Ideal Theory in BCK/BCI-Algebras Based on Soft Sets and 𝒩-Structures

Abstract

Based on soft sets and 𝒩-structures, the notion of (closed) 𝒩-ideal over a BCI-algebra is introduced, and related properties are investigated. Relations between 𝒩-BCI-algebras and 𝒩-ideals are established. Characterizations of a (closed) 𝒩-ideal over a BCI-algebra are provided. Conditions for an 𝒩-ideal to be an 𝒩-BCI-algebra are considered.

1. Introduction

To solve complicated problems in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as the probability theory, the theory of (intuitionistic) fuzzy sets, the theory of vague sets, the theory of interval mathematics, and the theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [1]. Maji et al. [2] and Molodtsov [1] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [1] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [2] described the application of soft set theory to a decision-making problem. Maji et al. [3] also studied several operations on the theory of soft sets. Chen et al. [4] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [5]. Roy and Maji [6] presented some results on an application of fuzzy soft sets in decision making problem. Feng et al. [7] provided a framework to combine fuzzy sets, rough sets, and soft sets all together, which gives rise to several interesting new concepts such as rough soft sets, soft rough sets, and soft rough fuzzy sets. Feng et al. [8] gave deeper insights into decision making based on fuzzy soft sets. Feng et al. [9] initiated the notion of soft rough sets, which can be seen as a generalized rough set model based on soft sets. Aygünoğlu et al. [10] introduced the notion of fuzzy soft group and studied its properties. Ali et al. [11] discussed new operations in soft set theory. Jun [12] applied the notion of soft set to BCK/BCI-algebras, and Jun et al. [13] considered applications of soft set theory in the ideals of 𝑑-algebras. Han et al. [14] discussed the fuzzy set theory of fated filters in 𝑅₀-algebras based on fuzzy points.

A (crisp) set 𝐴 in a universe 𝑋 can be defined in the form of its characteristic function 𝜇_𝐴∶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. [15] 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. [16] considered closed ideals in BCH-algebras based on 𝒩-structures. Jun et al. [17] introduced the notion of 𝒩-soft sets which are a soft set based on 𝒩-structures, and then they applied it to both a decision-making problem and a BCK/BCI-algebra.

In this paper, we introduce the notion of (closed) 𝒩-ideal over a BCI-algebra based on soft sets and 𝒩-structures and investigate related properties. We establish relations between 𝒩-BCI-algebras and 𝒩-ideals. We also provide characterizations of a (closed) 𝒩-ideal over a BCI-algebra and consider conditions for an 𝒩-ideal to be an 𝒩-BCI-algebra.

2. Preliminaries

Let 𝐾(𝜏) be the class of all algebras of type 𝜏=(2,0). An element 𝑋∈𝐾(𝜏) is called a BCI-algebra if it satisfies the following axioms:(a1)((𝑥∗𝑦)∗(𝑥∗𝑧))∗(𝑧∗𝑦)=0, (a2)(𝑥∗(𝑥∗𝑦))∗𝑦=0, (a3)𝑥∗𝑥=0, (a4)𝑥∗𝑦=𝑦∗𝑥=0⇒𝑥=𝑦

for all 𝑥,𝑦,𝑧∈𝑋. If a BCI-algebra 𝑋 satisfies(a5)0∗𝑥=0 for all 𝑥∈𝑋,

then we say that 𝑋 is a BCK-algebra.

We can define a partial ordering  ≤  on 𝑋 by∀𝑥,𝑦∈𝑋𝑥≤𝑦⟺𝑥∗𝑦=0. (2.1) In a BCK/BCI-algebra 𝑋, the following hold:(b1)𝑥∗0=𝑥, (b2)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦, (b3)0∗(0∗(0∗𝑥))=0∗𝑥, (b4)0∗(𝑥∗𝑦)=(0∗𝑥)∗(0∗𝑦)

for all 𝑥,𝑦,𝑧∈𝑋.

A nonempty subset 𝑆 of a BCK/BCI-algebra 𝑋 is called a subalgebra of 𝑋 if 𝑥∗𝑦∈𝑆 for all 𝑥,𝑦∈𝑆. A subset 𝐴 of a BCK/BCI-algebra 𝑋 is called an ideal of 𝑋 if it satisfies(c1)0∈𝐴, (c2)(∀𝑥,𝑦∈𝑋)(𝑥∗𝑦∈𝐴,𝑦∈𝐴⇒𝑥∈𝐴).

