In this paper, neutrosophic -structures are applied to p-ideals of BCI-algebras. In fact, we introduce the notion of neutrosophic -p-ideal in BCI-algebras, and investigate several properties. Further, we present characterizations of neutrosophic -p-ideal. Moreover, we consider relations between a neutrosophic -ideal and a neutrosophic -p-ideal. Also, we provide conditions for a neutrosophic -ideal to be a neutrosophic -p-ideal. Furthermore, it is proved that the neutrosophic -structure over Q is a neutrosophic -ideal of Q ⇔ G is a p-ideal of Q where G is a non-empty subset of a BCI-algebras Q.
The study of BCK/BCI-algebras was initiated by K. Iséki in 1966 as a generalization of the concept of set-theoretic difference and propositional calculus. Since then a great deal of literature has been produced on the theory of BCK/BCI-algebras. A remarkable feature of K. Iseki definition is that, its formulation is free from those of ring theoretical and lattice theoretical concepts. The concept of ideal has played an important role in the study of the theory of BCI-algebras. In a BCI-algebra, an ideal need not be subalgebra. If the ideal is also a subalgebra, then it has better algebraic properties. Zhang et al. [27] introduced the notion of a p-ideal in a BCI-algebra, and investigated related properties.
As a generalization of fuzzy set, intuitionistic fuzzy sets and logic, the notion of neutrosophic set is introduced by Smarandache [24, 25]. Since then, studies about “neutrosophy” and its derivatives, such as “neutrosophic logic”, “neutrosophic set”, “neutrosophic probability”, and “neutrosophic statistics”, have been applied in various domains, starting from computational intelligence, clustering, control, data analysis and data mining, decision making and support, design, human factors engineering and ergonomics, information processing and retrieval, knowledge representation and reasoning, all the way to image processing, medical diagnosis, optimization, pattern classification, production planning and scheduling, quality control, natural language processing, etc.
In a neutrosophic set, an element has three associated defining functions which are truth membership function (T), indeterminate membership function (I) and false membership function (F) defined on a universe of discourse. These three functions are independent completely. The neutrosophic set has vast applications in various fields such as decision making ([1, 2]), BCK/BCI-algebra ([12, 13]), semigroups ([14]) and graph theory ([7, 15]). Also, some other generalizations of fuzzy sets like interval type 2 fuzzy sets have been studied on various aspects in ([3–5]).
In order to provide mathematical tool for dealing with negative information, Jun et al. introduced the notion of negative-valued function, and constructed N-structures (see [11]). Later on, Khan et al. introduced the notion of neutrosophic N-structures, and it is applied to semigroups (see [14]) while Jun et al. applied neutrosophic N-structures to BCK/BCI-algebras (see [10]). Also, Song et al. studied a neutrosophic commutative N-ideal in BCK-algebras (see [26]). Recently, Muhiuddin et al. applied neutrosophic sets to BCK/BCI-algebras and semigroups on various aspects (see for e.g., [17–23]).
In the present paper, neutrosophic -structures are applied to p-ideals of BCI-algebras. The background of this study is displayed in the second section. In the third section, we introduce the concept of neutrosophic -p-ideal in BCI-algebras, and investigate several properties. Further, we give characterizations of neutrosophic -p-ideal. Also, we consider relations between a neutrosophic -ideal and a neutrosophic -p-ideal. In addition, we provide conditions for a neutrosophic -ideal to be a neutrosophic -p-ideal Finally, we conclude our work in the last section.
Preliminary
We begin the definition of BCI-algebra due to K. Iséki [9]. By a BCI-algebra we mean a system Q : = (Q, * , 0) with a binary operation * and a constant 0. We call the binary operation * on Q the * multiplication on Q, and the constant 0 of Q the zero element of Q.
