Double-Framed Soft Sets with Applications in BCK/BCI-Algebras

Abstract

In order to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties, the notion of double-framed soft sets is introduced, and applications in BCK/BCI-algebras are discussed. The notions of double-framed soft algebras in BCK/BCI-algebras are introduced, and related properties are investigated. Characterizations of double-framed soft algebras are considered. Product and int-uni structure of double-framed soft algebras are discussed, and several examples are provided.

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 a 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 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 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 due to 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. Jun et al. [9, 10] introduced the notion of intersectional soft sets and considered its applications to BCK/BCI-algebras. Also, Jun [11] discussed the union soft sets with applications in BCK/BCI-algebras. We refer the reader to the papers [12–24] for further information regarding algebraic structures/properties of soft set theory.

The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. To do this, we introduce the notion of double-framed soft sets and apply it to BCK/BCI-algebras. We discuss double-framed soft algebras in BCK/BCI-algebras and investigate related properties. We consider characterizations of double-framed soft algebras and deal with the product and int-uni structure of double-framed soft algebras. We provide several examples.

2. Preliminaries

A BCK/BCI-algebra is an important class of logical algebras introduced by K. 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 conditions: (),,(),()∗,∀𝑥∈𝑋)(𝑥∗0=𝑥∀𝑥,𝑦,𝑧∈𝑋𝑥≤𝑦⇒𝑥∗𝑧≤𝑦∗𝑧,𝑧∗𝑦≤𝑧∗𝑥∀𝑥,𝑦,𝑧∈𝑋𝑥∗𝑦∗𝑧=𝑥∗𝑧∗𝑦∀𝑥,𝑦,𝑧∈𝑋𝑥∗𝑧𝑦∗𝑧≤𝑥∗𝑦 (2.1)(2.2)(2.3)(2.4) where 𝑥≤𝑦 if and only if 𝑥∗𝑦=0.

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.

We refer the reader to the books [25, 26] for further information regarding BCK/BCI-algebras.

Molodtsov [4] defined the soft set in the following way: let 𝑈 be an initial universe set, and let 𝐸 be a set of parameters. Let 𝒫(𝑈) denote the power set of 𝑈 and 𝐴,𝐵,𝐶,…⊆𝐸.

Definition 2.1 (see [4]). A pair (𝛼,𝐴) is called a soft set over 𝑈, where 𝛼 is a mapping given by 𝛼∶𝐴→𝒫(𝑈). (2.5)

In other words, a soft set over 𝑈 is a parameterized family of subsets of the universe 𝑈. For 𝜀∈𝐴,  𝛼(𝜀) may be considered as the set of 𝜀-approximate elements of the soft set (𝛼,𝐴). Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [4].

3. Double-Framed Soft Algebras

In what follows, we take 𝐸=𝑋 as a set of parameters, which is a BCK/BCI-algebra and 𝐴,𝐵,𝐶,… as subalgebras of 𝐸 unless otherwise specified.

Definition 3.1. A double-framed pair ⟨(𝛼,𝛽);𝐴⟩ is called a double-framed soft set over 𝑈, where 𝛼 and 𝛽 are mappings from 𝐴 to 𝒫(𝑈).

Definition 3.2. A double-framed soft set ⟨(𝛼,𝛽);𝐴⟩ over 𝑈 is called a double-framed soft algebra over 𝑈 if it satisfies ∀𝑥,𝑦∈𝐴𝛼𝑥∗𝑦⊇𝛼(𝑥)∩𝛼𝑦,𝛽𝑥∗𝑦⊆𝛽(𝑥)∪𝛽𝑦. (3.1)

Example 3.3. Suppose that there are five houses in the initial universe set 𝑈 given by 𝑈=ℎ₁,ℎ₂,ℎ₃,ℎ₄,ℎ₅. (3.2) Let a set of parameters 𝐸={𝑒₀,𝑒₁,𝑒₂,𝑒₃} be a set of status of houses in which 𝑒₀ stands for the parameter “beautiful,” 𝑒₁ stands for the parameter “cheap,” 𝑒₂ stands for the parameter “in good location,” 𝑒₃ stands for the parameter “in green surroundings,”with the following binary operation: ∗𝑒₀𝑒₁𝑒₂𝑒₃𝑒₀𝑒₀𝑒₀𝑒₀𝑒₀𝑒₁𝑒₁𝑒₀𝑒₁𝑒₁𝑒₂𝑒₂𝑒₂𝑒₀𝑒₂𝑒₃𝑒₃𝑒₃𝑒₃𝑒₀ (3.3) Then (𝐸,∗,𝑒₀) is a BCK-algebra. For a subalgebra 𝐴:={𝑒₀,𝑒₂,𝑒₃} of 𝐸, consider a double-framed soft set ⟨(𝛼,𝛽);𝐴⟩ over 𝑈 as follows: (𝑈)ℎ𝛼∶𝐴→𝒫,𝑥↦¹,ℎ₂,ℎ₃,ℎ₄,ℎ₅if𝑥=𝑒₀,ℎ₁,ℎ₃,ℎ₄if𝑥=𝑒₂,ℎ₂,ℎ₃,ℎ₅if𝑥=𝑒₃,(𝑈)ℎ𝛽∶𝐴→𝒫,𝑥↦²if𝑥=𝑒₀,ℎ₁,ℎ₂,ℎ₃if𝑥=𝑒₂,ℎ₄,ℎ₅if𝑥=𝑒₃. (3.4) It is routine to verify that ⟨(𝛼,𝛽);𝐴⟩ is a double-framed soft algebra over 𝑈.

