Closed Int Soft 𝐵𝐶𝐼-Ideals and Int Soft c-𝐵𝐶𝐼-Ideals

Abstract

The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notions of closed intersectional soft 𝐵𝐶𝐼-ideals and intersectional soft commutative 𝐵𝐶𝐼-ideals are introduced, and related properties are investigated. Conditions for an intersectional soft 𝐵𝐶𝐼-ideal to be closed are provided. Characterizations of an intersectional soft commutative 𝐵𝐶𝐼-ideal are established, and a new intersectional soft c-𝐵𝐶𝐼-ideal from an old one is constructed.

1. Introduction

The real world is inherently uncertain, imprecise, and vague. Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [1]. In response to this situation Zadeh [2] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [3]. To solve complicated problem 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 be considered 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 probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [4]. Maji et al. [5] and Molodtsov [4] suggested that one reason for these difficulties may be the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [4] 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. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. Maji et al. [5] described the application of soft set theory to a decision making problem. Maji et al. [6] also studied several operations on the theory of soft sets. Aktaş and Çağman [7] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. Jun and Park [8] studied applications of soft sets in ideal theory of 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras. In 2012, Jun et al. [9, 10] introduced the notion of intersectional soft sets, and considered its applications to 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras. Independent of Jun et al.'s introduction, Çağman and Çitak [11] also studied soft int-group and its applications to group theory. Also, Jun [12] discussed the union soft sets with applications in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras. We refer the reader to the papers [13–26] for further information regarding algebraic structures/properties of soft set theory. Present authors [10] introduced the notion of int soft 𝐵𝐶𝐾/𝐵𝐶𝐼-ideals in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras. As a continuation of the paper [10], we introduce the notion of closed int soft 𝐵𝐶𝐼-ideals and int soft c-𝐵𝐶𝐼-ideals in 𝐵𝐶𝐼-algebras and investigate related properties. We discuss relations between a closed int soft 𝐵𝐶𝐼-ideal and an int soft 𝐵𝐶𝐼-ideal and provide conditions for an int soft 𝐵𝐶𝐼-ideal to be closed. We establish characterizations of an int soft c-𝐵𝐶𝐼-ideal and construct a new intersectional soft c-𝐵𝐶𝐼-ideal from an old one.

2. Preliminaries

A 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra is an important class of logical algebras introduced by Iséki and was extensively investigated by several researchers.

An algebra (𝑋;∗,0) of type (2,0) is called a 𝐵𝐶𝐼-algebra if it satisfies the following conditions: (I)(∀𝑥,𝑦,𝑧∈𝑋)(((𝑥∗𝑦)∗(𝑥∗𝑧))∗(𝑧∗𝑦)=0); (II)(∀𝑥,𝑦∈𝑋)((𝑥∗(𝑥∗𝑦))∗𝑦=0); (III)(∀𝑥∈𝑋)(𝑥∗𝑥=0); (IV)(∀𝑥,𝑦∈𝑋)(𝑥∗𝑦=0,𝑦∗𝑥=0⇒𝑥=𝑦).  If a 𝐵𝐶𝐼-algebra 𝑋 satisfies the following identity: (V)(∀𝑥∈𝑋)(0∗𝑥=0),  then 𝑋 is called a 𝐵𝐶𝐾-algebra. Any 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra 𝑋 satisfies the following axioms: (a1)(∀𝑥∈𝑋)(𝑥∗0=𝑥); (a2)(∀𝑥,𝑦,𝑧∈𝑋)(𝑥≤𝑦⇒𝑥∗𝑧≤𝑦∗𝑧,𝑧∗𝑦≤𝑧∗𝑥); (a3)(∀𝑥,𝑦,𝑧∈𝑋)((𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦); (a4)(∀𝑥,𝑦,𝑧∈𝑋)((𝑥∗𝑧)∗(𝑦∗𝑧)≤𝑥∗𝑦),  where 𝑥≤𝑦 if and only if 𝑥∗𝑦=0. In a 𝐵𝐶𝐼-algebra 𝑋, the following hold: (b1)(∀𝑥,𝑦∈𝑋)(𝑥∗(𝑥∗(𝑥∗𝑦))=𝑥∗𝑦); (b2) (∀𝑥,𝑦∈𝑋)(0∗(𝑥∗𝑦)=(0∗𝑥)∗(0∗𝑦)).