An algebra Q is called a BCI-algebra if it satisfies the following axioms:
((ϖ * ϱ) * (ϖ * z)) * (z * ϱ) =0,
(ϖ * (ϖ * ϱ)) * ϱ = 0,
ϖ * ϖ = 0,
ϖ * ϱ = ϱ * ϖ = 0 ⇒ ϖ = ϱ
for all ϖ, ϱ, z ∈ Q . If 0 * ϖ = 0 ∀ ϖ ∈ Q, then Q is called a BCK-algebra [16]. Further, we define ⪯ as follows:
If J is a subset of Q, then J is an p-ideal of Q if condition (6) holds and
Any p-ideal is an ideal, but the converse is not true (see [16]).
Lemma 2.1. ([6]) Let Q be a BCI-algebra. Then for ideal I of Q, the following hold:
Lemma 2.2. [[16]] If J is a subset of Q, then J is an p-ideal of Q if and only if J is an ideal of Q and
For more details on BCK/BCI-algebras, we refer the readers to [6, 16].
Let be a family of real numbers. Then we define
Notation: stands for “collection of functions from a set Q to [-1, 0]”. An ordered pair (Q, f) of Q and an -function f on Q is called an -structure (see [11]).
We denote QN, a neutrosophic -structure over a nonempty universe of discourse Q (see [14]) and defined as
where TN, IN and FN are -functions and stand for “negative truth membership function”, “negative indeterminacy membership function” and “negative falsity membership function” respectively, on Q.
Note that QN satisfies the following condition:
Definition 2.3. [[10]] A QN over Q is called a neutrosophic -subalgebra of Q if
Definition 2.4. [[10]] A QN over Q is called a neutrosophic -ideal of Q if
p-ideals based on Neutrosophic -Structures
Throughout our discussion, Q will denote a BCI-algebra unless otherwise mentioned.
Definition 3.1. A QN over Q is called a neutrosophic -ideal of Q if
Example 3.2. Let us take a BCI-algebra Q = {0, 1, 2, 3, 4} together with the binary operation “*” is given in Table 1.
Cayley table for the binary operation “*”
*
0
1
2
3
4
0
0
0
0
0
0
1
1
0
0
0
0
2
2
1
0
1
1
3
3
3
3
0
3
4
4
4
4
4
0
The neutrosophic -structure
over Q is a neutrosophic -ideal of Q.
Theorem 3.3.Every neutrosophic -ideal is a neutrosophic -ideal.
Proof. Let QN be a neutrosophic -ideal of Q. If we put z = 0 in (14) and by using (1), we have
for all ϖ, ϱ ∈ Q by (1). Consequently QN is a neutrosophic -ideal of Q.□
The following example shows that the converse of Theorem 3.3 is not true in general.
Example 3.4. Consider a BCI-algebra Q = {0, 1, 2, 3, 4} with the Cayley table which is given in Table 2.
Cayley table for the binary operation “*”
*
0
1
2
3
4
0
0
0
0
0
0
1
1
0
1
0
0
2
2
2
0
0
0
3
3
3
3
0
0
4
4
3
4
1
0
Let
be an -structure over Q where in [-1, 0], ξI ∈ [-1, 0) and belong to the interval “[-1, 0]”. Then QN is a neutrosophic -ideal of Q, but it is not a neutrosophic -ideal of Q since IN (1) ≱ ⋀ {IN ((1 * 3) * (2 * 3) , IN (2)}.
We consider characterizations of a neutrosophic -ideal.
Lemma 3.5.] Every neutrosophic -ideal QN of Q satisfies the following assertion:
Proposition 3.6.Every neutrosophic -ideal QN of Q satisfies the following inequalities
Proof. Assume that QN is a neutrosophic -ideal of Q. Note that in a BCI-algebra Q, we have ((ϖ * z) * (ϱ * z)) * (ϖ * ϱ) =0 for any ϖ, ϱ, z ∈ Q. By Theorem 3.3, QN is an -ideal of X. Hence,
for all ϖ, ϱ, z ∈ Q. Similarly, we can prove that IN ((ϖ * z) * (ϱ * z)) ≥ IN (ϖ * ϱ) and FN ((ϖ * z) * (ϱ * z)) ≤ FN (ϖ * ϱ) for all ϖ, ϱ, z ∈ Q.□
Theorem 3.7. Let QN be a neutrosophic -ideal of Q. If QN satisfies the following inequalities
then QN is a neutrosophic -ideal of Q.