Example 3.4. Consider the initial universe set 𝑈={ℎ₁,ℎ₂,ℎ₃,ℎ₄,ℎ₅} and a set of parameters 𝐸={𝑒₀,𝑒₁,𝑒₂,𝑒₃} which are given in Example 3.3. Define a binary operation ∗ on 𝐸 by the following Cayley table: ∗𝑒₀𝑒₁𝑒₂𝑒₃𝑒₀𝑒₀𝑒₀𝑒₀𝑒₀𝑒₁𝑒₁𝑒₀𝑒₀𝑒₁𝑒₂𝑒₂𝑒₂𝑒₀𝑒₂𝑒₃𝑒₃𝑒₃𝑒₃𝑒₀ (3.5) Consider a double-framed soft set ⟨(𝛼,𝛽);𝐸⟩ over 𝑈 as follows: 𝛼∶𝐸→𝒫(𝑈),𝑥↦𝑈if𝑥=𝑒₀,ℎ₁,ℎ₂,ℎ₅if𝑥=𝑒₁,ℎ₁,ℎ₂,ℎ₃,ℎ₅if𝑥=𝑒₂,ℎ₁,ℎ₄if𝑥=𝑒₃,(𝑈)ℎ𝛽∶𝐸→𝒫,𝑥↦³,ℎ₄if𝑥=𝑒₀,ℎ₂,ℎ₃,ℎ₄if𝑥=𝑒₁,ℎ₂,ℎ₃,ℎ₄,ℎ₅if𝑥=𝑒₂,ℎ₁,ℎ₃,ℎ₄if𝑥=𝑒₃. (3.6) Then ⟨(𝛼,𝛽);𝐸⟩ is a double-framed soft algebra over 𝑈.

Example 3.5. There are six women patients in the initial universe set 𝑈 given by 𝑈=ℎ₁,ℎ₂,ℎ₃,ℎ₄,ℎ₅,ℎ₆. (3.7) Let a set of parameters 𝐸={𝑒₀,𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑒₅,𝑒₆,𝑒₇} be a set of status of patients in which 𝑒₀ stands for the parameter “chest pain,” 𝑒₁ stands for the parameter “headache,” 𝑒₂ stands for the parameter “mental depression,” 𝑒₃ stands for the parameter “migraine,” 𝑒₄ stands for the parameter “neurosis,” 𝑒₅ stands for the parameter “omodynia,” 𝑒₆ stands for the parameter “period pains,” 𝑒₇ stands for the parameter “toothache,” with the following binary operation: ∗𝑒₀𝑒₁𝑒₂𝑒₃𝑒₄𝑒₅𝑒₆𝑒₇𝑒₀𝑒₀𝑒₀𝑒₀𝑒₀𝑒₄𝑒₄𝑒₄𝑒₄𝑒₁𝑒₁𝑒₀𝑒₀𝑒₀𝑒₅𝑒₄𝑒₄𝑒₄𝑒₂𝑒₂𝑒₂𝑒₀𝑒₀𝑒₆𝑒₆𝑒₄𝑒₄𝑒₃𝑒₃𝑒₂𝑒₁𝑒₀𝑒₇𝑒₆𝑒₅𝑒₄𝑒₄𝑒₄𝑒₄𝑒₄𝑒₄𝑒₀𝑒₀𝑒₀𝑒₀𝑒₅𝑒₅𝑒₄𝑒₄𝑒₄𝑒₁𝑒₀𝑒₀𝑒₀𝑒₆𝑒₆𝑒₆𝑒₄𝑒₄𝑒₂𝑒₂𝑒₀𝑒₀𝑒₇𝑒₇𝑒₆𝑒₅𝑒₄𝑒₃𝑒₂𝑒₁𝑒₀ (3.8) Then (𝐸,∗,𝑒₀) is a BCI-algebra. Consider a double-framed soft set ⟨(𝛼,𝛽);𝐸⟩ over 𝑈 as follows: (𝑈)ℎ𝛼∶𝐸→𝒫,𝑥↦¹,ℎ₂,ℎ₃,ℎ₄if𝑥∈𝑒₀,𝑒₁,𝑒₂,𝑒₃,ℎ₁,ℎ₃,ℎ₄,ℎ₅,ℎ₆if𝑥∈𝑒₄,𝑒₅,𝑒₆,𝑒₇,(𝑈)ℎ𝛽∶𝐸→𝒫,𝑥↦¹,ℎ₃,ℎ₅,ℎ₇if𝑥∈𝑒₀,𝑒₁,𝑒₂,𝑒₃,ℎ₁,ℎ₂,ℎ₄if𝑥∈𝑒₄,𝑒₅,𝑒₆,𝑒₇. (3.9) Then we have 𝛼𝑒₄∩𝛼𝑒₅=ℎ₁,ℎ₃,ℎ₄,ℎ₅,ℎ₆⊈ℎ₁,ℎ₂,ℎ₃,ℎ₄=𝛼𝑒₀=𝛼𝑒₄∗𝑒₅ (3.10) and/or 𝛽𝑒₅∪𝛽𝑒₆=ℎ₁,ℎ₂,ℎ₄⊉ℎ₁,ℎ₃,ℎ₅,ℎ₇=𝛽𝑒₀=𝛽𝑒₅∗𝑒₆. (3.11) Hence, ⟨(𝛼,𝛽);𝐸⟩ is not a double-framed soft algebra over 𝑈.