A 𝐵𝐶𝐼-algebra 𝑋 is said to be commutative (see [27]) if ∀𝑥,𝑦∈𝑋𝑥≤𝑦⇒𝑥=𝑦∗𝑦∗𝑥. (2.1)

Proposition 2.1. A 𝐵𝐶𝐼-algebra 𝑋 is commutative if and only if it satisfies ∀𝑥,𝑦∈𝑋𝑥∗𝑥∗𝑦=𝑦∗𝑦∗𝑥∗𝑥∗𝑦. (2.2)

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 𝐼 of a 𝐵𝐶𝐼-algebra 𝑋 is called a 𝐵𝐶𝐼-ideal of 𝑋 if it satisfies 0∈𝐼,(∀𝑥∈𝑋)∀𝑦∈𝐼𝑥∗𝑦∈𝐼⇒𝑥∈𝐼. (2.3)(2.4) A 𝐵𝐶𝐼-ideal 𝐼 of a 𝐵𝐶𝐼-algebra 𝑋 satisfies (∀𝑥∈𝑋)∀𝑦∈𝐼𝑥≤𝑦⇒𝑥∈𝐼. (2.5) A 𝐵𝐶𝐼-ideal 𝐼 of a 𝐵𝐶𝐼-algebra 𝑋 is said to be closed if it satisfies (∀𝑥∈𝑋)(𝑥∈𝐼⇒0∗𝑥∈𝐼). (2.6) A subset 𝐼 of a 𝐵𝐶𝐼-algebra 𝑋 is called a commutative 𝐵𝐶𝐼-ideal (briefly, 𝑐-𝐵𝐶𝐼-ideal) of 𝑋 (see [28]) if it satisfies (2.3) and 𝑥∗𝑦∗𝑧∈𝐼,𝑧∈𝐼⇒𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦∈𝐼 (2.7) for all 𝑥,𝑦,𝑧∈𝑋.

Proposition 2.2 (see [28]). A 𝐵𝐶𝐼-ideal 𝐼 of a 𝐵𝐶𝐼-algebra 𝑋 is commutative if and only if 𝑥∗𝑦∈𝐼 implies 𝑥∗((𝑦∗(𝑦∗𝑥))∗(0∗(0∗(𝑥∗𝑦))))∈𝐼.

Proposition 2.3 (see [28]). Let 𝐼 be a closed 𝐵𝐶𝐼-ideal of a 𝐵𝐶𝐼-algebra 𝑋. Then 𝐼 is commutative if and only if it satisfies ∀𝑥,𝑦∈𝑋𝑥∗𝑦∈𝐼⇒𝑥∗𝑦∗𝑦∗𝑥∈𝐼. (2.8)

Observe that every c-𝐵𝐶𝐼-ideal is a 𝐵𝐶𝐼-ideal, but the converse is not true (see [28]).

We refer the reader to the books [29, 30] for further information regarding 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras.

A soft set theory is introduced by Molodtsov [4], and Çağman and Enginoğlu [31] provided new definitions and various results on soft set theory.

In what follows, let 𝑈 be an initial universe set and 𝐸 be a set of parameters. We say that the pair (𝑈,𝐸) is a soft universe. Let 𝒫(𝑈) denote the power set of 𝑈 and 𝐴,𝐵,𝐶,…⊆𝐸.

Definition 2.4 (see [4, 31]). A soft set ℱ_𝐴 over 𝑈 is defined to be the set of ordered pairs ℱ_𝐴∶=𝑥,𝑓_𝐴(𝑥)∶𝑥∈𝐸,𝑓_𝐴(𝑥)∈𝒫(𝑈), (2.9) where 𝑓_𝐴∶𝐸→𝒫(𝑈) such that 𝑓_𝐴(𝑥)=∅ if 𝑥∉𝐴.
The function 𝑓_𝐴 is called the approximate function of the soft set ℱ_𝐴. The subscript 𝐴 in the notation 𝑓_𝐴 indicates that 𝑓_𝐴 is the approximate function of ℱ_𝐴.
In what follows, denote by 𝑆(𝑈) the set of all soft sets over 𝑈.
Let ℱ_𝐴∈𝑆(𝑈). For any subset 𝛾 of 𝑈, the 𝛾-inclusive set of ℱ_𝐴, denoted by ℱ^𝛾_𝐴, is defined to be the set ℱ^𝛾_𝐴∶=𝑥∈𝐴∣𝛾⊆𝑓_𝐴(𝑥). (2.10)

3. Closed Int Soft 𝐵𝐶𝐼-Ideals and Int Soft c-𝐵𝐶𝐼-Ideals

Definition 3.1 (see [10]). Assume that 𝐸 has a binary operation ↪. For any nonempty subset 𝐴 of 𝐸, a soft set ℱ_𝐴 over 𝑈 is said to be intersectional over 𝑈 if its approximate function 𝑓_𝐴 satisfies ∀𝑥,𝑦∈𝐴𝑥↪𝑦∈𝐴⇒𝑓_𝐴(𝑥)∩𝑓_𝐴𝑦⊆𝑓_𝐴𝑥↪𝑦. (3.1)