We refer the reader to the books [18, 19] for further information regarding BCK/BCI-algebras.

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.

Definition 2.1 (see [15]). By a subalgebra of a BCK/BCI-algebra 𝑋 based on 𝒩-function 𝑓 (briefly, 𝒩-subalgebra of 𝑋) are means an 𝒩-structure (𝑋,𝑓) in which 𝑓 satisfies the following assertion: ∀𝑥,𝑦∈𝑋𝑓𝑥∗𝑦≤𝑓(𝑥),𝑓𝑦. (2.2)

Definition 2.2 (see [15]). By an ideal of a BCK/BCI-algebra 𝑋 based on 𝒩-function 𝑓 (briefly, 𝒩-ideal of 𝑋) are means an 𝒩-structure (𝑋,𝑓) in which 𝑓 satisfies the following assertion: ∀𝑥,𝑦∈𝑋𝑓(0)≤𝑓(𝑥)≤𝑓𝑥∗𝑦,𝑓𝑦. (2.3)

Definition 2.3 (see [15]). Let 𝑋 be a BCI-algebra. An 𝒩-ideal (𝑋,𝑓) is said to be closed if it is also an 𝒩-subalgebra of 𝑋.

3. 𝒩-SoftBCK/BCI-Algebras and 𝒩-Soft Ideals

In what follows let 𝐸 denote a set of attributes unless otherwise specified. We will use the terminology “soft machine” which means that it produces a BCK/BCI-algebra.

Definition 3.1 (see [17]). Let 𝑋 be an initial universe set and 𝐸 a set of attributes. By an 𝒩-soft set over 𝑋 we mean a pair (𝑓,𝐴) where 𝐴⊂𝐸 and 𝑓 is a mapping from 𝐴 to ℱ(𝑋,[−1,0]); that is, for each 𝑎∈𝐴,𝑓(𝑎)∶=𝑓_𝑎 is an 𝒩-function on 𝑋.

Denote by 𝒩(𝑋,𝐸) the collection of all 𝒩-soft sets over 𝑋 with attributes from 𝐸, and we call it an 𝒩-soft class.

Definition 3.2 (see [17]). Let (𝑓,𝐴) and (𝑔,𝐵) be 𝒩-soft sets in 𝒩(𝑋,𝐸). Then (𝑓,𝐴) is called an 𝒩-soft subset of (𝑔,𝐵), denoted by (𝑓,𝐴)⊆(𝑔,𝐵), if it satisfies(i)𝐴⊆𝐵, (ii)(∀𝑒∈𝐴)(𝑓_𝑒⊆𝑔_𝑒,i.e.,𝑓_𝑒(𝑥)≤𝑔_𝑒(𝑥)∀𝑥∈𝑋).

Definition 3.3 (see [17]). Let (𝑓,𝐴) be an 𝒩-soft set over a BCK/BCI-algebra 𝑋 where 𝐴 is a subset of 𝐸. If there exists an attribute 𝑢∈𝐴 for which the 𝒩-structure (𝑋,𝑓_𝑢) is an 𝒩-subalgebra of 𝑋, then we say that (𝑓,𝐴) is an 𝒩-soft 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra related to the attribute 𝑢 (briefly, 𝑁_𝑢-soft 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra). If (𝑓,𝐴) is an 𝒩_𝑢-soft BCK/BCI-algebra for all 𝑢∈𝐴, we say that (𝑓,𝐴) is an 𝒩-soft BCK/BCI-algebra.

Definition 3.4. Let (𝑓,𝐴) be an 𝒩-soft set over a BCK/BCI-algebra 𝑋 where 𝐴 is a subset of 𝐸. If there exists an attribute 𝑢∈𝐴 for which the 𝒩-structure (𝑋,𝑓_𝑢) is an 𝒩-ideal of 𝑋, then we say that (𝑓,𝐴) is an 𝒩-soft ideal of 𝑋 related to the attribute 𝑢 (briefly, 𝒩_𝑢-soft ideal). If (𝑓,𝐴) is an 𝒩_𝑢-soft ideal of 𝑋 for all 𝑢∈𝐴, we say that (𝑓,𝐴) is an 𝒩-soft ideal of 𝑋.