Lemma 3.6. Every double-framed soft algebra ⟨(𝛼,𝛽);𝐴⟩ over 𝑈 satisfies the following condition: (∀𝑥∈𝐴)𝛼(𝑥)⊆𝛼(0),𝛽(𝑥)⊇𝛽(0). (3.12)

Proof. It is straightforward.

Proposition 3.7. For a double-framed soft algebra ⟨(𝛼,𝛽);𝐴⟩ over 𝑈, the following are equivalent: (1)(∀𝑥∈𝐴)(𝛼(𝑥)=𝛼(0),𝛽(𝑥)=𝛽(0)), (2)(∀𝑥,𝑦∈𝐴)(𝛼(𝑦)⊆𝛼(𝑥∗𝑦),𝛽(𝑦)⊇𝛽(𝑥∗𝑦)).

Proof. Assume that (2) is valid. Taking 𝑦=0 and using (2.1), we have 𝛼(0)⊆𝛼(𝑥∗0)=𝛼(𝑥) and 𝛽(0)⊇𝛽(𝑥∗0)=𝛽(𝑥). It follows from Lemma 3.6 that 𝛼(𝑥)=𝛼(0) and 𝛽(𝑥)=𝛽(0).
Conversely, suppose that 𝛼(𝑥)=𝛼(0) and 𝛽(𝑥)=𝛽(0) for all 𝑥∈𝐴. Using (3.1), we have 𝛼𝑦(0)𝑦(𝑥)𝑦,𝛽𝑦(0)𝑦(𝑥)𝑦,=𝛼∩𝛼=𝛼∩𝛼⊆𝛼𝑥∗𝑦=𝛽∪𝛽=𝛽∪𝛽⊇𝛽𝑥∗𝑦 (3.13) for all 𝑥,𝑦∈𝐴. This completes the proof.

Proposition 3.8. In a BCI-algebra 𝑋, every double-framed soft algebra ⟨(𝛼,𝛽);𝐴⟩ over 𝑈 satisfies the following condition: 𝛼(𝑥)𝑦,𝛽(𝑥)𝑦∀𝑥,𝑦∈𝐴𝑥∗0∗𝑦⊇𝛼∩𝛼𝑥∗0∗𝑦⊆𝛽∪𝛽. (3.14)

Proof. Using Lemma 3.6, we have 𝛼(𝑥)(𝑥)(0)𝑦(𝑥)𝑦,𝛽(𝑥)(𝑥)(0)𝑦(𝑥)𝑦,𝑥∗0∗𝑦⊇𝛼∩𝛼0∗𝑦⊇𝛼∩𝛼∩𝛼=𝛼∩𝛼𝑥∗0∗𝑦⊆𝛽∪𝛽0∗𝑦⊆𝛽∪𝛽∪𝛽=𝛽∪𝛽 (3.15) for all 𝑥,𝑦∈𝐴.

Let ⟨(𝛼,𝛽);𝐴⟩ and ⟨(𝑓,𝑔);𝐵⟩ be double-framed soft sets over a common universe 𝑈. Then ⟨(𝛼,𝛽);𝐴⟩ is called a double-framed soft subset of ⟨(𝑓,𝑔);𝐵⟩, denoted by ⟨(𝛼,𝛽);𝐴⟩⊆⟨(𝑓,𝑔);𝐵⟩, if (i)𝐴⊆𝐵, (ii)(∀𝑒∈𝐴)^{𝛼(𝑒)and𝑓(𝑒)areidenticalapproximations}_{𝛽(𝑒)and𝑔(𝑒)areidenticalapproximations}.

Theorem 3.9. Let ⟨(𝛼,𝛽);𝐴⟩ be a double-framed soft subset of a double-framed soft set ⟨(𝑓,𝑔);𝐵⟩. If ⟨(𝑓,𝑔);𝐵⟩ is a double-framed soft algebra over 𝑈, then so is ⟨(𝛼,𝛽);𝐴⟩.

Proof. Let 𝑥,𝑦∈𝐴. Then 𝑥,𝑦∈𝐵, and so 𝛼(𝑥)𝑦(𝑥)𝑦,𝛽(𝑥)𝑦(𝑥)𝑦.∩𝛼=𝑓∩𝑓⊆𝑓𝑥∗𝑦=𝛼𝑥∗𝑦∪𝛽=𝑔∪𝑔⊇𝑔𝑥∗𝑦=𝛽𝑥∗𝑦 (3.16) Hence, ⟨(𝛼,𝛽);𝐴⟩ is a double-framed soft algebra over 𝑈.

The converse of Theorem 3.9 is not true as seen in the following example.