Definition 3.2 (see [12]). Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Given a subalgebra 𝐴 of 𝐸, let ℱ_𝐴∈𝑆(𝑈). Then ℱ_𝐴 is called an intersectional soft 𝐵𝐶𝐼-ideal (briefly, int soft 𝐵𝐶𝐼-ideal) over 𝑈 if the approximate function 𝑓_𝐴 of ℱ_𝐴 satisfies (∀𝑥∈𝐴)𝑓_𝐴(0)⊇𝑓_𝐴(𝑥),∀𝑥,𝑦∈𝐴𝑓_𝐴(𝑥)⊇𝑓_𝐴𝑥∗𝑦∩𝑓_𝐴𝑦. (3.2)(3.3)

Definition 3.3. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Given a subalgebra 𝐴 of 𝐸, let ℱ_𝐴∈𝑆(𝑈). Then ℱ_𝐴 is called an intersectional soft commutative 𝐵𝐶𝐼-ideal (briefly, int soft c-𝐵𝐶𝐼-ideal) over 𝑈 if the approximate function 𝑓_𝐴 of ℱ_𝐴 satisfies (3.2) and 𝑓_𝐴𝑥∗𝑦∗𝑧∩𝑓_𝐴(𝑧)⊆𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦 (3.4) for all 𝑥,𝑦,𝑧∈𝐴.

Example 3.4. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋={0,𝑎,1,2,3} is a 𝐵𝐶𝐼-algebra with the following Cayley table: ∗0𝑎1230𝑎003211𝑎032121103232210333210 (3.5) For subsets 𝛾₁,𝛾₂, and 𝛾₃ of 𝑈 with 𝛾₁⊋𝛾₂⊋𝛾₃, let ℱ_𝐸∈𝑆(𝑈) in which its approximation function 𝑓_𝐸 is defined as follows: 𝑓_𝐸(𝑈)𝛾∶𝐸→𝒫,𝑥↦¹,if𝛾𝑥=0,₂,if𝛾𝑥=𝑎,₃,if𝑥∈{1,2,3}. (3.6) Then ℱ_𝐸 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈.

Theorem 3.5. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Then every int soft c-𝐵𝐶𝐼-ideal is an int soft 𝐵𝐶𝐼-ideal.

Proof. Let ℱ_𝐴 be an int soft c-𝐵𝐶𝐼-ideal over 𝑈 where 𝐴 is a subalgebra of 𝐸. Taking 𝑦=0 in (3.4) and using (a1) and (III) imply that 𝑓_𝐴(𝑥)=𝑓_𝐴(𝑥∗0)=𝑓_𝐴(𝑥∗((0∗(0∗𝑥))∗(0∗(0∗(𝑥∗0)))))⊇𝑓_𝐴((𝑥∗0)∗𝑧)∩𝑓_𝐴(𝑧)=𝑓_𝐴(𝑥∗𝑧)∩𝑓_𝐴(𝑧) (3.7) for all 𝑥,𝑧∈𝐴. Therefore ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈.

The following example shows that the converse of Theorem 3.5 is not true.

Example 3.6. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋={0,1,2,3,4} is a 𝐵𝐶𝐼-algebra with the following Cayley table: ∗01234010000021010032200043330044430 (3.8) Let 𝛾₁,𝛾₂, and 𝛾₃ be subsets of 𝑈 such that 𝛾₁⊋𝛾₂⊋𝛾₃. Let ℱ_𝐸∈𝑆(𝑈) in which its approximation function 𝑓_𝐸 is defined as follows: 𝑓_𝐸(𝑈)𝛾∶𝐸→𝒫,𝑥↦¹,if𝛾𝑥=0,₂,if𝛾𝑥=1,₃,if𝑥∈{2,3,4}. (3.9) Routine calculations show that ℱ_𝐸 is an int soft 𝐵𝐶𝐼-ideal over 𝑈. But it is not an int soft c-𝐵𝐶𝐼-ideal over 𝑈 since 𝑓_𝐸(2∗((3∗(3∗2))∗(0∗(0∗(2∗3)))))=𝛾₃⊇𝛾̸₁=𝑓_𝐸((2∗3)∗0)∩𝑓_𝐸(0). (3.10)
We provide conditions for an int soft 𝐵𝐶𝐼-ideal to be an int soft c-𝐵𝐶𝐼-ideal.