Example 3.5. Let 𝑈∶={apple,banana,carrot,peach,radish} be a universe, and consider a soft machine $ which produces the following products: apple$𝑥=apple∀𝑥∈𝑈,banana$𝑦=appleif𝑦∈banana,peaqch,radish,bananaif𝑦∈apple,carrot,carrot$𝑧=carrotif𝑧∈apple,banana,radish,appleif𝑧∈carrot,peach,peach$𝑢=peachif𝑢=apple,bananaif𝑢=carrot,appleif𝑢=peach,carrotif𝑢∈{banana,radish},radish$𝑣=appleif𝑣=radish,radishif𝑣∈apple,carrot,bananaif𝑣∈banana,peach. (3.1) Then 𝑈 is a BCK-algebra under the soft machine $. Consider a set of attributes 𝐴∶=cat,cow,dog,duck,horse,pig, (3.2) and let (𝑓,𝐴) be an 𝒩-soft set over the BCK-algebra 𝑈 with the tabular representation which is given by Table 1. Then 𝑓_cat,𝑓_cow,𝑓_dog,𝑓_duck, and 𝑓_fig are 𝒩-soft ideals over 𝑈. But 𝑓_hourse is not an 𝒩-soft ideal over 𝑈 since 𝑓_hourse(radish)=𝑓=−0.3>−0.4_hourseradish$peach,𝑓_hoursepeach. (3.3) In general, we know that horses like carrots best of all. In the above example, we know that (𝑓,𝐴) is not an 𝒩-soft ideal over 𝑈 based on attribute “horse.” This means that if a horse like a carrot better than the others, then (𝑓,𝐴) cannot be an 𝒩-soft ideal over 𝑈.

Table 1 

Tabular representation of (𝑓,𝐴).

Obviously, every 𝒩-soft ideal is an 𝒩-soft BCK-algebra in a BCK-algebra, but the converse is not true as seen in the following example.

Example 3.6. Let 𝑈∶=apple,banana,carrot,peach,radish (3.4) be a universe, and consider a soft machine ◊ which produces the following products: apple◊𝑥=appleif𝑥∈carrot,peach,radish,carrotif𝑥∈apple,banana,banana◊𝑦=bananaif𝑦∈apple,carrot,peach,radish,carrotif𝑥=banana,carrot◊𝑧=carrot∀𝑧∈𝑈,peach◊𝑢=peachif𝑢∈{carrot,radish},carrotif𝑢∈apple,peach,banana,radish◊𝑣=radishif𝑣∈carrot,peach,carrotif𝑣∈apple,banana,radish. (3.5) Then 𝑈 is a BCK-algebra under the soft machine ◊. Consider a set of parameters 𝐵∶=duck,horse,pig. (3.6) Let (𝑔,𝐵) be an 𝒩-soft set over the BCK-algebra 𝑈 with the tabular representation which is given by Table 2. It is easy to verify that (𝑔,𝐵) is an 𝒩-soft BCK-algebra. But it is not an 𝒩-soft ideal over 𝑈 because (𝑔,𝐵) is not an 𝒩_horse-soft ideal since 𝑔_horsepeach=−0.1>−0.4=𝑔_horsepeach◊appl𝑒,𝑔_horseapple. (3.7)
We discuss characterizations of an 𝒩-soft ideal.

Table 2 

Tabular representation of (𝑔,𝐵).

Theorem 3.7. For a BCK/BCI-algebra 𝑋, let (𝑓,𝐴) be an 𝒩-soft set in (𝑋,𝐸) such that 𝑓_𝑢(0)≤𝑓_𝑢(𝑥) for all 𝑥∈𝑋 and 𝑢∈𝐴. Then (𝑓,𝐴) is an 𝒩-soft ideal of 𝑋 if and only if the following assertion is valid: ∀𝑥,𝑦,𝑧∈𝑋(∀𝑢∈𝐴)𝑥∗𝑦≤𝑧⟹𝑓_𝑢(𝑥)≤𝑓_𝑢𝑦,𝑓_𝑢(𝑧). (3.8)