Example 3.10. Suppose that there are six houses in the initial universe set 𝑈 given by 𝑈=ℎ₁,ℎ₂,ℎ₃,ℎ₄,ℎ₅,ℎ₆. (3.17) Let a set of parameters 𝐸={𝑒₀,𝑒₁,𝑒₂,𝑒₃,𝑒₄} be a set of status of houses in which 𝑒₀ stands for the parameter “beautiful,” 𝑒₁ stands for the parameter “cheap,” 𝑒₂ stands for the parameter “in good location,” 𝑒₃ stands for the parameter “in green surroundings,” 𝑒₄ stands for the parameter “luxury,” with the following binary operation: ∗𝑒₀𝑒₁𝑒₂𝑒₃𝑒₄𝑒₀𝑒₀𝑒₀𝑒₀𝑒₃𝑒₃𝑒₁𝑒₁𝑒₀𝑒₁𝑒₄𝑒₃𝑒₂𝑒₂𝑒₂𝑒₀𝑒₃𝑒₃𝑒₃𝑒₃𝑒₃𝑒₃𝑒₀𝑒₀𝑒₄𝑒₄𝑒₃𝑒₄𝑒₁𝑒₀ (3.18) Then (𝐸,∗,𝑒₀) is a BCI-algebra. For a subalgebra 𝐴={𝑒₀,𝑒₁,𝑒₂}, define a double-framed soft set ⟨(𝛼,𝛽);𝐴⟩ over 𝑈 as follows: 𝛼∶𝐴→𝒫(𝑈),𝑥↦𝑈if𝑥=𝑒₀,ℎ₁,ℎ₃,ℎ₅if𝑥=𝑒₁,ℎ₂,ℎ₄,ℎ₆if𝑥=𝑒₂,(𝑈)ℎ𝛽∶𝐴→𝒫,𝑥↦⁶if𝑥=𝑒₀,ℎ₄,ℎ₆if𝑥=𝑒₁,ℎ₂,ℎ₄,ℎ₆if𝑥=𝑒₂. (3.19) It is routine to verify that ⟨(𝛼,𝛽);𝐸⟩ is a double-framed soft algebra over 𝑈. Take 𝐵=𝐸, and define a double-framed soft set ⟨(𝑓,𝑔);𝐵⟩ over 𝑈 as follows: 𝑓∶𝐵→𝒫(𝑈),𝑥↦𝑈if𝑥=𝑒₀,ℎ₁,ℎ₃,ℎ₅if𝑥=𝑒₁,ℎ₂,ℎ₄,ℎ₆if𝑥=𝑒₂,ℎ₁,ℎ₃if𝑥=𝑒₃,ℎ₂,ℎ₄if𝑥=𝑒₄,(𝑈)ℎ𝑔∶𝐵→𝒫,𝑥↦⁶if𝑥=𝑒₀,ℎ₄,ℎ₆if𝑥=𝑒₁,ℎ₂,ℎ₄,ℎ₆if𝑥=𝑒₂,ℎ₂,ℎ₃if𝑥=𝑒₃,ℎ₂,ℎ₅if𝑥=𝑒₄. (3.20) Then ⟨(𝑓,𝑔);𝐵⟩ is not a double-framed soft algebra over 𝑈 since 𝑓𝑒₂∗𝑒₄=𝑓𝑒₃=ℎ₁,ℎ₃⊉ℎ₂,ℎ₄=𝑓𝑒₂∩𝑓𝑒₄ (3.21) and/or 𝑔𝑒₄∗𝑒₃=𝑔𝑒₁=ℎ₄,ℎ₆⊈ℎ₂,ℎ₃,ℎ₅=𝑔𝑒₄∪𝑔𝑒₃. (3.22)

For two double-framed soft sets ⟨(𝛼,𝛽);𝐴⟩ and ⟨(𝑓,𝑔);𝐴⟩ over 𝑈, the double-framed soft int-uni set of ⟨(𝛼,𝛽);𝐴⟩ and ⟨(𝑓,𝑔);𝐴⟩ is defined to be a double-framed soft set ⟨(𝛼∩𝑓,𝛽∪𝑔);𝐴⟩ where 𝛼(𝑈)(𝑥)(𝑥),𝛽(𝑈)(𝑥)(𝑥).∩𝑓∶𝐴→𝒫,𝑥↦𝛼∩𝑓∪𝑔∶𝐴→𝒫,𝑥↦𝛽∪𝑔 (3.23) It is denoted by ⟨(𝛼,𝛽);𝐴⟩⊓⟨(𝑓,𝑔);𝐴⟩=⟨(𝛼∩𝑓,𝛽∪𝑔);𝐴⟩.

Theorem 3.11. The double-framed soft int-uni set of two double-framed soft algebras ⟨(𝛼,𝛽);𝐴⟩ and ⟨(𝑓,𝑔);𝐴⟩ over 𝑈 is a double-framed soft algebra over 𝑈.

Proof. For any 𝑥,𝑦∈𝐴, we have 𝛼⊇𝛼(𝑥)𝑦∩𝑓(𝑥)𝑦=𝛼(𝑥)(𝑥)∩𝛼𝑦𝑦=𝛼𝑥)∩𝛼𝑦,𝛽⊆𝛽(𝑥)𝑦∪𝑔(𝑥)𝑦=𝛽(𝑥)(𝑥)∪𝛽𝑦𝑦=𝛽(𝑥)∪𝛽𝑦.∩𝑓𝑥∗𝑦=𝛼𝑥∗𝑦∩𝑓𝑥∗𝑦∩𝛼∩𝑓∩𝑓∩𝑓∩𝑓(∩𝑓∪𝑔𝑥∗𝑦=𝛽𝑥∗𝑦∪𝑔𝑥∗𝑦∪𝛽∪𝑔∪𝑔∪𝑔∪𝑓∪𝑓 (3.24) Therefore, ⟨(𝛼,𝛽);𝐴⟩⊓⟨(𝑓,𝑔);𝐴⟩ is a double-framed soft algebra over 𝑈.