Theorem 3.7. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. For a subalgebra 𝐴 of 𝐸, let ℱ_𝐴∈𝑆(𝑈). Then the following are equivalent: (1)ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈;(2)ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈 and its approximate function 𝑓_𝐴 satisfies: ∀𝑥,𝑦∈𝐴𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦⊇𝑓_𝐴𝑥∗𝑦. (3.11)

Proof. Assume that ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈. Then ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈 (see Theorem 3.5). If we take 𝑧=0 in (3.4) and use (a1) and (3.2), then we have (3.11).
Conversely, let ℱ_𝐴 be an int soft 𝐵𝐶𝐼-ideal over 𝑈 such that its approximate function 𝑓_𝐴 satisfies (3.11). Then 𝑓_𝐴(𝑥∗𝑦)⊇𝑓_𝐴((𝑥∗𝑦)∗𝑧)∩𝑓_𝐴(𝑧) for all 𝑥,𝑦,𝑧∈𝐴 by (3.3), which implies from (3.11) that 𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦⊇𝑓_𝐴𝑥∗𝑦∗𝑧∩𝑓_𝐴(𝑧) (3.12) for all 𝑥,𝑦,𝑧∈𝐴. Therefore ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈.

Definition 3.8. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Given a subalgebra 𝐴 of 𝐸, let ℱ_𝐴∈𝑆(𝑈). An int soft 𝐵𝐶𝐼-ideal ℱ_𝐴 over 𝑈 is said to be closed if the approximate function 𝑓_𝐴 of ℱ_𝐴 satisfies (∀𝑥∈𝐴)𝑓_𝐴(0∗𝑥)⊇𝑓_𝐴(𝑥). (3.13)

Example 3.9. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋={0,1,2,𝑎,𝑏} is a 𝐵𝐶𝐼-algebra with the following Cayley table: ∗012𝑎𝑏01000𝑎𝑎2101𝑏𝑎𝑎220𝑎𝑎𝑏𝑎𝑎𝑎00𝑏𝑎𝑏10 (3.14) Let {𝛾₁,𝛾₂,𝛾₃,𝛾₄,𝛾₅} be a class of subsets of 𝑈 which is a poset with the following Hasse diagram: (3.15) Let ℱ_𝐸∈𝑆(𝑈) in which its approximation function 𝑓_𝐸 is defined as follows: 𝑓_𝐸(𝑈)𝛾∶𝐸→𝒫,𝑥↦⁵,if𝛾𝑥=0,₂,if𝛾𝑥=1,₄,if𝛾𝑥=2,₃,if𝛾𝑥=𝑎,₁,if𝑥=𝑏. (3.16) Then ℱ_𝐸 is a closed int soft 𝐵𝐶𝐼-ideal over 𝑈.

Example 3.10. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋={2^𝑛∣𝑛∈ℤ} is a 𝐵𝐶𝐼-algebra with a binary operation “÷” (usual division). Let ℱ_𝐸∈𝑆(𝑈) in which its approximation function 𝑓_𝐸 is defined as follows: 𝑓_𝐸(𝑈)𝛾∶𝐸→𝒫,𝑥↦¹,if𝛾𝑛≥0,₂,if𝑛<0, (3.17) where 𝛾₁ and 𝛾₂ are subsets of 𝑈 with 𝛾₁⊋𝛾₂. Then ℱ_𝐸 is an int soft 𝐵𝐶𝐼-ideal over 𝑈 which is not closed since 𝑓_𝐸1÷2³=𝑓_𝐸2⁻³=𝛾₂⊉𝛾₁=𝑓_𝐸2³. (3.18)

Theorem 3.11. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Then an int soft 𝐵𝐶𝐼-ideal over 𝑈 is closed if and only if it is an int soft algebra over 𝑈.

Proof. Let ℱ_𝐴 be an int soft 𝐵𝐶𝐼-ideal over 𝑈. If ℱ_𝐴 is closed, then 𝑓_𝐴(0∗𝑥)⊇𝑓_𝐴(𝑥) for all 𝑥∈𝐴. It follows from (3.3) that 𝑓_𝐴𝑥∗𝑦⊇𝑓_𝐴𝑥∗𝑦∗𝑥∩𝑓_𝐴(𝑥)=𝑓_𝐴0∗𝑦∩𝑓_𝐴(𝑥)⊇𝑓_𝐴(𝑥)∩𝑓_𝐴𝑦 (3.19) for all 𝑥,𝑦∈𝐴. Hence ℱ_𝐴 is an int soft algebra over 𝑈.
Conversely, let ℱ_𝐴 be an int soft 𝐵𝐶𝐼-ideal over 𝑈 which is also an int soft algebra over 𝑈. Then 𝑓_𝐴(0∗𝑥)⊇𝑓_𝐴(0)∩𝑓_𝐴(𝑥)=𝑓_𝐴(𝑥) (3.20) for all 𝑥∈𝐴. Therefore ℱ_𝐴 is closed.

