EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 646-671 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bipolar Soft Generalized Topological Structures and Their Application in Decision Making Hind Y. Saleh1, Baravan A. Asaad2,3,∗, Ramadhan A. Mohammed1 1 Department of Mathematics, College of Basic Education, University of Duhok, Duhok-42001, Iraq 2 Department of Computer Science, College of Science, Cihan University-Duhok, Duhok-42001, Iraq 3 Department of Mathematics, Faculty of Science, University of Zakho, Zakho-42002, Iraq Abstract. The basic of bipolar soft set theory stands for a mathematical instrument that brings together the soft set theory and bipolarity. Its definition is based on two soft sets, a set that provides positive information and other that gives negative. This paper mainly aims at defining a new bipolar soft generalized topological space; setting out of the point that the collection of bipolar soft sets forms the basis for the definition of the new concept is defined. Added to that, an investigation has been made of the four concepts of bipolar soft generalized, namely ˜̃g-interior,˜̃g-closure, ˜̃g-exterior and ˜̃g-boundary. Furthermore, the main properties of bipolar soft generalized topological space (BSGT S) are established. This paper also attends to the discussion of the relations between these new definitions and the application of the given bipolar soft generalized topological spaces in a decision-making problem where an algorithm for this application has been suggested. Finally, to clarify and substantiate what the current work subsumes, some examples have been provided. 2020 Mathematics Subject Classifications: 03E75, 54A05 Key Words and Phrases: Soft set, BSS, BSGT S, bipolar soft ˜̃g-open ( bipolar soft ˜̃g-closed) set, bipolar soft ˜̃g-interior, bipolar soft ˜̃g-closure, decision making 1. Introduction For formal modeling, reasoning, and computing, the majority of the traditional tools are characterized by being crisp, deterministic, and precise. Yet in the domains of eco- nomics, engineering, environment, social science, medical science, etc., many complicated problematic issues exist. As such, to solve or model them, the typical methods based on the case, in particular, may lack suitability. On this basis, a set of theories has been ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4353 Email addresses: hind.saleh@uod.ac (H. Y. Saleh), baravan.asaad@uoz.edu.krd (B. A. Asaad), ramadhan.hajani@uod.ac (R. A. Mohammed) https://www.ejpam.com 646 © 2022 EJPAM All rights reserved. H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 647 proposed to tackle these issues. In 1999, Molodtsov [27] adopted the soft set theory that was designed to solve sophisticated problems. Molodtsov’s theory has been implemented in several branches of mathematics. Exam- ples are decision making problems, medical science, social sciences, operation research etc. The theory has been further improved by other researchers like Maji et al. [24] in terms of defining the operation family of special information systems. Based on this, the operations of the soft set were redefined by