# Hybrid Structures Applied to Ideals in BCI-Algebras

Authors: G. Muhiuddin; D. Al-Kadi; A. Mahboob
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, the notion of hybrid structure is applied to the ideal theory in BCI-algebras. In fact, we introduce the notions of hybridp-ideal, hybrid h-ideal, and hybrid a-ideal in BCI-algebras and investigate their related properties. Furthermore, we show that every hybrid p-ideal (or h-ideal or a-ideal) is a hybrid ideal in a BCI-algebra but converse need not be true in general and in support, and we exhibit counter examples for each case. Moreover, we consider characterizations of hybrid p-ideal, hybrid h-ideal, and hybrid a-ideal in BCI-algebras.


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## Body

## 1. Introduction

Imai and Iséki [1, 2] initiated the study of “BCK/BCI-algebras” in 1966 as a generalization of the notions of set-theoretical difference and propositional calculus. A great deal of literature has been developed on the theory of BCK/BCI-algebras since then, in particular, more focus has been placed on the “ideal theory” of BCK/BCI-algebras. In BCK/BCI-algebras, different kinds of ideals on different aspects have been studied (see, for example, [3–6]).Fuzzy sets, introduced by Zadeh [7], deal with potential uncertainties, related to the imprecision of states, perceptions, and preferences. Molodtsov [8] proposed the concept of a “soft set” as a new mathematical framework for dealing with uncertainties, free of the difficulties that have disrupted normal theoretical approaches. Molodtsov pointed out a number of ways to develop soft sets. Molodtsov applied soft set theory in a variety of ways, such as smoothness of functions, game theory, operational research, Riemann integration, Perron integration, probability, and measurement theory (see [8–10]). Algebraic structures such as BCK/BCI-algebra [11], d-algebras [12], group [13], semigroup [14], ring [15], semiring [16], and decision-making [17, 18] are theoretically applied by soft set theory. Muhiuddin et al. (see, for example, [19–23]) investigated the fuzzy set theoretical approach to the BCK/BCI-algebras on various aspects. Further concepts related to this analysis in different fields have also been studied in [24–34].In a system of parameters, Jun et al. [35] proposed the concept of “hybrid structure” over the initial universe set by merging fuzzy sets and soft sets. The idea of a hybrid subalgebra, a hybrid field, and a hybrid linear space was introduced with this notion. The hybrid structure theory and its applications to BCK/BCI-algebras and semigroups have recently been studied in (see [36–40] and references).The objective of this paper is to introduce the notions of hybridp-ideal, hybrid h-ideal, and hybrid a-ideal in BCI-algebras and investigate their related properties. Furthermore, we show that hybrid (p-ideal, h-ideal, and a-ideal) are all hybrid ideals, but converse need not be true in general and in support, and we exhibit counter examples. Also, we provide conditions for a hybrid p-ideal (or hybrid h-ideal or hybrid a-ideal) to be a hybrid ideal.