Proof. Suppose that (𝑓,𝐴) is an 𝒩-soft ideal of 𝑋. Let 𝑢∈𝐴 and 𝑥,𝑦,𝑧∈𝑋 be such that 𝑥∗𝑦≤𝑧. Then (𝑥∗𝑦)∗𝑧=0, and so 𝑓_𝑢𝑥∗𝑦≤𝑓_𝑢𝑥∗𝑦∗𝑧,𝑓_𝑢(𝑧=𝑓_𝑢)(0),𝑓_𝑢(𝑧)=𝑓_𝑢(𝑧). (3.9) It follows that 𝑓_𝑢(𝑥)≤⋁{𝑓_𝑢(𝑥∗𝑦),𝑓_𝑢(𝑦)}≤⋁{𝑓_𝑢(𝑧),𝑓_𝑢(𝑦)}.
Conversely, assume that (3.8) holds. Note that 𝑥∗(𝑥∗𝑦)≤𝑦 for all 𝑥,𝑦∈𝑋. It follows from (3.8) that 𝑓_𝑢(𝑥)≤𝑓_𝑢𝑥∗𝑦,𝑓_𝑢𝑦 (3.10) for all 𝑥,𝑦∈𝑋 and 𝑢∈𝐴. Hence (𝑓,𝐴) is an 𝒩_𝑢-soft ideal of 𝑋 for all 𝑢∈𝐴, and so (𝑓,𝐴) is an 𝒩-soft ideal of 𝑋.

Lemma 3.8 (see [17]). Every 𝒩-soft BCK/BCI-algebra (𝑓,𝐴) over a BCK/BCI-algebra 𝑋 satisfies the following inequality: (∀𝑥∈𝑋)(∀𝑢∈𝐴)𝑓_𝑢(0)≤𝑓_𝑢(𝑥). (3.11)

Theorem 3.9. Let (𝑓,𝐴) be an 𝒩-BCK/BCI-algebra of a BCK/BCI-algebra 𝑋. Then (𝑓,𝐴) is an 𝒩-soft ideal of 𝑋 if and only if it satisfies (3.8).

Proof. Necessity is by Theorem 3.7. Conversely, assume that assertion (3.8) is valid. Since 𝑥∗(𝑥∗𝑦)≤𝑦 for all 𝑥,𝑦∈𝑋, it follows that 𝑓_𝑢(𝑥)≤⋁{𝑓_𝑢(𝑥∗𝑦),𝑓_𝑢(𝑦)} for all 𝑥,𝑦∈𝑋 and 𝑢∈𝐴. Combining this and Lemma 3.8, we know that (𝑓,𝐴) is an 𝒩-soft ideal of 𝑋.

Proposition 3.10. Every 𝒩-soft ideal (𝑓,𝐴) of a BCI-algebra 𝑋 satisfies the following inequality: (∀𝑥∈𝑋)(∀𝑢∈𝐴)𝑓_𝑢(0∗(0∗𝑥))≤𝑓_𝑢(𝑥). (3.12)

Proof. Let (𝑓,𝐴) be an 𝒩-soft ideal of a 𝐵𝐶𝐼-algebra 𝑋. Then 𝑓_𝑢(0∗(0∗𝑥))≤𝑓_𝑢((0∗(0∗𝑥))∗𝑥),𝑓_𝑢(𝑥=𝑓_𝑢)(0),𝑓_𝑢(𝑥)=𝑓_𝑢(𝑥) (3.13) for all 𝑥∈𝑋 and 𝑢∈𝐴.

The following example shows that there exists an attribute 𝑢∈𝐴 such that an 𝒩_𝑢-soft ideal of a BCI-algebra may not be an 𝒩_𝑢-soft BCI-algebra.

Example 3.11. Let 𝑈=ℚ−{0} be a universe where ℚ is the set of all rational numbers. Let ★ be a soft machine which is established by 𝑥∀𝑥,𝑦∈𝑈𝑥★𝑦=𝑦. (3.14) Then 𝑈 is a BCI-algebra under the soft machine ★. Let (𝑓,𝐴) be an 𝒩-soft set in 𝒩(𝑈,𝐸) which is defined by (∀𝑥∈𝑈)(∀𝑢∈𝐴)𝑓_𝑢(𝑥)=−0.6if𝑥∈ℤ−{0}−0.2otherwise. (3.15) Then (𝑓,𝐴) is an 𝒩-soft ideal over 𝑈, but it is not an 𝒩-soft BCI-algebra over 𝑈 since 𝑓_𝑢(3★2)=−0.2>−0.6=𝑓_𝑢(3),𝑓_𝑢(2). (3.16)