For a double-framed soft set ⟨(𝛼,𝛽);𝐴⟩ over 𝑈 and two subsets 𝛾 and 𝛿 of 𝑈, the 𝛾-inclusive set and the 𝛿-exclusive set of ⟨(𝛼,𝛽);𝐴⟩, denoted by 𝑖_𝐴(𝛼;𝛾) and 𝑒_𝐴(𝛽;𝛿), respectively, are defined as follows: 𝑖_𝐴(𝑥,𝑒𝛼;𝛾∶=𝑥∈𝐴∣𝛾⊆𝛼)_𝐴𝛽;𝛿∶=𝑥∈𝐴∣𝛿⊇𝛽(𝑥), (3.25)(3.26) respectively. The set 𝐷𝐹_𝐴𝛼,𝛽_(𝛾,𝛿)∶=𝑥∈𝐴∣𝛾⊆𝛼(𝑥),𝛿⊇𝛽(𝑥) (3.27) is called a double-framed including set of ⟨(𝛼,𝛽);𝐴⟩. It is clear that 𝐷𝐹_𝐴𝛼,𝛽_(𝛾,𝛿)=𝑖_𝐴𝛼;𝛾∩𝑒_𝐴𝛽;𝛿. (3.28)

Theorem 3.12. For a double-framed soft set ⟨(𝛼,𝛽);𝐸⟩ over 𝑈, the following are equivalent: (1)⟨(𝛼,𝛽);𝐸⟩ is a double-framed soft algebra over 𝑈,(2)for every subsets 𝛾 and 𝛿 of 𝑈 with 𝛾∈Im(𝛼) and 𝛿∈Im(𝛽), the 𝛾-inclusive set and the 𝛿-exclusive set of ⟨(𝛼,𝛽);𝐸⟩ are subalgebras of 𝐸.

Proof. Assume that ⟨(𝛼,𝛽);𝐸⟩ is a double-framed soft algebra over 𝑈. Let 𝑥,𝑦∈𝐸 be such that 𝑥,𝑦∈𝑖_𝐸(𝛼;𝛾) and 𝑥,𝑦∈𝑒_𝐸(𝛽;𝛿) for every subsets 𝛾 and 𝛿 of 𝑈 with 𝛾∈Im(𝛼) and 𝛿∈Im(𝛽). It follows from (3.1) that 𝛼𝑥∗𝑦⊇𝛼(𝑥)∩𝛼𝑦⊇𝛾,𝛽𝑥∗𝑦⊆𝛽(𝑥)∪𝛽𝑦⊆𝛿. (3.29) Hence, 𝑥∗𝑦∈𝑖_𝐸(𝛼;𝛾) and 𝑥∗𝑦∈𝑒_𝐸(𝛽;𝛿), and therefore, 𝑖_𝐸(𝛼;𝛾) and 𝑒_𝐸(𝛽;𝛿) are subalgebras of 𝐸.
Conversely suppose that (2) is valid. Let 𝑥,𝑦∈𝐸 be such that 𝛼(𝑥)=𝛾_𝑥,  𝛼(𝑦)=𝛾_𝑦,  𝛽(𝑥)=𝛿_𝑥 and 𝛽(𝑦)=𝛿_𝑦. Taking 𝛾=𝛾_𝑥∩𝛾_𝑦 and 𝛿=𝛿_𝑥∪𝛿_𝑦 implies that 𝑥,𝑦∈𝑖_𝐸(𝛼;𝛾) and 𝑥,𝑦∈𝑒_𝐸(𝛽;𝛿). Hence, 𝑥∗𝑦∈𝑖_𝐸(𝛼;𝛾) and 𝑥∗𝑦∈𝑒_𝐸(𝛽;𝛿), which imply that 𝛼𝑥∗𝑦⊇𝛾=𝛾_𝑥∩𝛾_𝑦(𝑥)𝑦,𝛽=𝛼∩𝛼𝑥∗𝑦⊆𝛿=𝛿_𝑥∪𝛿_𝑦=𝛽(𝑥)∪𝛽𝑦. (3.30) Therefore, ⟨(𝛼,𝛽);𝐸⟩ is a double-framed soft algebra over 𝑈.

Corollary 3.13. If ⟨(𝛼,𝛽);𝐸⟩ is a double-framed soft algebra over 𝑈, then the double-framed including set of ⟨(𝛼,𝛽);𝐸⟩ is a subalgebra 𝑋.

For any double-framed soft set ⟨(𝛼,𝛽);𝐸⟩ over 𝑈, let ⟨(𝛼^∗,𝛽^∗);𝐸⟩ be a double-framed soft set over 𝑈 defined by 𝛼^∗(𝑈)𝛼(𝑥)∶𝐸→𝒫,𝑥↦if𝑥∈𝑖_𝐸,𝛽𝛼;𝛾𝜂otherwise,_∗(𝑈)𝛽(𝑥)∶𝐸→𝒫,𝑥↦if𝑥∈𝑒_𝐸𝛽;𝛿,𝜌otherwise, (3.31) where 𝛾,𝛿,𝜂, and 𝜌 are subsets of 𝑈 with 𝜂⊊𝛼(𝑥) and 𝜌⊋𝛽(𝑥).