## 2. Preliminaries

An algebraZ=Z;∗,0 of type 2,0 is a BCI-algebra if it satisfies for all s,t,r∈Z:(1)K1s∗t∗s∗r∗r∗t=0,K2s∗s∗t∗t=0,K3s∗s=0,K4s∗t=0andt∗s=0⇒s=t.If aBCI-algebra Z satisfies(2)K50∗s=0,then Z is a BCK−algebra.AnyBCK/BCI-algebra Z satisfies the following conditions:(3)P1s∗0=s,P2s∗t∗r=s∗r∗t,P3s≤t⇒s∗r≤t∗randr∗t≤r∗s,P4s∗r∗t∗r≤s∗t,where s≤t if and only if s∗t=0. Note that Z,≤ is a partially ordered set.AnyBCI-algebra Z satisfies the following conditions [41, 42]:(4)P50∗0∗s∗t=0∗t∗s,P60∗s∗t=0∗s∗0∗t,P7s∗s∗s∗t=s∗t,P80∗0∗s∗t=0∗t∗0∗s.For more details onBCK-algebras and BCI-algebras, we refer the readers to [43–45].A subset∅≠T of a BCK/BCI-algebra Z is called a subalgebra if t∗s∈T∀t,s∈T and is called an ideal of Z if 0∈T and ∀t,s∈Z,t∗s∈T,s∈T implies t∈T. Furthermore, a subset ∅≠T of BCI-algebra Z is called p-ideal (resp. q-ideal and a-ideal) if 0∈T and ∀t,s,r∈Z,t∗r∗s∗r∈T,s∈T implies t∈T (resp. t∗s∗r∈T,s∈T implies t∗r∈T, and t∗r∗0∗s∈T,r∈T implies s∗t∈T).Definition 1 (see [35]).
For a set of parametersZ, an initial universe set U, a power set of the initial set PU, and the unit interval I, a hybrid structure (briefly, HS) in Z over U is defined to be a mapping gδ˜:=g˜;δ˜:Z⟶PU×I;z↦g˜z;δ˜z, where g˜:Z⟶PU and δ˜:Z⟶I are mappings.Definition 2 (see [35]).
LetZ be a BCK/BCI-algebra. For a HS gδ˜ in Z over U, gδ˜ is said to be a hybrid subalgebra of Z if the following statements are valid:(5)∀s,t∈Zg˜s∗t⊇g˜s∩g˜t,δ˜s∗t≤∨δ˜s,δ˜t.Definition 3 (see [39]).
A HSgδ˜ in Z over U is said to be a hybrid ideal of Z over U if(6)HI1∀s∈Zg˜0⊇g˜s,δ˜0≤δ˜s,HI2∀s,t∈Zg˜s⊇g˜s∗t∩g˜t,δ˜s≤∨δ˜s∗t,δ˜t.Proposition 1 (see [39]).
Letgδ˜=g˜,δ˜ be a hybrid ideal of Z. If the inequality s≤t holds in Z, then g˜s⊇g˜tandδ˜s≤δ˜t.