Definition 3.12. Let (𝑓,𝐴) be an 𝒩-soft set over a BCI-algebra 𝑋, where 𝐴 is a subset of 𝐸. If there exists an attribute 𝑢∈𝐴 for which the 𝒩-structure (𝑋,𝑓_𝑢) is a closed 𝒩-ideal of 𝑋, then we say that (𝑓,𝐴) is a closed 𝒩-soft ideal over 𝑋 related to the attribute 𝑢 (briefly, closed 𝒩_𝑢-soft ideal). If (𝑓,𝐴) is a closed 𝒩_𝑢-soft ideal over 𝑋 for all 𝑢∈𝐴, we say that (𝑓,𝐴) is a closed 𝒩-soft ideal over 𝑋.

Example 3.13. Suppose there are five colors in the universe 𝑈, that is, 𝑈∶=white,blackish,reddish,green,yellow. (3.17) Let ⊎ be a soft machine to mix two colors according to order in such a way that we have the following results: 𝑥⊎white=𝑥∀𝑥∈𝑈,𝑥⊎blackish=whiteif𝑥∈{white,blackish},greenif𝑥∈green,yellow,𝑥if𝑥=reddish,𝑥⊎reddish=whiteif𝑥∈{reddish},𝑥if𝑥∈white,blackish,green,yellow,𝑥⊎green=greenif𝑥∈{white,reddish},yellowif𝑥=blackish,whiteif𝑥=green,blackishif𝑥=yellow,𝑥⊎yellow=greenif𝑥∈{white,blackish,reddish},whiteif𝑥∈green,yellow. (3.18) Then 𝑈 is a BCI-algebra under the soft machine ⊎. Consider a set of attributes 𝐴∶={beautiful,fine,moderate}, (3.19) and let (𝑓,𝐴) be an 𝒩-soft set over 𝑈 with the tabular representation which is given by Table 3. Then (𝑓,𝐴) is a closed 𝒩-soft ideal over 𝑈.

Table 3 

Tabular representation of (𝑓,𝐴).

Theorem 3.14. A closed 𝒩-soft ideal of a BCI-algebra 𝑋 related to an attribute is an 𝒩-soft BCI-algebra over 𝑋 related to the same attribute.

Proof. Let (𝑓,𝐴) be a closed 𝒩-soft ideal over a BCI-algebra 𝑋 related to an attribute 𝑢∈𝐴. Then 𝑓_𝑢(0∗𝑥)≤𝑓_𝑢(𝑥) for all 𝑥∈𝑋. It follows that 𝑓_𝑢𝑥∗𝑦≤𝑓_𝑢𝑥∗𝑦∗𝑥,𝑓_𝑢(𝑥)=𝑓_𝑢(𝑥∗𝑥)∗𝑦,𝑓_𝑢(𝑥=𝑓_𝑢)0∗𝑦,𝑓_𝑢(𝑥)≤𝑓_𝑢𝑦,𝑓_𝑢(𝑥) (3.20) for all 𝑥,𝑦∈𝑋. Hence (𝑓,𝐴) is an 𝒩_𝑢-soft BCI-algebra over 𝑋.

We provide a characterization of a closed 𝒩-soft ideal.

Theorem 3.15. For an 𝒩-soft ideal (𝑓,𝐴) over a BCI-algebra 𝑋, the following are equivalent:(1)(𝑓,𝐴) is closed,(2)(𝑓,𝐴) is an 𝒩-soft BCI-algebra over 𝑋.

Proof. (1) ⇒ (2). It follows from Theorem 3.14.
(2) ⇒ (1). Suppose that (𝑓,𝐴) is an 𝒩-soft BCI-algebra over 𝑋. Then 𝑓_𝑢(0)≤𝑓_𝑢(𝑥) for all 𝑥∈𝑋 and 𝑢∈𝐴, and so 𝑓_𝑢(0∗𝑥)≤𝑓_𝑢(0),𝑓_𝑢(𝑥)≤𝑓_𝑢(𝑥),𝑓_𝑢(𝑥)=𝑓_𝑢(𝑥). (3.21) Thus (𝑓,𝐴) is a closed 𝒩_𝑢-soft ideal over 𝑋 for all 𝑢∈𝐴, and therefore (𝑓,𝐴) is a closed 𝒩-soft ideal over 𝑋.