Theorem 3.14. If ⟨(𝛼,𝛽);𝐸⟩ is a double-framed soft algebra over 𝑈, then so is ⟨(𝛼^∗,𝛽^∗);𝐸⟩.

Proof. Assume that ⟨(𝛼,𝛽);𝐸⟩ is a double-framed soft algebra over 𝑈. Then 𝑖_𝐸(𝛼;𝛾) and 𝑒_𝐸(𝛽;𝛿) are subalgebras of 𝐸 for every subsets 𝛾 and 𝛿 of 𝑈 with 𝛾∈Im(𝛼) and 𝛿∈Im(𝛽). Let 𝑥,𝑦∈𝐸. If 𝑥,𝑦∈𝑖_𝐸(𝛼;𝛾), then 𝑥∗𝑦∈𝑖_𝐸(𝛼;𝛾). Thus, 𝛼^∗𝑥∗𝑦=𝛼𝑥∗𝑦⊇𝛼(𝑥)∩𝛼𝑦=𝛼^∗(𝑥)∩𝛼^∗𝑦. (3.32) If 𝑥∉𝑖_𝐸(𝛼;𝛾) or 𝑦∉𝑖_𝐸(𝛼;𝛾), then 𝛼^∗(𝑥)=𝜂 or 𝛼^∗(𝑦)=𝜂. Hence, 𝛼^∗𝑥∗𝑦⊇𝜂=𝛼^∗(𝑥)∩𝛼^∗𝑦. (3.33) Now, if 𝑥,𝑦∈𝑒_𝐸(𝛽;𝛿), then 𝑥∗𝑦∈𝑒_𝐸(𝛽;𝛿). Thus, 𝛽^∗𝑥∗𝑦=𝛽𝑥∗𝑦⊆𝛽(𝑥)∪𝛽𝑦=𝛽^∗(𝑥)∪𝛽^∗𝑦. (3.34) If 𝑥∉𝑒_𝐸(𝛽;𝛿) or 𝑦∉𝑒_𝐸(𝛽;𝛿), then 𝛽^∗(𝑥)=𝜌 or 𝛽^∗(𝑦)=𝜌. Hence, 𝛽^∗𝑥∗𝑦⊆𝜌=𝛽^∗(𝑥)∪𝛽^∗𝑦. (3.35) Therefore, ⟨(𝛼^∗,𝛽^∗);𝐸⟩ is a double-framed soft algebra over 𝑈.

The following example shows that the converse of Theorem 3.14 is not true in general.

Example 3.15. Suppose that there are ten houses in the initial universe set 𝑈 given by 𝑈=ℎ₁,ℎ₂,ℎ₃,ℎ₄,ℎ₅,ℎ₆,ℎ₇,ℎ₈,ℎ₉,ℎ₁₀. (3.36) Let a set of parameters 𝐸={𝑒₀,𝑒₁,𝑒₂,𝑒₃} be a set of status of houses in which 𝑒₀ stands for the parameter “beautiful,” 𝑒₁ stands for the parameter “cheap,” 𝑒₂ stands for the parameter “in good location,” 𝑒₃ stands for the parameter “in green surroundings,” with the following binary operation: ∗𝑒₀𝑒₁𝑒₂𝑒₃𝑒₀𝑒₀𝑒₁𝑒₂𝑒₃𝑒₁𝑒₁𝑒₀𝑒₃𝑒₂𝑒₂𝑒₂𝑒₃𝑒₀𝑒₁𝑒₃𝑒₃𝑒₂𝑒₁𝑒₀ (3.37) Then (𝐸,∗,𝑒₀) is a BCI-algebra. Consider a double-framed soft set ⟨(𝛼,𝛽);𝐸⟩ over 𝑈 as follows: 𝛼∶𝐸→𝒫(𝑈),𝑥↦𝑈if𝑥=𝑒₀,ℎ₂,ℎ₄,ℎ₆,ℎ₈,ℎ₁₀if𝑥=𝑒₁,ℎ₃,ℎ₆,ℎ₉if𝑥=𝑒₂,ℎ₈if𝑥=𝑒₃,(𝑈)ℎ𝛽∶𝐸→𝒫,𝑥↦⁶if𝑥=𝑒₀,ℎ₃,ℎ₆,ℎ₉if𝑥=𝑒₁,ℎ₂,ℎ₄,ℎ₆,ℎ₈,ℎ₁₀if𝑥=𝑒₂,𝑈if𝑥=𝑒₃. (3.38) Note that 𝑖_𝐸(𝛼;𝛾)={𝑒₀,𝑒₁}=𝑒_𝐸(𝛽;𝛿) for 𝛾={ℎ₂,ℎ₄,ℎ₆,ℎ₈,ℎ₁₀} and 𝛿={ℎ₃,ℎ₆,ℎ₉}. Let ⟨(𝛼^∗,𝛽^∗);𝐸⟩ be a double-framed soft set over 𝑈 defined by 𝛼^∗(𝑈)𝛼(𝑥)∶𝐸→𝒫,𝑥↦if𝑥∈𝑖_𝐸,𝛽𝛼;𝛾∅otherwise,_∗(𝑈)𝛽(𝑥)∶𝐸→𝒫,𝑥↦if𝑥∈𝑒_𝐸𝛽;𝛿,𝑈otherwise, (3.39) that is, 𝛼^∗∶𝐸→𝒫(𝑈),𝑥↦𝑈if𝑥=𝑒₀,ℎ₂,ℎ₄,ℎ₆,ℎ₈,ℎ₁₀if𝑥=𝑒₁,∅if𝑥∈𝑒₂,𝑒₃,𝛽^∗(𝑈)ℎ∶𝐸→𝒫,𝑥↦⁶if𝑥=𝑒₀,ℎ₃,ℎ₆,ℎ₉if𝑥=𝑒₁,𝑈if𝑥∈𝑒₂,𝑒₃. (3.40) It is routine to verify that ⟨(𝛼^∗,𝛽^∗);𝐸⟩ is a double-framed soft algebra over 𝑈. But ⟨(𝛼,𝛽);𝐸⟩ is not a double-framed soft algebra over 𝑈 since 𝛼(𝑒₁)∩𝛼(𝑒₂)⊈𝛼(𝑒₃)=𝛼(𝑒₁∗𝑒₂) and/or 𝛽(𝑒₁)∪𝛽(𝑒₂)⊉𝛽(𝑒₃)=𝛽(𝑒₁∗𝑒₂).
Let ⟨(𝛼,𝛽);𝐴⟩ and ⟨(𝛼,𝛽);𝐵⟩ be double-framed soft sets over 𝑈. The (𝛼_∧,𝛽_∨)-product of ⟨(𝛼,𝛽);𝐴⟩ and ⟨(𝛼,𝛽);𝐵⟩ is defined to be a double-framed soft set ⟨(𝛼_𝐴∧𝐵,𝛽_𝐴∨𝐵);𝐴×𝐵⟩ over 𝑈 in which 𝛼_𝐴∧𝐵(𝑈),(𝑥)𝑦,𝛽∶𝐴×𝐵→𝒫𝑥,𝑦↦𝛼∩𝛼_𝐴∨𝐵∶𝐴×𝐵→𝒫(𝑈),𝑥,𝑦↦𝛽(𝑥)∪𝛽𝑦. (3.41)