Çaǧman and Enginoglu [14] who further constructed, by using the soft set theory, a uni-int decision making method. Finally, the soft sets were compared to both fuzzy and rough sets by Aktas and Çaǧman [1]. Later on, some properties and applications of the soft set theory have been investigated by many researchers (see [9], [13], [32], [34], [35], [37], [41], [42]). For the time being, two definitions of the soft topological spaces exist. The concept of soft topological spaces on a universe set was first defined by Shabir and Naz [37]. Likewise, for the demonstration of the notion of soft topological spaces, the soft sets were also, Çaǧman [15]. This was followed by a excess of researches that tackled the soft topological spaces (see [2], [3], [4], [5], [6], [7], [8], [10], [11], [12], [18], [19], [22], [23], [25], [26], [33], [38], [40]). The notion of generalized neighborhood system and generalized topological spaces was defined by Császár ([16],[17]) who further studied a set of its basic properties, namely con- tinuous functions, associated by interior and closure operations on generalized topological spaces, and compared his findings to those of the usual topology. On their part, Thomas and John [39] constructed the notion of soft generalized topological spaces (SGT Ss) via soft generalized open sets over an initial universe with a fixed set of parameters, and stud- ied some of their properties such as compactness and separation axioms. The generalized topology differs from that based on its axioms. According to Császár, a family of subsets of Ω stands for a generalized topology on Ω when the empty set and arbitrary union of its members are included. It is worth noting that soft sets theory, not sets, forms the basis for the soft generalized topological spaces. In 2013, the bipolar soft set structure which may form the source of more general and clear results than the soft set structure was investigated by Shabir and Naz [38]. Varied definitions of the bipolar soft set and the basic operations such as intersection, union and complementation were put forward by Shabir and Naz [38] and Karaaslan and Karatas [22]. Based on Dubois and Prada [18], decision making is constructed on two sides, namely negative and positive. A number of definitions, operations, and applications on bipolar soft sets have been investigated in ([4], [22], [23], [40]). Added to that, Öztürk [31] studied the concepts of closure and interior operations, basis and subspace in bipolar soft topological spaces. There has been an expansion of the definition of bipolar soft topological spaces defined in [36] by Fadel and Dzul-Kifli [20] who have attended to the key concepts and properties and put forward some illustrative examples. Additionally, there has been further works on the topological structures on bipolar soft sets, (see [21],[22]). For instance, Musa and Asaad ([28], [29], [30]) introduced a new idea concerning bipolar soft sets by extending the hypersoft sets named bipolar hypersoft sets. They further investigated bipolar hypersoft topological spaces and some of their operations and properties. The coming parts are organized as