## 3. Hybrid Ideals in BCI-Algebras

Throughout the following sectionsZ denotes BCI-algebra unless stated otherwise.Definition 4.
A HSgδ˜ in Z over U is said to be a hybrid p-ideal of Z over U if it satisfies HI1 and HI3∀s,t,r∈Zg˜s⊇g˜s∗r∗t∗r∩g˜t,δ˜s≤∨δ˜s∗r∗t∗r,δ˜t.Example 1.
LetU=γ1,γ2,γ3,γ4 be the initial universe set. On a set of parameters Z=f,s,n,t, we define the operation ∗ in Table 1.
Then,Z,∗,f is a BCI-algebra. Let gδ˜ be a HS in Z over U which is given by Table 2.
By routine verificationgδ˜ is a hybrid p-ideal of Z over U.Table 1
Cayley table of the binary operation∗.
∗fsntffsntssftnnntfsttnsfTable 2
Table representation of hybrid structuregδ˜.
Zg˜δ˜fU0.3sγ1,γ2,γ30.4nγ1,γ20.5tγ1,γ20.5Theorem 1.
For anyBCI-algebra, every hybrid p-ideal is a hybrid ideal.Proof.
Suppose thatgδ˜ is a hybrid p-ideal of Z over U. Since s∗0=s for all s∈Z, we have(7)g˜s⊇g˜s∗0∗t∗0∩g˜t⊇g˜s∗t∩g˜t,δ˜s≤∪δ˜s∗0∗t∗0,δ˜t,≤∪δ˜s∗t,δ˜t.for all s,t∈Z. Hence, gδ˜ is a hybrid ideal of Z over U.
As shown in the following example, the converse of Theorem1 is not generally valid.Example 2.
LetU=γ1,γ2,γ3,γ4,γ5 be the initial universe set and Z=f,s,n,t,e be the set of parameters. We define the binary operation ∗ on Z by Table 3.
Then,Z,∗,f is a BCI-algebra. Now, define a HS gδ˜ in Z over U which is given by Table 4.
It is easy to check thatgδ˜ is a hybrid ideal of Z over U but not hybrid p-ideal as(8)g˜n=γ2,γ4⊇γ1,γ2,γ3,γ4=g˜n∗n∗s∗n∩g˜s=g˜f∗s∩g˜s=g˜f∩g˜s.Table 3
Cayley table of the binary operation∗.
∗fsnteffffttssfsetnnnfttttttffeetesfTable 4
Table representation of hybrid structuregδ˜.
Zg˜δfU0.4sγ1,γ2,γ3,γ40.8nγ2,γ40.5tγ2,γ40.7eγ2,γ40.8Theorem 2.
Ifgδ˜ is a hybrid p-ideal of Z over U, then(9)g˜s⊇g˜0∗0∗s,δ˜s≤δ˜0∗0∗s,for all s∈Z.Proof.
Letgδ˜ be a hybrid p-ideal of Z over U. Then,(10)g˜s⊇g˜s∗r∗t∗r∩g˜t,δ˜s≤∨δ˜s∗r∗t∗r,δ˜t,for all s,t,r∈Z. Substituting s for r and 0 for t in (10), then(11)g˜s⊇g˜g˜s∗s∗0∗s,g˜0=g˜0∗0∗s∩g˜0=g˜0∗0∗s,δ˜s≤∪δ˜s∗s∗0∗s,δ˜0=∪δ˜0∗0∗s,δ˜0=δ˜0∗0∗s.Theorem 3.
Ifgδ˜ is a hybrid p-ideal of Z over U, then(12)g˜s=g˜0∗0∗s,δ˜s=δ˜0∗0∗s,for all s∈Z.Proof.
ByK2, 0∗0∗s≤s. Therefore, by Proposition 1,(13)g˜0∗0∗s⊇g˜s,δ˜0∗0∗s≤δ˜s.
Thus, by using (9), we have(14)g˜s=g˜0∗0∗s,δ˜s=δ˜0∗0∗s.Theorem 4.
Ifgδ˜ is a hybrid p-ideal of Z over U, then(15)g˜s∗r∗t∗r⊇g˜s∗t,δ˜s∗r∗t∗r≤δ˜s∗t,for all s,t,r∈Z.Proof.
Letgδ˜ be a hybrid p-ideal of Z over U. Note that s∗r∗t∗r≤s∗t holds in Z. It implies that s∗r∗t∗r∗s∗t=0. Since gδ˜ is a hybrid ideal of Z over U, we have(16)g˜s∗r∗t∗r⊇g˜s∗r∗t∗r∗s∗t∩g˜s∗t=g˜0∩g˜s∗t=g˜s∗t,δ˜s∗r∗t∗r≤∪δ˜s∗r∗t∗r∗s∗t,δ˜s∗t=∪δ˜0,δ˜s∗t=δ˜s∗t,for all s,t,r∈Z.
The following theorem gives a condition for a hybrid ideal to be a hybridp-ideal.Theorem 5.