Theorem 3.16. For any BCK/BCI-algebras 𝐸 and 𝐹 as sets of parameters, let ⟨(𝛼,𝛽);𝐸⟩ and ⟨(𝛼,𝛽);𝐹⟩ be double-framed soft algebras over 𝑈. Then the (𝛼_∧,𝛽_∨)-product of ⟨(𝛼,𝛽);𝐸⟩ and ⟨(𝛼,𝛽);𝐹⟩ is also a double-framed soft algebra over 𝑈.

Proof. Note that (𝐸×𝐹,⊛,(0,0)) is a BCK/BCI-algebra. For any (𝑥,𝑦),(𝑎,𝑏)∈𝐸×𝐹, we have 𝛼_𝐸∧𝐹𝑥,𝑦⊛(𝑎,𝑏)=𝛼_𝐸∧𝐹()⊇(𝛼(𝑥)(𝑎∩𝛼𝑦(𝑏)=𝛼(𝑥)𝑦∩(𝛼(𝑎)(𝑏𝑥∗𝑎,𝑦∗𝑏=𝛼𝑥∗𝑎∩𝛼𝑦∗𝑏∩𝛼))∩𝛼∩𝛼∩𝛼))=𝛼_𝐸∧𝐹𝑥,𝑦∩𝛼_𝐸∧𝐹(),𝛽𝑎,𝑏_𝐸∨𝐹𝑥,𝑦⊛(𝑎,𝑏)=𝛽_𝐸∨𝐹()⊆𝛽(𝑥)(𝑎)∪𝛽𝑦(𝑏)=𝛽(𝑥)𝑦∪𝛽(𝑎)(𝑏)𝑥∗𝑎,𝑦∗𝑏=𝛽𝑥∗𝑎∪𝛽𝑦∗𝑏∪𝛽∪𝛽∪𝛽∪𝛽=𝛽_𝐸∨𝐹𝑥,𝑦∪𝛽_𝐸∨𝐹(𝑎,𝑏). (3.42) Hence, ⟨(𝛼_𝐸∧𝐹,𝛽_𝐸∨𝐹);𝐸×𝐹⟩ is a double-framed soft algebra over 𝑈.

Theorem 3.16 is illustrated from the following example.