We consider a condition for an 𝒩-soft ideal to be closed.

Theorem 3.16. Given an attribute 𝑢∈𝐴, if an 𝒩_𝑢-soft ideal (𝑓,𝐴) over a BCI-algebra 𝑋 satisfies the following assertion: (∀𝑥∈𝑋)𝑓_𝑢(0∗𝑥)≤𝑓_𝑢(𝑥), (3.22) then (𝑓,𝐴) is a closed 𝒩_𝑢-soft ideal over 𝑋.

Proof. For all 𝑥,𝑦∈𝑋, we have (𝑥∗𝑦)∗𝑥=0∗𝑦. Hence 𝑓_𝑢𝑥∗𝑦≤𝑓_𝑢𝑥∗𝑦∗𝑥,𝑓_𝑢(𝑥=𝑓_𝑢)0∗𝑦,𝑓_𝑢(𝑥≤𝑓_𝑢)(𝑥),𝑓_𝑢𝑦. (3.23) Thus (𝑓,𝐴) is an 𝒩_𝑢-soft BCI-algebra over 𝑋. It follows from Theorem 3.15 that (𝑓,𝐴) is a closed 𝒩_𝑢-soft ideal over 𝑋.

Corollary 3.17. If an 𝒩-soft ideal (𝑓,𝐴) over a BCI-algebra 𝑋 satisfies the following assertion: (∀𝑥∈𝑋)(∀𝑢∈𝐴)𝑓_𝑢(0∗𝑥)≤𝑓_𝑢(𝑥), (3.24) then (𝑓,𝐴) is a closed 𝒩-soft ideal over 𝑋.

Definition 3.18 (see [17]). Let (𝑓,𝐴) and (𝑔,𝐵) be two 𝒩-soft sets in (𝑋,𝐸). The union of (𝑓,𝐴) and (𝑔,𝐵) is defined to be the 𝒩-soft set (ℎ,𝐶) in (𝑋,𝐸) satisfying the following conditions:(i)𝐶=𝐴∪𝐵, (ii) for all 𝑥∈𝐶, ℎ_𝑥=𝑓_𝑥if𝑔𝑥∈𝐴⧵𝐵,_𝑥if𝑓𝑥∈𝐵⧵𝐴,_𝑥∪𝑔_𝑥if𝑥∈𝐴∩𝐵. (3.25) In this case, we write (𝑓,𝐴)∪(𝑔,𝐵)=(ℎ,𝐶).

Lemma 3.19 (see [15]). If (𝑋,𝑓) and (𝑋,𝑔) are 𝒩-ideals of a BCK/BCI -algebra 𝑋, then the union (𝑋,𝑓∪𝑔) of (𝑋,𝑓) and (𝑋,𝑔) is an 𝒩-ideal of 𝑋.

Theorem 3.20. If (𝑓,𝐴) and (𝑔,𝐵) are 𝒩-soft ideals over a BCK/BCI-algebra 𝑋, then the union of (𝑓,𝐴) and (𝑔,𝐵) is an 𝒩-soft ideal over 𝑋.

Proof. Let (𝑓,𝐴)∪(𝑔,𝐵)=(ℎ,𝐶) be the union of (𝑓,𝐴) and (𝑔,𝐵). Then 𝐶=𝐴∪𝐵. For any 𝑥∈𝐶, if 𝑥∈𝐴⧵𝐵 (resp., 𝑥∈𝐵⧵𝐴), then (𝑋,ℎ_𝑥)=(𝑋,𝑓_𝑥) (resp., (𝑋,ℎ_𝑥)=(𝑋,𝑔_𝑥)) is an 𝒩-ideal of 𝑋. If 𝐴∩𝐵≠∅, then (𝑋,ℎ_𝑥)=(𝑋,𝑓_𝑥∪𝑔_𝑥) is an 𝒩-ideal of 𝑋 for all 𝑥∈𝐴∩𝐵 by Lemma 3.19. Therefore (ℎ,𝐶) is an 𝒩-soft ideal over a BCK/BCI-algebra 𝑋.