Letgδ˜ be a hybrid ideal of Z over U satisfying the following condition:(17)g˜s∗t⊇g˜s∗r∗t∗r,δ˜s∗t≤δ˜s∗r∗t∗r,for all s,t,r∈Z. Then, gδ˜ is a hybrid p-ideal of Z over U.Proof.
Suppose thatgδ˜ is a hybrid ideal of Z satisfying (17). Then,(18)g˜s⊇g˜s∗t∩g˜t⊇g˜s∗r∗t∗r∩g˜t,δ˜s≤∪δ˜s∗t,δ˜t≤∪δ˜s∗r∗t∗r,δ˜t,for all s,t,r∈Z, as required.Definition 5.
A HSgδ˜ in Z over U is said to be a hybrid h-ideal of Z over U if it satisfies HI1 and HI4∀s,t,r∈Zg˜s∗r⊇g˜s∗t∗r∩g˜t,δ˜s∗r≤∪δ˜s∗t∗r,δ˜t.Example 3.
LetU=γ1,γ2,γ3,γ4,γ5 be the initial universe set. As a set of parameters, we consider Z=f,s,n,t,e. The operation ∗ is defined on Z by Table 5.
Then,Z,∗,f is a BCI-algebra. Define a HS gδ˜ in Z over U by the following Table 6.
It can be easily checked thatgδ˜ is a hybrid h-ideal of Z over U.Table 5
Cayley table of the binary operation∗.
∗fsnteffffttssfsetnnnfttttttffeetesfTable 6
Table representation of hybrid structuregδ˜.
Zg˜δ˜fU0.1sγ2,γ40.7nγ1,γ2,γ3,γ40.3tγ2,γ3,γ40.5eγ2,γ40.7Theorem 6.
For anyBCI-algebra, every hybrid h-ideal is a hybrid ideal.Proof.
Proof. Letgδ˜ be a hybrid h-ideal of Z over U. Since s∗0=s for all s∈Z, we have(19)g˜s∗0⊇g˜s∗t∗0∩g˜tg˜s⊇g˜s∗t∩g˜tandδ˜s∗0≤∪δ˜s∗t∗0,δ˜tδ˜s≤∪δ˜s∗t,δ˜t,for all s,t∈Z. Hence, gδ˜ is a hybrid ideal of Z over U.
In general, the converse of Theorem6 is not valid, as the following example shows.Example 4.
LetU=γ1,γ2,γ3,γ4,γ5 be the initial universe set. For a set of parameters Z=p,s,t,w,q, we define a binary operation by the Cayley table in Table 7.
Then,Z,∗,p is a BCI-algebra. Define a HS gδ˜ in Z over U by Table 8.
It is routine to verify thatgδ˜ is a hybrid ideal but not hybrid h-ideal since(20)g˜q∗t=g˜w=γ2,γ3⊇U=g˜q∗p∗t∩g˜p=g˜q∗q∩g˜p=g˜p∩g˜p,δ˜q∗t=δ˜w=0.4≤∪δ˜q∗p∗t,δ˜p=∪δ˜q∗q,δ˜p=∪δ˜p,δ˜p=0.2.Table 7
Cayley table of the binary operation∗.
∗pstwqpppqwtsspqwttttpqwwwwtpqqqqwtpTable 8
Table representation of hybrid structuregδ˜.
Zg˜δ˜pU0.2sγ1,γ2,γ3,γ40.3tγ2,γ30.4wγ2,γ30.4qγ2,γ30.4Theorem 7.
Letgδ˜ be a hybrid h-ideal of Z over U. Then,(21)g˜0∗s⊇g˜s,δ˜0∗s≤δ˜s,for all s∈Z.Proof.
Letgδ˜ be a hybrid h-ideal of Z over U. Then,(22)g˜s∗r⊇g˜s∗t∗r∩g˜t,δ˜s∗r≤∪δ˜s∗t∗r,δ˜t.
Substitutings by 0, r by s, and t by s, we have(23)g˜0∗s⊇g˜0∗s∗s∩g˜s=g˜0∩g˜s=g˜s,δ˜0∗s≤∪δ˜0∗s∗s,δ˜s=∪δ˜0,δ˜s=δ˜s.Definition 6.
A HSgδ˜ in Z over U is said to be a hybrid a-ideal of Z over U if it satisfies HI1 and HI5∀s,t,r∈Zg˜t∗s⊇g˜s∗r∗0∗t∩g˜r,δ˜t∗s≤∪δ˜s∗r∗0∗t,δ˜r..Example 5.
Considergδ˜ represented in Example 1. It is routine to verify that gδ˜ is a hybrid a-ideal of Z over U.Theorem 8.
Every hybrida-ideal is both a hybrid subalgebra and a hybrid ideal of Z over U.Proof.
Letgδ˜ be a hybrid a-ideal of Z over U. By substituting t=r=0 in HI5 and using K3 and P1, we obtain(24)g˜0∗s⊇g˜s∗0∗0∗0∩g˜0=g˜s∗0∩g˜0=g˜s,δ˜0∗s≤∪δ˜s∗0∗0∗0,δ˜0=δ˜s,for all s,t,r∈Z.