Let 𝑋 be a 𝐵𝐶𝐼-algebra and 𝐵(𝑋)∶={𝑥∈𝑋∣0≤𝑥}. For any 𝑥∈𝑋 and 𝑛∈ℕ, we define 𝑥^𝑛 by 𝑥¹=𝑥,𝑥^𝑛+1=𝑥∗0∗𝑥^𝑛. (3.21) If there is an 𝑛∈ℕ such that 𝑥^𝑛∈𝐵(𝑋), then we say that 𝑥 is of finite periodic (see [32]), and we denote its period |𝑥| by |𝑥|=min𝑛∈ℕ∣𝑥^𝑛∈𝐵(𝑋). (3.22) Otherwise, 𝑥 is of infinite period and denoted by |𝑥|=∞.

Theorem 3.12. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra in which every element is of finite period. Then every int soft 𝐵𝐶𝐼-ideal over 𝑈 is closed.

Proof. Let ℱ_𝐸 be an int soft 𝐵𝐶𝐼-ideal over 𝑈. For any 𝑥∈𝐸, assume that |𝑥|=𝑛. Then 𝑥^𝑛∈𝐵(𝑋). Note that 0∗𝑥^𝑛−1∗𝑥=0∗0∗0∗𝑥^𝑛−1=()∗∗𝑥0∗𝑥0∗0∗𝑥^𝑛−1=0∗𝑥∗0∗𝑥^𝑛−1=0∗𝑥^𝑛=0, (3.23) and so 𝑓_𝐸((0∗𝑥^𝑛−1)∗𝑥)=𝑓_𝐸(0)⊇𝑓_𝐸(𝑥) by (3.2). It follows from (3.3) that 𝑓_𝐸0∗𝑥^𝑛−1⊇𝑓_𝐸0∗𝑥^𝑛−1∗𝑥∩𝑓_𝐸(𝑥)⊇𝑓_𝐸(𝑥). (3.24) Also, note that 0∗𝑥^𝑛−2∗𝑥=0∗0∗0∗𝑥^𝑛−2=()∗∗𝑥0∗𝑥0∗0∗𝑥^𝑛−2=0∗𝑥∗0∗𝑥^𝑛−2=0∗𝑥^𝑛−1, (3.25) which implies from (3.24) that 𝑓_𝐸0∗𝑥^𝑛−2∗𝑥=𝑓_𝐸0∗𝑥^𝑛−1⊇𝑓_𝐸(𝑥). (3.26) Using (3.3), we have 𝑓_𝐸0∗𝑥^𝑛−2⊇𝑓_𝐸0∗𝑥^𝑛−2∗𝑥∩𝑓_𝐸(𝑥)⊇𝑓_𝐸(𝑥). (3.27) Continuing this process, we have 𝑓_𝐸(0∗𝑥)⊇𝑓_𝐸(𝑥) for all 𝑥∈𝐸. Therefore ℱ_𝐸 is closed.

Lemma 3.13 (see [10]). Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Given a subalgebra 𝐴 of 𝐸, let ℱ_𝐴∈𝑆(𝑈). If ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈, then the approximate function 𝑓_𝐴 satisfies the following condition: ∀𝑥,𝑦,𝑧∈𝐴𝑥∗𝑦≤𝑧⇒𝑓_𝐴(𝑥)⊇𝑓_𝐴𝑦∩𝑓_𝐴(𝑧). (3.28)

Proposition 3.14. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Given a subalgebra 𝐴 of 𝐸, let ℱ_𝐴∈𝑆(𝑈). If the approximate function 𝑓_𝐴 of ℱ_𝐴 satisfies (3.2) and (3.28), then ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈.

Proof. Note that 𝑥∗(𝑥∗𝑦)≤𝑦 by (II), and thus 𝑓_𝐴(𝑥)⊇𝑓_𝐴(𝑥∗𝑦)∩𝑓_𝐴(𝑦) by (3.28). Therefore ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈.

Theorem 3.15. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. For a subalgebra 𝐴 of 𝐸, let ℱ_𝐴 be a closed int soft 𝐵𝐶𝐼-ideal over 𝑈. Then the following are equivalent: (1)ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈;(2)the approximate function 𝑓_𝐴 of ℱ_𝐴 satisfies: ∀𝑥,𝑦∈𝐴𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥⊇𝑓_𝐴𝑥∗𝑦. (3.29)