Definition 3.21 (see [17]). Let (𝑓,𝐴) and (𝑔,𝐵) be two 𝒩-soft sets in (𝑋,𝐸). The intersection of (𝑓,𝐴) and (𝑔,𝐵) is the 𝒩-soft set (ℎ,𝐶) in (𝑋,𝐸), where 𝐶=𝐴∪𝐵 and for every 𝑥∈𝐶, ℎ_𝑥=𝑓_𝑥if𝑔𝑥∈𝐴⧵𝐵,_𝑥if𝑓𝑥∈𝐵⧵𝐴,_𝑥∩𝑔_𝑥if𝑥∈𝐴∩𝐵. (3.26)

In this case, we write (𝑓,𝐴)∩(𝑔,𝐵)=(ℎ,𝐶).

Theorem 3.22. Let (𝑓,𝐴) and (𝑔,𝐵) be 𝒩-soft ideals over a BCK/BCI-algebra 𝑋. If 𝐴 and 𝐵 are disjoint, then the intersection of (𝑓,𝐴) and (𝑔,𝐵) is an 𝒩-soft ideal over 𝑋.

Proof. Let (𝑓,𝐴)∩(𝑔,𝐵)=(ℎ,𝐶) be the intersection of (𝑓,𝐴) and (𝑔,𝐵). Then 𝐶=𝐴∪𝐵. Since 𝐴∩𝐵=∅, if 𝑥∈𝐶, then either 𝑥∈𝐴⧵𝐵 or 𝑥∈𝐵⧵𝐴. If 𝑥∈𝐴⧵𝐵, then (𝑋,ℎ_𝑥)=(𝑋,𝑓_𝑥) is an 𝒩-ideal of 𝑋. If 𝑥∈𝐵⧵𝐴, then (𝑋,ℎ_𝑥)=(𝑋,𝑔_𝑥) is an 𝒩-ideal of 𝑋. Hence (ℎ,𝐶) is an 𝒩-soft ideal over a 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra 𝑋.

The following example shows that Theorem 3.22 is not valid if 𝐴 and 𝐵 are not disjoint.

Example 3.23. Let 𝑈 be an initial universe set that consists of “white,” “blackish,” “reddish,” “green” and “yellow.” Consider a soft machine “¥’’ which produces the following products: white¥𝑥=whiteif𝑥∈{white,blackish},𝑥if𝑥∈reddish,green,yellow,blackish¥𝑦=blackishif𝑦=white,whiteif𝑦=blackish,𝑦if𝑦∈reddish,green,yellow,reddish¥𝑧=whiteif𝑧=reddish,reddishif𝑧∈{white,blackish},yellowif𝑧=green,greenif𝑧=yellow,green¥𝑢=whiteif𝑢=green,greenif𝑢∈{white,blackish},yellowif𝑢=reddish,reddishif𝑢=yellow,yellow¥𝑣=whiteif𝑣=yellow,reddishif𝑣=green,greenif𝑣=reddish,yellowif𝑣∈{white,blackish}. (3.27) Then 𝑈 is a BCI-algebra under the soft machine ¥. Consider sets of attributes: 𝐴∶={beautiful,fine,smart},𝐵∶={smart,chaste}. (3.28) Let (𝑓,𝐴) and (𝑔,𝐵) be 𝒩-soft sets over 𝑈 with the tabular representations which are given by Tables 4 and 5, respectively. Then (𝑓,𝐴) and (𝑔,𝐵) are 𝒩-soft ideals over 𝑈. But we have 𝑓_smart∩𝑔_smart(reddish)==−0.5>−0.6f_smart∩𝑔_smartreddish¥green,𝑓_smart∩𝑔_smartgreen, (3.29) and so (𝑓,𝐴)∩(𝑔,𝐵) is not an 𝒩-soft ideal over 𝑈.

Table 4 

Tabular representation of (𝑓,𝐴).

Table 5 

Tabular representation of (𝑔,𝐵).

4. Conclusions

We have introduced the notion of (closed) 𝒩-ideal over a BCI-algebra based on soft sets and 𝒩-structures. We have investigated several properties and established relations between 𝒩-𝐵𝐶𝐼-algebras and 𝒩-ideals. We have provided characterizations of a (closed) 𝒩-ideal over a BCI-algebra and considered conditions for an 𝒩-ideal to be an 𝒩-BCI-algebra. Based on these results, we will apply the 𝒩-structure to the other type of ideals/filters in BCK/BCI-algebras, MV-algebras, MTL-algebras, 𝑅₀-algebras BL-algebras, and so forth, in the future study.

Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.