Example 3.17. Consider the initial universe set 𝑈 given by 𝑈=ℎ₁,ℎ₂,ℎ₃,ℎ₄,ℎ₅,ℎ₆, (3.43) which consists of six women patients, and take five parameters as follows: 𝑒₀ stands for the parameter “chest pain,” 𝑒₁ stands for the parameter “headache,” 𝑒₂ stands for the parameter “mental depression,” 𝑒₃ stands for the parameter “migraine,” 𝑒₄ stands for the parameter “neurosis.” Let 𝐸={𝑒₀,𝑒₁,𝑒₂} and 𝐹={𝑒₀,𝑒₃,𝑒₄} with the following Cayley tables: ∗𝑒₀𝑒₁𝑒₂𝑒₀𝑒₀𝑒₀𝑒₀𝑒₁𝑒₁𝑒₀𝑒₁𝑒₂𝑒₂𝑒₂𝑒₀∗𝑒₀𝑒₃𝑒₄𝑒₀𝑒₀𝑒₀𝑒₀𝑒₃𝑒₃𝑒₀𝑒₀𝑒₄𝑒₄𝑒₄𝑒₀ (3.44) Then 𝐸×𝐹=𝑒₀,𝑒₀,𝑒₀,𝑒₃,𝑒₀,𝑒₄,𝑒₁,𝑒₀,𝑒₁,𝑒₃,𝑒₁,𝑒₄,𝑒₂,𝑒₀,𝑒₂,𝑒₃,𝑒₂,𝑒₄ (3.45) is a BCK-algebra with the following Cayley table: ∗𝑒₀,𝑒₀𝑒₀,𝑒₃𝑒₀,𝑒₄𝑒₁,𝑒₀𝑒₁,𝑒₃𝑒₁,𝑒₄𝑒₂,𝑒₀𝑒₂,𝑒₃𝑒₂,𝑒₄𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₃𝑒₀,𝑒₃𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₃𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₃𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₄𝑒₀,𝑒₄𝑒₀,𝑒₄𝑒₀,𝑒₀𝑒₀,𝑒₄𝑒₀,𝑒₄𝑒₀,𝑒₀𝑒₀,𝑒₄𝑒₀,𝑒₄𝑒₀,𝑒₀𝑒₁,𝑒₀𝑒₁,𝑒₀𝑒₁,𝑒₀𝑒₁,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₁,𝑒₀𝑒₁,𝑒₀𝑒₁,𝑒₀𝑒₁,𝑒₃𝑒₁,𝑒₃𝑒₁,𝑒₀𝑒₁,𝑒₀𝑒₀,𝑒₃𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₁,𝑒₃𝑒₁,𝑒₀𝑒₁,𝑒₀𝑒₁,𝑒₄𝑒₁,𝑒₄𝑒₁,𝑒₄𝑒₁,𝑒₀𝑒₀,𝑒₄𝑒₀,𝑒₄𝑒₀,𝑒₀𝑒₁,𝑒₄𝑒₁,𝑒₄𝑒₁,𝑒₀𝑒₂,𝑒₀𝑒₂,𝑒₀𝑒₂,𝑒₀𝑒₂,𝑒₀𝑒₂,𝑒₀𝑒₂,𝑒₀𝑒₂,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₂,𝑒₃𝑒₂,𝑒₃𝑒₂,𝑒₀𝑒₂,𝑒₀𝑒₂,𝑒₃𝑒₂,𝑒₀𝑒₂,𝑒₀𝑒₀,𝑒₃𝑒₀,𝑒₀𝑒₀,𝑒₀𝑒₂,𝑒₄𝑒₂,𝑒₄𝑒₂,𝑒₄𝑒₂,𝑒₀𝑒₂,𝑒₄𝑒₂,𝑒₄𝑒₂,𝑒₀𝑒₀,𝑒₄𝑒₀,𝑒₄𝑒₀,𝑒₀ (3.46) Let ⟨(𝛼,𝛽);𝐸⟩ and ⟨(𝛼,𝛽);𝐹⟩ be double-framed soft sets over 𝑈 given by (𝑈)ℎ𝛼∶𝐸→𝒫,𝑥↦¹,ℎ₂,ℎ₃,ℎ₄,ℎ₅if𝑥∈𝑒₀,𝑒₁,ℎ₁,ℎ₂if𝑥=𝑒₂,(𝑈)ℎ𝛽∶𝐸→𝒫,𝑥↦³,ℎ₆if𝑥∈𝑒₀,𝑒₁,ℎ₂,ℎ₃,ℎ₆if𝑥=𝑒₂,(𝑈)ℎ𝛼∶𝐹→𝒫,𝑥↦¹,ℎ₂,ℎ₃,ℎ₄,ℎ₅if𝑥=𝑒₀,ℎ₂,ℎ₃,ℎ₄if𝑥=𝑒₃,ℎ₃,ℎ₄if𝑥=𝑒₄,(𝑈)ℎ𝛽∶𝐹→𝒫,𝑥↦³,ℎ₆if𝑥∈𝑒₀,𝑒₁,ℎ₁,ℎ₃,ℎ₄,ℎ₆if𝑥=𝑒₂, (3.47)(3.48) respectively. Then ⟨(𝛼,𝛽);𝐸⟩ and ⟨(𝛼,𝛽);𝐹⟩ are double-framed soft algebras over 𝑈. Moreover, the (𝛼_∧,𝛽_∨)-product of ⟨(𝛼,𝛽);𝐸⟩ and ⟨(𝛼,𝛽);𝐹⟩ is a double-framed soft algebra over 𝑈.

4. Conclusion

We have introduced the notion of double-framed soft algebras in a BCK/BCI-algebras and provided several examples. We have considered the characterization of double-framed soft algebra. We have shown the following.(1)Every double-framed soft subset of a double-framed soft algebra is also a double-framed soft algebra. (2)The double-framed soft int-uni set of two double-framed soft algebras is a double-framed soft algebra. (3)The (𝛼_∧,  𝛽_∨)-product of double-framed soft algebras ⟨(𝛼,𝛽);𝐸⟩ and ⟨(𝛼,𝛽);𝐹⟩ is also a double-framed soft algebra.

Given a double-framed soft algebra, we have made a new double-framed soft algebra. We have provided an example which shows that the new constructed double-framed soft set is a double-framed soft algebra, but the given double-framed soft set is not a double-framed soft algebra.

Future research will focus on applying the idea/result in this paper to the ideal theory of BCK/BCI-algebras and related algebraic structures.

Acknowledgment

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