Proof. Assume that ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈. Note that ∗∗≤∗∗=∗∗𝑥∗𝑦∗𝑦∗𝑥𝑥∗𝑦∗𝑦∗𝑥0∗0∗𝑥∗𝑦𝑦∗𝑦∗𝑥0∗0∗𝑥∗𝑦𝑦∗𝑦∗𝑥𝑦∗𝑦∗𝑥𝑦∗𝑦∗𝑥0∗0∗𝑥∗𝑦=0∗0∗0∗𝑥∗𝑦=0∗𝑥∗𝑦 (3.30) for all 𝑥,𝑦∈𝐴. Using Lemma 3.13, (3.11), and (3.13), we have 𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥⊇𝑓_A𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦∩𝑓_𝐴0∗𝑥∗𝑦⊇𝑓_𝐴𝑥∗𝑦∩𝑓_𝐴0∗𝑥∗𝑦=𝑓_𝐴𝑥∗𝑦 (3.31) for all 𝑥,𝑦∈𝐴. Now suppose that the approximate function 𝑓_𝐴 of ℱ_𝐴 satisfies (3.29). Since ∗∗≤∗∗,𝑥∗𝑦∗𝑦∗𝑥0∗0∗𝑥∗𝑦𝑥∗𝑦∗𝑦∗𝑥𝑦∗𝑦∗𝑥𝑦∗𝑦∗𝑥0∗0∗𝑥∗𝑦≤0∗0∗𝑥∗𝑦 (3.32) it follows from Lemma 3.13, (3.13), and (3.29) that 𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦⊇𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∩𝑓_𝐴0∗0∗𝑥∗𝑦⊇𝑓_𝐴𝑥∗𝑦∩𝑓_𝐴0∗0∗𝑥∗𝑦=𝑓_𝐴𝑥∗𝑦 (3.33) for all 𝑥,𝑦∈𝐴. By Theorem 3.7, ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈.

Theorem 3.16. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a commutative 𝐵𝐶𝐼-algebra. Then every closed int soft 𝐵𝐶𝐼-ideal is an int soft c-𝐵𝐶𝐼-ideal.

Proof. Let ℱ_𝐴 be a closed int soft 𝐵𝐶𝐼-ideal over 𝑈 where 𝐴 is a subalgebra of 𝐸. Using (a3), (b1), (I), (III), and Proposition 2.1, we have ∗=∗=∗=∗=∗≤.𝑥∗𝑦∗𝑦∗𝑥𝑥∗𝑦𝑥∗𝑥∗𝑦𝑦∗𝑦∗𝑥𝑦∗𝑦∗𝑥∗𝑥∗𝑦𝑦∗𝑦∗𝑥𝑦∗𝑦∗𝑦∗𝑥𝑦∗𝑥∗𝑥∗𝑦𝑦∗𝑥𝑦∗𝑥∗𝑥∗𝑦𝑥∗𝑥∗𝑦∗𝑥=0∗𝑥∗𝑦 (3.34) It follows from Lemma 3.13 and (3.13) that 𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥⊇𝑓_𝐴𝑥∗𝑦∩𝑓_𝐴0∗𝑥∗𝑦=𝑓_𝐴𝑥∗𝑦 (3.35) for all 𝑥,𝑦∈𝐴. Therefore, by Theorem 3.15, ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈.

Using the notion of 𝛾-inclusive sets, we consider a characterization of an int soft c-𝐵𝐶𝐼-ideal.

Lemma 3.17 (see [25]). Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Given a subalgebra 𝐴 of 𝐸, let ℱ_𝐴∈𝑆(𝑈). Then the following are equivalent: (1)ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈;(2)the nonempty 𝛾-inclusive set of ℱ_𝐴 is a 𝐵𝐶𝐼-ideal of 𝐴 for any 𝛾⊆𝑈.

Theorem 3.18. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Given a subalgebra 𝐴 of 𝐸, let ℱ_𝐴∈𝑆(𝑈). Then the following are equivalent: (1)ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈;(2)the nonempty 𝛾-inclusive set of ℱ_𝐴 is a c-𝐵𝐶𝐼-ideal of 𝐴 for any 𝛾⊆𝑈.

Proof. Assume that ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈. Then ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈 by Theorem 3.5. Hence ℱ^𝛾_𝐴 is a 𝐵𝐶𝐼-ideal of 𝐴 for all 𝛾⊆𝑈 by Lemma 3.17. Let 𝛾⊆𝑈 and 𝑥,𝑦∈𝐴 be such that 𝑥∗𝑦∈ℱ^𝛾_𝐴. Then 𝑓_𝐴(𝑥∗𝑦)⊇𝛾, and so 𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦⊇𝑓_𝐴𝑥∗𝑦⊇𝛾 (3.36) by Theorem 3.7. Thus 𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦∈ℱ^𝛾_𝐴. (3.37) It follows from Proposition 2.2 that ℱ^𝛾_𝐴 is a c-𝐵𝐶𝐼-ideal of 𝐴.
Conversely, suppose that the nonempty 𝛾-inclusive set of ℱ_𝐴 is a c-𝐵𝐶𝐼-ideal of 𝐴 for any 𝛾⊆𝑈. Then ℱ^𝛾_𝐴 is a 𝐵𝐶𝐼-ideal of 𝐴 for all 𝛾⊆𝑈. Hence ℱ_𝐴 is an int soft 𝐵𝐶𝐼-ideal over 𝑈 by Lemma 3.17. Let 𝑥,𝑦∈𝐴 be such that 𝑓_𝐴(𝑥∗𝑦)=𝛾. Then 𝑥∗𝑦∈ℱ^𝛾_𝐴, and so 𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦∈ℱ^𝛾_𝐴 (3.38) by Proposition 2.2. Hence 𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦⊇𝛾=𝑓_𝐴𝑥∗𝑦. (3.39) It follows from Theorem 3.7 that ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈.