follows: In Section 2, some related preliminaries H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 648 are briefly recalled. In Section 3, the new concept of bipolar soft topological spaces called bipolar soft generalized topological spaces have been firstly defined. This has been followed by the presentation of the basic properties of bipolar soft generalized topological spaces on an initial bipolar soft set. In addition, the definitions of the notions of bipolar soft˜̃g-open sets, bipolar soft ˜̃g-close sets, bipolar soft ˜̃g-closure, bipolar soft ˜̃g-interior, bipolar soft ˜̃g-exterior and bipolar soft ˜̃g-boundary have been provided along the study of their properties and the investigation of the relation between such concepts. In Section 4, this has been followed by presenting the application of bipolar soft generalized topological spaces in a decision making problem. Finally, a binary information table has been utilized in our attempt to analogously represent the bipolar soft generalized topological spaces. Section 5 concludes this paper. 2. Preliminaries In this section, we introduce some basic concepts about bipolar soft sets. In this paper, let Ω be an initial universe, Υ(Ω) be denoted the collection of all subsets of Ω and ϖ be a set of parameters. Let ς, σ ⊆ ϖ and BSS(Ω) be the set of all bipolar soft sets over Ω with parameters ϖ. Now, we mention the main definitions of bipolar soft sets and its related topics that we need through the paper. Definition 1. [24] Let ς = {ϱ1, ϱ2, ..., ϱn} be a set of parameters. The Not set of ς denoted by ¬ς = {¬ϱ1,¬ϱ2, ...,¬ϱn} for all i, ¬ϱi = Not ϱi. Definition 2. [38] A triple (Θ,Λ, ς) is called a bipolar soft set on Ω, where Θ and Λ are mappings defined by Θ : ς −→ Υ(Ω) and Λ : ¬ς −→ Υ(Ω) such that Θ(ϱ) ∩ Λ(¬ϱ) = ϕ for all ϱ ∈ ς and ¬ϱ ∈ ¬σ. In other words, a bipolar soft set (Θ,Λ, ς) can be written as (Θ,Λ, ς) = {(ϱ,Θ(ϱ),Λ(¬ϱ)) : ϱ ∈ ς,Θ(ϱ) ∩ Λ(¬ϱ) = ϕ}. Definition 3. [38] For any two bipolar soft sets (Θ1,Λ1, ς) and (Θ2,Λ2, σ), we say that (Θ1,Λ1, ς) is a bipolar soft subset of (Θ2,Λ2, σ) if: (i) ς ⊆ σ and, (ii) Θ1(ϱ) ⊆ Θ2(ϱ) and Λ2(¬ϱ) ⊆ Λ1(¬ϱ) for all ϱ ∈ ς and ¬ϱ ∈ ¬ς. This relationship is denoted by (Θ1,Λ1, ς) ˜̃⊆(Θ2,Λ2, ς). Similarly, we say that (Θ1,Λ1, ς) is a bipolar soft superset of (Θ2,Λ2, σ), denoted by (Θ1,Λ1, ς) ˜̃⊇ (Θ2,Λ2, σ), if (Θ2,Λ2, σ) is a bipolar soft subset of (Θ1,Λ1, ς). Definition 4. [38] Two bipolar soft sets (Θ1,Λ1, ς) and (Θ2,Λ2, σ) are said to be equal, denoted by (Θ1,Λ1, ς) = (Θ2,Λ2, σ), if (Θ1,Λ1, ς) is a bipolar soft subset of (Θ2,Λ2, σ) and (Θ2,Λ2, σ) is a bipolar soft subset of (Θ1,Λ1, ς). H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 649 Definition 5. [38] The complement of a bipolar soft set (Θ,Λ, ς) is denoted by (Θ,Λ, ς)c and defined by (Θ,Λ, ς)c=(Θc,Λc, ς) where Θc and Λc are mappings given by Θc(ϱ) = Λ(¬ϱ) and Λc(¬ϱ) = Θ(ϱ) for all ϱ ∈ ς and ¬ϱ ∈ ¬ς. Definition 6. [38] A relative null bipolar soft set (Φ, ˜̃ Ω, ς) is a bipolar soft set (Θ,Λ, ς) if Θ(ϱ) = ϕ for all ϱ ∈ ς and Λ(¬ϱ) = Ω for all ¬ϱ ∈ ¬ς. Definition 7. [38] A relative absolute bipolar soft set ( ˜̃ Ω,Φ, ς) is a bipolar soft set (Θ,Λ, ς) if