Proof. If the inequality (17) holds. Using inequality (12), we have
for all ϖ, ϱ, z ∈ Q. This completes the proof.□
Proposition 3.8.If QN be a neutrosophic -ideal of Q, then following hold
Proof. Assume that QN is a neutrosophic -ideal of Q. If we substitute z : = ϖ and ϱ : =0 in (14), then
for all ϖ, ϱ, z ∈ Q. □
Proposition 3.9.Every neutrosophic -ideal QN of Q satisfies the following conditions
Proof. Assume that QN is a neutrosophic -ideal of Q. If we substitute ϖ : =0 * (0 * ϖ) and y : = ϖ in (12), then
for all ϖ ∈ Q.□
Theorem 3.10. Let QN be a neutrosophic -ideal of Q. If QN satisfies the following inequalities
then QN is a neutrosophic -ideal of Q.
Proof. If the inequality (20) holds. Using (4), (5) and Proposition 3.9, we have
for all ϖ, ϱ, z ∈ Q. This completes the proof.□
For any neutrosophic -structure QN over Q and α, β, γ ∈ [-1, 0] are s.t. -3 ≤ α + β + γ ≤ 0. Define the following sets.
The set
is called the (α, β, γ)-level set of QN (see [10]). Note that
Note that we call TNα, INβ and FNγ level p-ideals of QN.
Theorem 3.11.Let QN be a neutrosophic -ideal in Q. Then TNα, INβ and FNγ are p-ideals of Q ∀ α, β, γ ∈ [-1, 0] under condition -3 ≤ α + β + γ ≤ 0 whenever they are non-empty.
Proof. Let TNα, INβ and FNγ be non-empty ∀ α, β, γ ∈ [-1, 0] s.t. -3 ≤ α + β + γ ≤ 0. Then ϖ ∈ TNα, ϱ ∈ INβ and z ∈ FNγ for some ϖ, ϱ, z ∈ Q. Thus TN (0) ≤ TN (ϖ) ≤ α, IN (0) ≥ IN (ϱ) ≥ β, and FN (0) ≤ FN (z) ≤ γ, that is, 0 ∈ TNα ∩ INβ ∩ FNγ. Let (ϖ * z) * (ϱ * z) ∈ TNα and ϱ ∈ TNα. Then TN ((ϖ * z) * (ϱ * z)) ≤ α and TN (ϱ) ≤ α, which imply that
that is, ϖ ∈ TNα. If (r * t) * (s * t) ∈ INβ and s ∈ INβ, then IN ((r * t) * (s * t)) ≥ β and IN (s) ≥ β. Thus
and so r ∈ INβ. Finally, suppose that (j * l) * (k * l) ∈ FNγ and k ∈ FNγ. Then FN ((j * l) * (k * l)) ≤ γ and FN (k) ≤ γ. Thus
that is, j ∈ FNγ. Therefore TNα, INβ and FNγ are p-ideals of Q.□
Corollary 3.12.Let QN be a neutrosophic -structure over Q and let α, β, γ ∈ [-1, 0] be such that -3 ≤ α + β + γ ≤ 0. If QN is a neutrosophic -ideal of Q, then the nonempty (α, β, γ)-level set of QN is a p-ideal of Q.
Proof. Proof is obvious.□
Lemma 3.13. ([10]) Let QN be a neutrosophic -structure over Q and assume that TNα, INβ and FNγ are ideals of Q for all α, β, γ ∈ [-1, 0] with -3 ≤ α + β + γ ≤ 0. Then QN is a neutrosophic -ideal of Q.
Theorem 3.14.In QN over Q, let TNα, INβ and FNγ are p-ideals of Q ∀ α, β, γ ∈ [-1, 0] under condition -3 ≤ α + β + γ ≤ 0. Then QN is a neutrosophic -ideal of Q.