The c-𝐵𝐶𝐼-ideals ℱ^𝛾_𝐴 in Theorem 3.18 are called the inclusive c-𝐵𝐶𝐼-ideals of ℱ_𝐴.

Theorem 3.19. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Let ℱ_𝐸,𝒢_𝐸∈𝑆(𝑈) such that (i)(∀𝑥∈𝐸)(𝑓_𝐸(𝑥)⊇𝑔_𝐸(𝑥)); (ii)ℱ_𝐸 and 𝒢_𝐸 are int soft 𝐵𝐶𝐼-ideals over 𝑈. If ℱ_𝐸 is closed and 𝒢_𝐸 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈, then ℱ_𝐸 is also an int soft c-𝐵𝐶𝐼-ideal over 𝑈.

Proof. Assume that ℱ_𝐸 is closed and 𝒢_𝐸 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈. Let 𝛾 be a subset of 𝑈 such that ℱ^𝛾_𝐸≠∅≠𝒢^𝛾_𝐸. Then ℱ^𝛾_𝐸 and 𝒢^𝛾_𝐸 are 𝐵𝐶𝐼-ideals of 𝐸 and obviously ℱ^𝛾_𝐸⊇𝒢^𝛾_𝐸. Let 𝑥∈ℱ^𝛾_𝐸. Then 𝑓_𝐸(𝑥)⊇𝛾, and so 𝑓_𝐸(0∗𝑥)⊇𝑓_𝐸(𝑥)⊇𝛾 since ℱ_𝐸 is closed. Thus 0∗𝑥∈ℱ^𝛾_𝐸, and thus ℱ^𝛾_𝐸 is a closed 𝐵𝐶𝐼-ideal of 𝐸. Since 𝒢_𝐸 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈, it follows from Theorem 3.18 that 𝒢^𝛾_𝐸 is a c-𝐵𝐶𝐼-ideal of 𝐸. Let 𝑥,𝑦∈𝐸 be such that 𝑥∗𝑦∈ℱ^𝛾_𝐸. Then 0∗(𝑥∗𝑦)∈ℱ^𝛾_𝐸. Since (𝑥∗(𝑥∗𝑦))∗𝑦=0∈𝒢^𝛾_𝐸, it follows from Proposition 2.2 that ∗=∗∗𝑥∗𝑥∗𝑦𝑦∗𝑦∗𝑥∗𝑥∗𝑦𝑥∗𝑥∗𝑦𝑦∗𝑦∗𝑥∗𝑥∗𝑦0∗0∗𝑥∗𝑥∗𝑦∗𝑦∈𝒢^𝛾_𝐸⊆ℱ^𝛾_𝐸, (3.40)and so from (a3) that 𝑥∗𝑦∗𝑦∗𝑥∗𝑥∗𝑦∗𝑥∗𝑦∈ℱ^𝛾_𝐸. (3.41) Hence 𝑥∗(𝑦∗(𝑦∗(𝑥∗(𝑥∗𝑦))))∈ℱ^𝛾_𝐸 by (2.4). Note that ∗≤∗≤∗≤𝑥∗𝑦∗𝑦∗𝑥𝑥∗𝑦∗𝑦∗𝑥∗𝑥∗𝑦𝑦∗𝑦∗𝑥∗𝑥∗𝑦𝑦∗𝑦∗𝑥𝑦∗𝑥𝑦∗𝑥∗𝑥∗𝑦𝑥∗𝑥∗𝑦∗𝑥=0∗𝑥∗𝑦∈ℱ^𝛾_𝐸. (3.42) Using (2.5) and (2.4), we have 𝑥∗(𝑦∗(𝑦∗𝑥))∈ℱ^𝛾_𝐸. Hence ℱ^𝛾_𝐸 is a c-𝐵𝐶𝐼-ideal of 𝐸. Therefore ℱ_𝐸 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈 by Theorem 3.18.

