be a mapping from the set L into the set Q. Let g˜μ be a hybrid structure of a BCH-algebra L over U. Then, the preimage of g˜μ is defined as φ−1g˜μ=φ−1g˜,φ−1μ in L with the membership function and nonmembership function given by φ−1g˜q=g˜φq and φ−1μq=μφq. It can be shown that φ−1g˜μ is a hybrid structure of a BCH-algebra L over U.Definition 7.
A mappingφ:L⟶Q is called a homomorphism of a BCH-algebra if φq∗l=φq∗φl, for all q,l∈L. Note that if φ:L⟶Q is a homomorphism of a BCH-algebra, then φ0=0.Proposition 14.
Letφ:L⟶Q be a homomorphism of BCH-algebras. If g˜μ=g˜,μ is a hybrid subalgebra of a BCH-algebra Q over U, then the preimage φ−1g˜μ=φ−1g˜,φ−1μ of g˜μ under φ is a hybrid subalgebra of a BCH-algebra L over U.Proof.
Assume thatg˜μ=g˜;μ is a hybrid subalgebra of a BCH-algebra Q over U and let q,l∈L. Then,(28)φ−1g˜q∗l=g˜φq∗l=g˜φq∗φl⊇g˜φq∩g˜φl=φ−1g˜q∩φ−1g˜l,φ−1μq∗l=μφq∗l=μφq∗φl≤∨μφq,μφl=∨φ−1μq,φ−1μl.
Therefore,φ−1g˜μ=φ−1g˜,φ−1μ is a hybrid subalgebra of L.


## 4. Conclusion

The present work is devoted to the study of hybrid subalgebras ofBCH-algebras introduced, and related properties are investigated. Furthermore, some characterizations of hybrid subalgebras of BCH-algebras are given. Also, we stated and proved some theorems in hybrid subalgebras of BCH-algebras. Finally, the homomorphic images and inverse images of fuzzy BCH-subalgebras are studied and discussed. To extend these results, one can further study these notions on different algebras such as rings, hemirings, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, EQ-algebras, d-algebras, Q-algebras, and lattice implication algebras. Some important issues for future work are as follows: (1) to develop strategies for obtaining more valuable results and (2) to apply these notions and results for studying related notions in other algebraic (hybrid) structures.



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*Source: 8960437-2021-07-16.xml*