Proof. If TNα, INβ and FNγ are p-ideals of Q implying ideals of Q. Therefore by Lemma 3.13, QN is a neutrosophic -ideal of Q. Let ϖ ∈ Q and α, β, γ ∈ [-1, 0] under condition -3 ≤ α + β + γ ≤ 0 s.t. TN (0 * (0 * ϖ)) = α, IN (0 * (0 * ϖ)) = β and FN (0 * (0 * ϖ)) = γ. Then 0 * (0 * ϖ) ∈ TNα ∩ INβ ∩ FNγ. Since TNα ∩ INβ ∩ FNγ is an implicative ideal of Q, it follows from Lemma 2.2 that ϖ ∈ TNα ∩ INβ ∩ FNγ. Hence
Therefore by Theorem 3.10, QN is a neutrosophic -ideal of Q.□
For any fixed numbers ξT, ξF ∈ [-1, 0), ξI ∈ (-1, 0], and a non-empty subset G of Q, define a neutrosophic -structure over Q by
where TNG, ING and FNG are -functions on Q which are given as follows:
and
If be a neutrosophic -ideal, then , and are implicative p-ideals of Q by Theorem 3.11, and so G is an implicative p-ideal of Q. Again, assume that G is an implicative ideal of Q. Thus, we get 0 ∈ G, and so TNG (0) = ξT ≤ TNG (ϖ), ING (0) = ξI ≥ ING (x), and FNG (0) = ξF ≤ FNG (ϖ) for all ϖ ∈ Q. For any ϖ, ϱ, z ∈ Q, Let us take the following cases:
Case 1. (ϖ * ϱ) * (ϱ * z) ∈ G and ϱ ∈ G,
Case 2. (ϖ * ϱ) * (ϱ * z) ∈ G and ϱ ∉ G,
Case 3. (ϖ * ϱ) * (ϱ * z) ∉ G and ϱ ∈ G,
Case 4. (ϖ * ϱ) * (ϱ * z) ∉ G and ϱ ∉ G.
Case 1 shows that ϖ ∈ G, and therefore
Hence
If Case 2 is valid, then TNG (ϱ) =0, ING (ϱ) = -1 and FNG (ϱ) =0. Thus
Similarly, we can show Case 3.
Finally for Case 4, we have
Hence is a neutrosophic-ideal of Q. Thus, we have
Theorem 3.15. over Q is a neutrosophic -ideal of Q ⇔ G is a p-ideal of Q where G is a non-empty subset of Q.
Lemma 3.16. ([15]) Let A and B be ideals of Q such that A ⊆ B. If A is an implicative ideal of Q, then so is B.
Let QN and QM be two neutrosophic -structures over Q. We define a relation (= , ≤ , =) between QN and QM as follows:
Let
and
be neutrosophic -ideals of Q such that QN (= , ≤ , =) QM, and let QN be a neutrosophic -ideal of Q. Then TNα, INβ and FNγ are implicative p-ideals of Q ∀ α, β, γ ∈ [-1, 0] by Theorem 3.3. The condition QN (= , ≤ , =) QM implies that TNα = TMα, INβ ⊆ IMβ and FNγ = FMγ. It follows from Lemma 3.16 that TMα, IMβ and FMγ are implicative ideals of Q ∀ α, β, γ ∈ [-1, 0]. Thus by Theorem 3.14, QM is a neutrosophic -ideal of Q. Hence we obtain the following theorem.
Theorem 3.17. Let
and
be neutrosophic -ideals of Q such that QN (= , ≤ , =) QM. If QN is a neutrosophic -ideal of Q, then so is QM.
Lemma 3.18. [[10]] Let QN be a neutrosophic -ideal of Q. Then following hold
Lemma 3.19.Let QN be a neutrosophic -structure. If QN satisfies (23), then QN is a neutrosophic -ideal of Q.