Theorem 3.20. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Let ℱ_𝐸∈𝑆(𝑈) and define a soft set ℱ^∗_𝐸 over 𝑈 by 𝑓^∗_𝐸(𝑈)𝑓∶𝐸→𝒫,𝑥↦^𝐸(𝑥),𝑖𝑓𝑥∈ℱ^𝛾_𝐸,𝛿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (3.43) where 𝛾 and 𝛿 are subset of 𝑈 with 𝛿⊊𝑓_𝐸(𝑥). If ℱ_𝐸 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈, then so is ℱ^∗_𝐸.

Proof. If ℱ_𝐸 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈, then ℱ^𝛾_𝐸 is a c-𝐵𝐶𝐼-ideal of 𝐴 for any 𝛾⊆𝑈. Hence 0∈ℱ^𝛾_𝐸, and so 𝑓^∗_𝐸(0)=𝑓_𝐸(0)⊇𝑓_𝐸(𝑥)⊇𝑓^∗_𝐸(𝑥) for all 𝑥∈𝐴. Let 𝑥,𝑦,𝑧∈𝐴. If (𝑥∗𝑦)∗𝑧∈ℱ^𝛾_𝐸 and 𝑧∈ℱ^𝛾_𝐸, then 𝑥∗((𝑦∗(𝑦∗𝑥))∗(0∗(0∗(𝑥∗𝑦))))∈ℱ^𝛾_𝐸 and so 𝑓^∗_𝐸𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦=𝑓_𝐸𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦⊇𝑓_𝐸𝑥∗𝑦∗𝑧∩𝑓_𝐸(𝑧)=𝑓^∗_𝐸𝑥∗𝑦∗𝑧∩𝑓^∗_𝐸(𝑧). (3.44) If (𝑥∗𝑦)∗𝑧∉ℱ^𝛾_𝐸 or 𝑧∉ℱ^𝛾_𝐸, then 𝑓^∗_𝐸((𝑥∗𝑦)∗𝑧)=𝛿 or 𝑓^∗_𝐸(𝑧)=𝛿. Hence 𝑓^∗_𝐸𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦⊇𝛿=𝑓^∗_𝐸𝑥∗𝑦∗𝑧∩𝑓^∗_𝐸(𝑧). (3.45) This shows that ℱ^∗_𝐸 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈.

Theorem 3.21. Let (𝑈,𝐸)=(𝑈,𝑋) where 𝑋 is a 𝐵𝐶𝐼-algebra. Then any c-𝐵𝐶𝐼-ideal of 𝐸 can be realized as an inclusive c-𝐵𝐶𝐼-ideal of some int soft c-𝐵𝐶𝐼-ideal over 𝑈.

Proof. Let 𝐴 be a c-𝐵𝐶𝐼-ideal of 𝐸. For any subset 𝛾⊊𝑈, let ℱ_𝐴 be a soft set over 𝑈 defined by 𝑓_𝐴∶𝐸→𝒫(𝑈),𝑥↦𝛾,if𝑥∈𝐴,∅,if𝑥∉𝐴. (3.46) Obviously, 𝑓_𝐴(0)⊇𝑓_𝐴(𝑥) for all 𝑥∈𝐸. For any 𝑥,𝑦,𝑧∈𝐸, if (𝑥∗𝑦)∗𝑧∈𝐴 and 𝑧∈𝐴 then 𝑥∗((𝑦∗(𝑦∗𝑥))∗(0∗(0∗(𝑥∗𝑦))))∈𝐴. Hence 𝑓_𝐴𝑥∗𝑦∗𝑧∩𝑓_𝐴(𝑧)=𝛾=𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦. (3.47) If (𝑥∗𝑦)∗𝑧∉𝐴 or 𝑧∉𝐴 then 𝑓_𝐴((𝑥∗𝑦)∗𝑧)=∅ or 𝑓_𝐴(𝑧)=∅. It follows that 𝑓_𝐴𝑥∗𝑦∗𝑦∗𝑥∗0∗0∗𝑥∗𝑦⊇∅=𝑓_𝐴𝑥∗𝑦∗𝑧∩𝑓_𝐴(𝑧). (3.48) Therefore ℱ_𝐴 is an int soft c-𝐵𝐶𝐼-ideal over 𝑈, and clearly ℱ^𝛾_𝐴=𝐴. This completes the proof.

4. Conclusion

We have introduced the notions of closed int soft 𝐵𝐶𝐼-ideals and int soft commutative 𝐵𝐶𝐼-ideals, and investigated related properties. We have provided conditions for an int soft 𝐵𝐶𝐼-ideal to be closed, and established characterizations of an int soft commutative 𝐵𝐶𝐼-ideal. We have constructed a new int soft c-𝐵𝐶𝐼-ideal from old one.

On the basis of these results, we will apply the theory of int soft sets to the another type of ideals, filters, and deductive systems in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras, Hilbert algebras, MV-algebras, MTL-algebras, BL-algebras, and so forth, in future study.

Acknowledgments

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