Again settings=r=0 in HI5, using P1,K3,HI1, and above equations, we obtain(25)g˜t∗0⊇g˜0∗0∗0∗t∩g˜0=g˜0∗0∗t∩g˜0=g˜0∗0∗t⊇g˜0∗t,which implies g˜t⊇g˜0∗t(26)δ˜t∗0≤∪δ˜0∗0∗t,δ˜0=δ˜0∗0∗t≤δ˜0∗t,which implies δ˜t≤δ˜0∗t for all t∈Z. Then, from HI5 and P1, it implies that(27)g˜t⊇g˜0∗t⊇g˜t∗r∗0∗0∩g˜r=g˜t∗r∩g˜r,δ˜t≤δ˜0∗t≤∪δ˜t∗r∗0∗0,δ˜r≤∪δ˜t∗r,δ˜r,for all t,r∈Z. Hence, gδ˜ is a hybrid ideal of Z over U.
Now, for anyt,r∈Z and using above equations, we have(28)g˜t∗r⊇g˜t∗r∗t∩g˜t=g˜0∗r∩g˜t⊇g˜t∩g˜r,δ˜t∗r≤∪δ˜t∗r∗t,δ˜t=∪δ˜0∗r,δ˜t≤∪δ˜t,δ˜r.
Therefore,gδ˜ is a hybrid subalgebra of Z over U. The converse of Theorem 8 is not generally valid as seen in the following example.Example 6.
LetU=γ1,γ2,γ3,γ4,γ5 be the initial universe set. For a set of parameters Z=p,s,t, we define a binary operation by the Cayley table in Table 9.
Then,Z,∗,p is a BCI-algebra. Let gδ˜ be a HS in Z over U which is given by Table 10.
Then, it directs to show thatgδ˜ is a hybrid subalgebra and hybrid ideal but not hybrid a-ideal as(29)g˜s∗p=g˜s=γ1,γ2,γ3⊇U=g˜p∗p∗p∗s∩g˜p=g˜p∗p∩g˜p=g˜p∩g˜p,δ˜s∗p=δ˜s=0.7≤∪δ˜p∗p∗p∗s,δ˜p=∪δ˜p∗p,δ˜p=∪δ˜p,δ˜p=0.6.Table 9
Cayley table of the binary operation∗.
∗pstppppsspstttpTable 10
Table representation of hybrid structuregδ˜.
Zg˜δ˜pU0.6sγ1,γ2,γ30.7tγ1,γ2,γ30.7Theorem 9.
Ifgδ˜ is a hybrid a-ideal of Z over U, then g˜s∗r∗0∗t⊇g˜s∗r∗t and δ˜s∗r∗0∗t≤δ˜s∗r∗t for all s,t,r∈Z.Proof.
Suppose thatgδ˜ is a hybrid a-ideal of Z over U. Since s∗r∗0∗t=s∗r∗r∗t∗r≤s∗r∗t, s∗r∗0∗t∗s∗r∗t=0.
By Theorem8, gδ˜ is a hybrid ideal of Z over U. Thus,(30)g˜s∗r∗0∗t⊇g˜s∗r∗0∗t∗s∗r∗t∩g˜s∗r∗t=g˜0∩g˜s∗r∗t⊇g˜s∗r∗t,δ˜s∗r∗0∗t≤∪δ˜s∗r∗0∗t∗s∗r∗t,δ˜s∗r∗t=∪δ˜0,δ˜s∗r∗t≤δ˜s∗r∗t,for every s,t,r∈Z.Theorem 10.
Letgδ˜ be a hybrid ideal of Z over U satisfying the following inequality:(31)g˜t∗s⊇g˜s∗0∗t,δ˜t∗s≤δ˜s∗0∗t,for all s,t∈Z, then gδ˜ is a hybrid a-ideal of Z over U.Proof.
Sincegδ˜ is a hybrid ideal of Z over U,(32)g˜t∗s⊇g˜s∗0∗t⊇g˜s∗0∗t∗r∩g˜r⊇g˜s∗r∗0∗t∩g˜r,δ˜t∗s≤δ˜s∗0∗t≤∪δ˜s∗0∗t∗r,δ˜r≤∪δ˜s∗r∗0∗t,δ˜r,for all s,t,r∈Z. Hence, gδ˜ is a hybrid a-ideal of Z over U.


## 4. Conclusion

In this paper, we apply the notion of hybrid structure to the ideal theory inBCI-algebras. In fact, we have discussed the notions of hybrid p-ideal, hybrid h-ideal, and a-ideal in BCI-algebras and investigated several related properties. Furthermore, it has been shown that every hybrid p-ideal (or h-ideal or a-ideal) is a hybrid ideal in a BCI-algebra but converse need not be true in general, and in support, examples for each case are provided. Moreover, characterizations of hybrid p-ideal, hybrid h-ideal, and hybrid a-ideal in BCI-algebras are discussed.



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