Proof. We have 0 * ϖ ⪯ ϖ ∀ ϖ ∈ X, then by (23), we obtain TN (0) ≤ TN (ϖ) , IN (0) ≥ IN (ϖ) , FN (0) ≤ FN (ϖ) ∀ ϖ ∈ X. Note that ϖ * ((ϖ * ϱ) * (ϱ * z)) ≤ ϱ for all ϖ, ϱ, z ∈ X. It follows from (23) that TN (ϖ) ≤ ⋁ {TN ((ϖ * z) * (ϱ * z)) , TN (ϱ)}, IN (ϖ) ≥ ⋀ {IN ((ϖ * z) * (ϱ * z)) , IN (ϱ)}, and FN (ϖ) ≤ ⋁ {FN ((ϖ * z) * (ϱ * z)) , FN (ϱ)} for all ϖ, ϱ, z ∈ X. Therefore QN is a neutrosophic -ideal of Q.□
Theorem 3.20. Let QN be a neutrosophic -structure. Then following assertions are equivalent.
QN is a neutrosophic -ideal of Q.
QN is a neutrosophic -subalgebra of Q which satisfies
∀ ϖ, ϱ, z, r, s ∈ Q.
Proof. Assume that QN is a neutrosophic -ideal of Q. Then by Theorem 3.3, QN is a neutrosophic -ideal of Q implying that a neutrosophic -s Q by [10, Theorem 7]. Let ϖ, ϱ, z, r, s ∈ Q be such that ((ϖ * z) * (ϱ * z)) * r ⪯ s. By Theorem 3.7 and Lemma 3.19, we have
Conversely, let QN be a neutrosophic -subalgebra of Q which satisfies the condition (24). Clearly, QN satisfies (13). Let ϖ, r, s ∈ Q be such that ϖ * r ⪯ s. Then (0 * (0 * ϖ)) * r ⪯ s, and so
for all ϖ, r, s ∈ Q. Therefore QN is a neutrosophic -ideal of Q by Lemma 3.19. Since (ϖ * ϱ) * ((ϖ * z) * (ϱ * z)) ⪯0, it follows from (24) and (13) that is
Therefore QN is a neutrosophic -ideal of Q by Theorem 3.7.□
Conclusion
Algebraic structures play an important 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 neutrosophic set theory. Neutrosophic set theory introduced by Smarandache [24], is an important mathematical tool to deal with uncertainties, fuzzy or vague objects and has vast applications in real life situations. Neutrosophic set, logic, probability, statistics are generalizations of fuzzy and intuitionistic fuzzy set and logic. Neutrosophic theory is gaining significant attention in solving various real problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. It has been applied in computational intelligence, multi-criteria (group) decision making, image processing, medical diagnosis, fault diagnosis, control design, civil engineering etc. In this paper, we presented an application of neutrosophic set theory in an algebraic structure, called a BCI-algebra. In fact, using the notion of neutrosophic -structures, we have discussed the notion of neutrosophic -p-ideal in BCI-algebras, and investigated several properties. Further, we have given characterizations of neutrosophic -p-ideal. Also, we have considered relations between a neutrosophic -ideal and a neutrosophic -p-ideal. Moreover, we have provided conditions for a neutrosophic -ideal to be a neutrosophic -p-ideal.
We hope that this work will provide a deep impact on the upcoming research in this field and other neutrosophic algebraic studies to open up new horizons of interest and innovations. Indeed, this work may serve as a foundation for further study of neutrosophic set theory in related algebraic structures. To extend these results, one can further study these notions to different algebras such as BL-algebras, MTL-algebras, R-algebras, MV-algebras, EQ-algebras and lattice implication algebras etc. Some important issues for future work are: (1) to develop strategies for obtaining more valuable results, (2) to apply these notions and results for studying related notions in other algebraic (neutrosophic) structures.
Footnotes
Acknowledgements
The authors would like to express their sincere thanks to the anonymous reviewers for their valuable comments and helpful suggestions which greatly improved the quality of this paper.
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