Θ(ϱ) = Ω for all ϱ ∈ ς and Λ(¬ϱ) = ϕ for all ¬ϱ ∈ ¬ς. Definition 8. [38] The bipolar soft intersection between two bipolar soft sets (Θ1,Λ1, ς) and (Θ2,Λ2, σ) is the bipolar soft set (χ,Ψ, κ) where κ = ς ∩ σ is a nonempty set and for all ϱ ∈ κ, χ(ϱ) = Θ1(ϱ) ∩Θ2(e) and Ψ(¬ϱ) = Λ1(¬ϱ) ∪ Λ2(¬ϱ). It is denoted by (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, σ) = (χ,Ψ, κ). Definition 9. [38] The bipolar soft union between two bipolar soft sets (Θ1,Λ1, ς) and (Θ2,Λ2, σ) is the bipolar soft set (χ,Ψ, κ) where κ = ς ∪ σ is a nonempty set and for all ϱ ∈ κ, χ(ϱ) = Θ1(ϱ) ∪Θ2(e) and Ψ(¬ϱ) = Λ1(¬ϱ) ∩ Λ2(¬ϱ). It is denoted by (Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, σ) = (χ,Ψ, κ). Definition 10. [38] The bipolar soft union between two bipolar soft sets (Θ1,Λ1, ς) and (Θ2,Λ2, σ) is the bipolar soft set (χ,Ψ, κ) where κ = ς ∪ σ is a nonempty set and for all ϱ ∈ κ, χ(ϱ) =  Θ1(ϱ), ϱ ∈ ς − σ, Θ2(ϱ), ϱ ∈ σ − ς, Θ1(ϱ) ∪Θ2(ϱ), ϱ ∈ ς ∩ σ. Ψ(¬ϱ) =  Θ1(¬ϱ), ¬ϱ ∈ ¬ς − ¬σ, Θ2(¬ϱ), ¬ϱ ∈ ¬σ − ¬ς, Θ1(¬ϱ) ∩Θ2(¬ϱ), ¬ϱ ∈ ¬ς ∩ ¬σ. It is denoted by (Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, σ) = (χ,Ψ, κ). Definition 11. [38] The bipolar soft intersection between two bipolar soft sets (Θ1,Λ1, ς) and (Θ2,Λ2, σ) is the bipolar soft set (χ,Ψ, κ) where κ = ς ∪ σ is a nonempty set and for all ϱ ∈ κ, χ(ϱ) =  Θ1(ϱ), ϱ ∈ ς − σ, Θ2(ϱ), ϱ ∈ σ − ς, Θ1(ϱ) ∩Θ2(ϱ), ϱ ∈ ς ∩ σ. H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 650 Ψ(¬ϱ) =  Θ1(¬ϱ), ¬ϱ ∈ ¬ς − ¬σ, Θ2(¬ϱ), ¬ϱ ∈ ¬σ − ¬ς, Θ1(¬ϱ) ∪Θ2(¬ϱ), ¬ϱ ∈ ¬ς ∩ ¬σ. It is denoted by (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, σ) = (χ,Ψ, κ). Definition 12. [38] Let g̃ be the family of soft sets of Ω, then g̃ is said to be soft generalized topology (SGT ) on Ω if (i) (Φ, ς) belong to g̃. (ii) The union of any members of soft sets in g̃ belongs to g̃. Then (Ω, g̃, ς) is called a soft generalized topological space (SGT S) over Ω. Every member of g̃ is called a soft g̃-open set. The complement of a soft g̃-open set is soft g̃-closed. Proposition 1. [38] If (Θ1,Λ1, ς),(Θ2,Λ2, σ) ˜̃∈ BSS(Ω), then: (i) ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, σ)) c = (Θ1,Λ1, ς) c ˜̃∩ (Θ2,Λ2, σ) c. (ii) ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, σ)) c = (Θ1,Λ1, ς) c ˜̃∪ (Θ2,Λ2, σ) c. (iii) ((Θ1,Λ1, ς) c)c = (Θ1,Λ1, ς). (iv) (Φ, ˜̃ Ω, ς) ˜̃⊆ (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, σ) c ˜̃⊆ (Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, σ) c ˜̃⊆ ( ˜̃ Ω,Φ, ς). Definition 13. [20] The bipolar soft difference between two bipolar soft sets (Θ1,Λ1, ς) and (Θ2,Λ2, σ) is the bipolar soft set (Θ,Λ, κ) where κ = ς ∪ σ is defined as (Θ1,Λ1, ς) ˜̃\ (Θ2,Λ2, σ) = (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, σ) c. 3. Main Results In this section, we introduce the bipolar soft generalized topological spaces and we investigate some concepts and properties such as bipolar soft ˜̃g-interior, bipolar soft ˜̃g- closure, bipolar soft ˜̃g-exterior and bipolar soft ˜̃g-boundary. Definition 14. Let ˜̃g be the collection of bipolar soft subsets over Ω, then ˜̃g is said to be a bipolar soft generalized topology (BSGT ) on Ω if it satisfies the following conditions: (i) (Φ, ˜̃ Ω, ς) ˜̃∈ ˜̃g. (ii) If (Θj ,Λj , ς) ˜̃∈ ˜̃g for all j ∈ J , then ˜̃⋃ j∈J (Θj ,Λj , ς) ˜̃∈ ˜̃g. Then (Ω, ˜̃g, ς,¬ς) is called a bipolar soft generalized topological space (BSGT S) over Ω. H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 651 Definition 15. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S, if ˜̃g is the collection of all possible bipolar soft sets which can be defined over Ω, then ˜̃g is called the discrete BSGT on Ω. Definition 16. A BSGT ˜̃g is said to be strong if ( ˜̃ Ω,Φ, ς) ˜̃∈ ˜̃g. Definition 17. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S, then the members of ˜̃g are said to be bipolar soft ˜̃g-open sets in Ω. Clearly (Φ, ˜̃ Ω, ς) is bipolar soft ˜̃g-open. Definition 18. Let (Ω, ˜̃g1, ς,¬ς) and let (Ω, ˜̃g2, ς,¬ς) be a BSGT Ss. Then: (i) If ˜̃g1 ˜̃⊆ ˜̃g2 then ˜̃g2 is bipolar soft finer than ˜̃g1. (ii) If either ˜̃g1 ˜̃⊆ ˜̃g2 or ˜̃g2 ˜̃⊆ ˜̃g1, then ˜̃g1 is bipolar soft comparable with ˜̃g2. Example 1. Let Ω = {ω1, ω2, ω3, ω4, ω5, ω6, ω7, ω8} be the universal set which are eight categories of people that are living in Duhok city. It can be defined by: Ω ={SyrianRefugees, TurkishRefugees, IranianRefugees,HostCommunity, IDPs, Residents,Returnees, Foreigners}. Let ς = {ϱ1, ϱ2, ϱ3, ϱ4, ϱ5, ϱ6, ϱ7, ϱ8} be the set of parameters, where ϱi, i = 1, 2, ...7, 8, stands for parameters ”Hard Working”, ”Negligent”, ”Flexibility”, ”Rigidity”, ”Self-Confidence”, ”Shyness”, ”Skillful” and ”Unskillful” respectively. It is regarded as positive description and non positive description which belong to each category. The eight categories of people wish to find a job, to employ in government institute or work in a company in Duhok city. Now, we can divide the set ς into two parts ς1 = {ϱ1, ϱ3, ϱ5, ϱ7} and ς2 = {ϱ2, ϱ4, ϱ6, ϱ8} and the bijective function f : ς1 → ς2 can be defined as f(ϱi) = ¬ϱi = ϱi+1 for i = 1, 3, 5, 7. Here the notion ¬ϱi means Not ϱi for all i = 1, 3, 5, 7. Now, we can describe the following BSGT S ˜̃g = {(Φ, ˜̃Ω, ς), (Θ1,Λ1, ς), (Θ2,Λ2, ς), (Θ3,Λ3, ς)} offers to select some workers and employ them in tourism companies in Duhok city, where (Θ1,Λ1, ς) ={(ϱ1, {ω1, ω3, ω4}, {ω2, ω6}), (ϱ3, {ω2, ω5, ω7}, {ω1, ω3, ω8}), (ϱ5, {ω3, ω4}, {ω1, ω2, ω5, ω8}), (ϱ7, {ω5, ω6, ω7, ω8}, {ω2, ω3})}, (Θ2,Λ2, ς) ={(ϱ1, {ω1, ω2, ω4}, {ω3, ω5, ω6, ω7}), (ϱ3, {ω2, ω5}, {ω1, ω3, ω4, ω8}), (ϱ5, {ω1, ω3, ω4}, {ω2, ω5, ω7, ω8}), (ϱ7, {ω5}, {ω2, ω3, ω4})}and (Θ3,Λ3, ς) ={(ϱ1, {ω1, ω2, ω3, ω4}, {ω6}), (ϱ3, {ω2, ω5, ω7}, {ω1, ω3, ω8}), (ϱ5, {ω1, ω3, ω4}, {ω2, ω5, ω8}), (ϱ7, {ω5, ω6, ω7, ω8}, {ω2, ω3})}. Each (Θ,Λ, ς) in ˜̃g can be depicted as a table. Each category includes positive description αi and negative description βj and it can be represented by (αi, βj). If the description exists in a category, then it is considered as 1, otherwise, it is 0. Tabular representation of bipolar soft sets (Θ1,Λ1, ς), (Θ2,Λ2, ς) and (Θ3,Λ3, ς) are given in Tables 1,2 and 3. H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 652 Table 1: Tabular form of the bipolar soft set (Θ1,Λ1, ς) (Θ1,Λ1, ς) (Θ1,Λ1)(ϱ1) (Θ1,Λ1)(ϱ3) (Θ1,Λ1)(ϱ5) (Θ1,Λ1)(ϱ7) ω1 (1,0) (0,1) (0,1) (0,0) ω2 (0,1) (1,0) (0,1) (0,1) ω3 (1,0) (0,1) (1,0) (0,1) ω4 (1,0) (0,0) (1,0) (0,0) ω5 (0,0) (1,0) (0,1) (1,0) ω6 (0,1) (0,0) (0,0) (1,0) ω7 (0,0) (1,0) (0,0) (0,1) ω8 (0,0) (0,1) (0,1) (1,0) Table 2: Tabular form of the bipolar soft set (Θ2,Λ2, ς) (Θ2,Λ2, ς) (Θ2,Λ2)(ϱ1) (Θ2,Λ2)(ϱ3) (Θ2,Λ2)(ϱ5) (Θ2,Λ2)(ϱ7) ω1 (1,0) (0,1) (1,0) (0,0) ω2 (1,0) (1,0) (0,1) (0,1) ω3 (0,1) (0,1) (1,0) (0,1) ω4 (1,0) (0,1) (1,0) (0,1) ω5 (0,1) (0,0) (0,1) (1,0) ω6 (0,1) (0,0) (0,0) (0,0) ω7 (0,1) (0,0) (0,1) (0,0) ω8 (0,0) (0,1) (0,1) (0,0) Table 3: Tabular form of the bipolar soft set (Θ3,Λ3, ς) (Θ3,Λ3, ς) (Θ3,Λ3)(ϱ1) (Θ3,Λ3)(ϱ3) (Θ3,Λ3)(ϱ5) (Θ3,Λ3)(ϱ7) ω1 (1,0) (0,1) (1,0) (0,0) ω2 (1,0) (1,0) (0,1) (0,1) ω3 (1,0) (0,1) (1,0) (0,1) ω4 (1,0) (0,0) (1,0) (0,0) ω5 (0,0) (1,0) (0,1) (1,0) ω6 (0,1) (0,0) (0,0) (1,0) ω7 (0,0) (1,0) (0,0) (1,0) ω8 (0,0) (0,1) (0,1) (1,0) Theorem 1. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S, then g̃ = {(Θ, ς) : (Θ,Λ, ς) ˜̃∈ ˜̃g} is SGT . Proof. Suppose that (Ω, ˜̃g, ς,¬ς) is a BSGT S. Then (Φ, ˜̃ Ω, ς) ˜̃∈ ˜̃g implies that (Φ, ς)∈̃ g̃. Let {(Θi, ς) : i ∈ I} belongs to g̃. Since (Θi,Λi, ς) ˜̃∈ ˜̃g for all i ∈ I, so that ˜̃⋃ i∈I(Θi,Λi, ς)˜̃∈ ˜̃g. Thus, ⋃̃i∈I(Θi, ς) ∈̃ g̃. Hence g̃ defines a SGT . The following example shows that the converse of Theorem 1 is not true. Example 2. Let Ω = {ω1, ω2, ω3, ω4} and ς = {ϱ1, ϱ2}. Suppose that g̃ = {(Φ, ς), (Θ1, ς), (Θ2, ς), (Θ3, ς), (Θ4, ς)} and H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 653 ¬g̃ = {(Φ, ς), (Λ1,¬ς), (Λ2,¬ς), (Λ3,¬ς), (Λ4,¬ς)} are two soft generalized topologies de- fined on Ω, where (Θ1, ς) = {(ϱ1, {ω2}), (ϱ2, {ω1})}, (Θ2, ς) = {(ϱ1, {ω1}), (ϱ2, {ω3})}, (Θ3, ς) = {(ϱ1, {ω2}), (ϱ2, {ω1, ω3})}, (Θ4, ς) = {(ϱ1, {ω1, h2}), (ϱ2, {ω1, ω3})}, and (Λ1,¬ς) = {(¬ϱ1, {ω1, ω3, ω4}), (¬ϱ2, {ω2, ω4})}, (Λ2,¬ς) = {(¬ϱ1, {ω3}), (¬ϱ2, {ω4})}, (Λ3,¬ς) = {(¬ϱ1, {ω3, ω4}), (¬ϱ2, {ω2})}, (Λ4,¬ς) = {(¬ϱ1, {ω3, ω4}), (¬ϱ2, {ω2, ω4})}. Then ˜̃g = {(Φ, ˜̃Ω, ς), (Θ1,Λ1, ς), (Θ2,Λ2, ς), (Θ3,Λ3, ς), (Θ4,Λ4, ς}, where (Θ1,Λ1, ς), (Θ2,Λ2, ς), (Θ3,Λ3, ς) and (Θ4,Λ4, ς) are bipolar soft sets defined as follows (Θ1,Λ1, ς) = {(ϱ1, {ω2}, {ω1, ω3, ω4}), (ϱ2, {ω1}, {ω2, ω4})}, (Θ2,Λ2, ς) = {(ϱ1, {ω1}, {ω3}), (ϱ2, {ω3}, {ω4})}, (Θ3,Λ3, ς) = {(ϱ1, {ω2}.{ω3, ω4}), (ϱ2, {ω1, ω3}, {ω2})}, (Θ4,Λ4, ς) = {(ϱ1, {ω1, ω2}, {ω3, ω4}), (ϱ2, {ω1, ω3}, {ω2, ω4})}. Thus, (Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς) = {(ϱ1, {ω1, ω2}, {ω3}), (ϱ2, {ω1, ω3}, {ω4})} ˜̃ /∈ ˜̃g. Therefore,˜̃g is not BSGT . The following theorem shows when that converse of Theorem 1 is true. Theorem 2. Let (Ω, g̃, ς) be a SGT . Then the collection ˜̃g consisting of bipolar soft sets (Θ,Λ, ς) such that (Θ, ς) ∈̃ g̃ and Λ(¬ϱ) = Ω \Θ(ϱ) for all ¬ϱ ∈ ¬ς, defines a BSGT on Ω. Proof. (i) Since (Φ, ς) ∈̃ g̃, then Ω(¬ϱ) = Ω \ Φ(ϱ) = Ω \ ϕ = Ω and hence (Φ, ˜̃ Ω, ς) ˜̃∈ ˜̃g. (ii) Let {(Θi,Λi, ς) : i ∈ I} ˜̃∈ ˜̃g. Then {(Θi, ς) : i ∈ I} ∈̃ g̃ and Λi(¬ϱ) = Ω \ Θi(ϱ). Now, since g̃ is a SGT , then ⋃̃ i∈I(Θi, ς) ∈̃ g̃. Let (Θ, ς) = ⋃̃ i∈I(Θi, ς) ∈̃ g̃, then ς(¬ϱ) = Ω \ ( ⋃ i∈IΘi(ϱ)) = ⋂ i∈IΛi(¬ϱ). Thus, ˜̃⋃ i∈I(Θi,Λi, ς) ˜̃∈ ˜̃g. Therefore, the proof is completed. Theorem 3. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and {˜̃gi}i∈I be an indexed family of BSGT s. Then ˜̃⋂ i∈I ˜̃g is a BSGT , where each ˜̃gi is bipolar soft finer than ˜̃⋂ i∈I ˜̃gi for each i. H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 654 Proof. Since each {˜̃gi}, i ∈ I is a BSGT over Ω, the bipolar soft set (Φ, ˜̃ Ω, ς) ˜̃∈ ˜̃gi, i ∈ I and hence (Φ, ˜̃ Ω, ς) ˜̃∈ ˜̃⋂ i∈I ˜̃gi. Let {(Θj ,Λj , ς) : j ∈ J } be a family of bipolar soft sets in ˜̃⋂ i∈I ˜̃gi. Then each (Θj ,Λj , ς) belongs to each ˜̃gi. But ˜̃gi being BSGT is closed under arbitrary bipolar soft unions. So ˜̃⋃ j∈J (Θj ,Λj , ς) ˜̃∈ ˜̃⋂ i∈I ˜̃gi. Hence ˜̃⋂ i∈I ˜̃gi is a BSGT define on Ω. Clearly each ˜̃gi, i ∈ I, is bipolar soft finer than ˜̃⋂ i∈I ˜̃gi. Remark 1. Let (Ω, ˜̃g1, ς,¬ς) and (Ω, ˜̃g2, ς,¬ς) be BSGT Ss. Then (Ω, ˜̃g1 ˜̃∩ ˜̃g2, ς,¬ς) may not be a BSGT S as shown by the following example. Example 3. Let Ω = {ω1, ω2, ω3, ω4, ω5} and ς = {ϱ1, ϱ2}. Let (Ω, ˜̃g1, ς,¬ς), (Ω, ˜̃g2, ς,¬ς)˜̃∈ BSGT Ss where ˜̃g1={(Φ, ˜̃Ω, ς), (Θ1,Λ1, ς), (Θ2,Λ2, ς)}, and ˜̃g2 = {(Φ, ˜̃Ω, ς), (Θ3,Λ3, ς), (Θ4,Λ4, ς)}, where (Θ1,Λ1, ς) = {(ϱ1, {ω1, ω2}, {ω5}), (ϱ2, {ω2}, {ω1, ω4, ω5})}, (Θ2,Λ2, ς) = {(ϱ1, {ω1, ω2}, {ω5}), (ϱ2, {ω2}, {ω1, ω4})}, (Θ3,Λ3, ς) = {(ϱ1, {ω3, ω4}, {ω5}), (ϱ2, {ω3}, {ω4})}and (Θ4,Λ4, ς) = {(ϱ1, {ω3, ω4}, {ω5}), (ϱ2, {ω3}, {ω4, ω5})}. Now, ˜̃g1 ˜̃∪ ˜̃g2 = {(Φ, ˜̃Ω, ς), (Θ1,Λ1, ς), (Θ2,Λ2, ς), (Θ3,Λ3, ς), (Θ4,Λ4, ς)}, then ˜̃g1 ˜̃∪ ˜̃g2 is not BSGT S since (Θ2,Λ2, ς) ˜̃∈ ˜̃g1 ˜̃∪ ˜̃g2 and (Θ3,Λ3, ς) ˜̃∈ ˜̃g1 ˜̃∪ ˜̃g2, but (Θ2,Λ2, ς) ˜̃∪ (Θ3,Λ3, ς) = {(ϱ1, {ω1, ω2, ω3, ω4}, {ω5}), (ϱ2, {ω2, ω3}, {ω4})} ˜̃ /∈ ˜̃g1 ˜̃∪ ˜̃g2. Theorem 4. Let ˜̃ ζ be a family of bipolar soft sets defined on a universal set Ω, then there exists a unique BSGT ˜̃g such that it is the smallest BSGT containing ˜̃ ζ. Proof. Consider the collection of all BSGT s on Ω which contains ˜̃ ζ (as subset of Υ(Ω)) surely contains ˜̃ ζ. Now, let ˜̃g be the intersection of the members of this collection. By Theorem 3, ˜̃g is a BSGT , it contains ˜̃ ζ and clearly it is the smallest BSGT containing ˜̃ ζ, for any such BSGT will be member of the collection of BSGT s just considered, and hence bipolar soft finer than its intersections ˜̃g. Uniqueness of ˜̃g is trivial. Definition 19. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then the bipolar soft ˜̃g-interior of (Θ,Λ, ς), denoted by i˜̃g(Θ,Λ, ς), is the bipolar soft union of all bipolar soft ˜̃g-open subsets of (Θ,Λ, ς). In other words, i˜̃g(Θ,Λ, ς) is a largest bipolar soft ˜̃g-open set contained in (Θ,Λ, ς), so, we can write as H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 655 i˜̃g(Θ,Λ, ς) = ˜̃⋃{(χ, ψ, ς): (χ, ψ, ς) ˜̃∈ ˜̃g, (χ, ψ, ς) ˜̃⊆ (Θ,Λ, ς)}. Here, are some properties of i˜̃g(Θ,Λ, ς). Theorem 5. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ1,Λ1, ς),(Θ2,Λ2, ς) ˜̃∈ BSS(Ω). Then (i) i˜̃g(Θ1,Λ1, ς) ˜̃⊆ (Θ1,Λ1, ς). (ii) (Θ1,Λ1, ς) is bipolar soft ˜̃g-open if and only if i˜̃g(Θ1,Λ1, ς) = (Θ1,Λ1, ς). (iii) i˜̃g(i˜̃g(Θ1,Λ1, ς) = i˜̃g(Θ1,Λ1, ς). (iv) If (Θ1,Λ1, ς) ˜̃⊆ (Θ2,Λ2, ς), then i˜̃g(Θ1,Λ1, ς) ˜̃⊆ i˜̃g(Θ2,Λ2, ς). (v) i˜̃g((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊆ i˜̃g(Θ1,Λ1, ς) ˜̃∩ i˜̃g(Θ2,Λ2, ς). (vi) i˜̃g((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)) ˜̃⊇ i˜̃g(Θ1,Λ1, ς) ˜̃∪ i˜̃g(Θ2,Λ2, ς). (vii) i˜̃g (Φ, ˜̃Ω, ς) = (Φ, ˜̃ Ω, ς). Proof. (i) Since i˜̃g(Θ1,Λ1, ς) = ˜̃⋃{(Θj ,Λj , ς) : (Θj ,Λj , ς) ˜̃∈ ˜̃g, (Θj ,Λj , ς) ˜̃⊆ (Θ1,Λ1, ς), j ∈ J }. Then Θj(ϱ) ⊆ Θ1(ϱ) and Λ1(¬ϱ) ⊆ ΛJ (¬ϱ) for all j ∈ J . So, ⋃ j∈J Θj(ϱ) ⊆ Θ1(ϱ) and Λ1(¬ϱ) ⊆ ⋂ j∈J Λj(¬ϱ). Therefore i˜̃g (Θ1,Λ1, ς) ˜̃⊆ (Θ1,Λ1, ς). (ii) Let (Θ1,Λ1, ς) be a bipolar soft ˜̃g-open set. Then (Θ1,Λ1, ς) is the largest bipolar soft ˜̃g-open set contained in (Θ1,Λ1, ς). From (i), we have i˜̃g (Θ1,Λ1, ς) ˜̃⊆ (Θ1,Λ1, ς). Therefore, i˜̃g (Θ1,Λ1, ς) = (Θ1,Λ1, ς). Conversely, assume that i˜̃g (Θ1,Λ1, ς) = (Θ1,Λ1, ς). Since i˜̃g (Θ1,Λ1, ς) is a bipolar soft union of all bipolar soft ˜̃g-open subsets of (Θ1,Λ1, ς) and ˜̃g is closed under arbi- trary bipolar soft union, then i˜̃g(Θ1,Λ1, ς) is bipolar soft ˜̃g-open. Thus, (Θ1,Λ1, ς) is bipolar soft ˜̃g-open. (iii) Since i˜̃g (Θ1,Λ1, ς) is a bipolar soft ˜̃g-open set. Thus by (ii), i˜̃g (i˜̃g (Θ1,Λ1, ς)) = i˜̃g (Θ1,Λ1, ς). (iv) Suppose (Θ1,Λ1, ς) ˜̃⊆ (Θ2,Λ2, ς). From (i), i˜̃g(Θ1,Λ1, ς) ˜̃⊆ (Θ1,Λ1, ς). Therefore, i˜̃g(Θ1,Λ1, ς) ˜̃⊆ (Θ2,Λ2, ς). Now, i˜̃g(Θ1,Λ1, ς) is a bipolar soft ˜̃g-open set contained in (Θ2,Λ2, ς). So, it is contained in bipolar soft ˜̃g-interior, and since i˜̃g (Θ2,Λ2, ς) is H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 656 the largest bipolar soft ˜̃g-open set contained in (Θ2,Λ2, ς). Therefore, i˜̃g (Θ1,Λ1, ς)˜̃⊆ i˜̃g (Θ2,Λ2, ς). (v) Since (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς) ˜̃⊆ (Θ1,Λ1, ς) and (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς) ˜̃⊆ (Θ2,Λ2, ς). So, by (iv), i˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊆ i˜̃g (Θ1,Λ1, ς) and i˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς))˜̃⊆ i˜̃g (Θ2,Λ2, ς). Hence, i˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊆ i˜̃g (Θ1,Λ1, ς) ˜̃∩ i˜̃g(Θ2,Λ2, ς). (vi) We know that (Θ1,Λ1, ς) ˜̃⊆ (Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς) and (Θ2,Λ2, ς) ˜̃⊆ (Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς). Then by (v), i˜̃g (Θ1,Λ1, ς) ˜̃⊆ i˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)) and i˜̃g (Θ2,Λ2, ς)˜̃⊆ i˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)). So, i˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)) ˜̃⊇ i˜̃g (Θ1,Λ1, ς) ˜̃∪ i˜̃g(Θ2,Λ2, ς). (vii) The proof is trivial. In the next example, we will show that the equality of parts (v) and (vi) in Theorem 5 do not hold. Example 4. Let Ω = {ω1, ω2, ω3, ω4, ω5}, ς = {ϱ3, ϱ4} and˜̃g = {(Φ, ˜̃Ω, ς), (Θ1,Λ1, ς), (Θ2,Λ2, ς), (Θ3,Λ3, ς), (Θ4,Λ4, ς), (Θ5,Λ5, ς)}, where (Θ1,Λ1, ς) = {(ϱ3, {ω1, ω3, ω5}, {ω2}), (ϱ4, {ω1, ω3, ω5}, {ω2})}, (Θ2,Λ2, ς) = {(ϱ3, {ω4}, {ω1, ω2}), (ϱ4, {ω4}, {ω1, ω2})}, ((Θ3,Λ3, ς) = {(ϱ3, {ω4}, {ω2, ω3}), (ϱ4, {ω4}, {ω2, ω3})}, (Θ4,Λ4, ς) = {(ϱ3, {ω1, ω3, ω4, ω5}, {ω2}), (ϱ4, {ω1, ω3, ω4, ω5}, {ω2})} and (Θ5,Λ5, ς) = {(ϱ3, {ω4}, {ω2}), (ϱ4, {ω4}, {ω2})}. To show the converse of (v). Let (χ1, ψ1, ς) = {(ϱ3, {ω4, ω5}, {ω1, ω2}), (ϱ4, {ω4, ω5}, {ω1, ω2})} and (Θ3,Λ3, ς) = (χ2, ψ2, ς) = {(ϱ3, {ω4}, {ω2, ω3}), (ϱ4, {ω4}, {ω2, ω3})}. So, i˜̃g(χ1, ψ1, ς) = (Θ2,Λ2, ς) and i˜̃g(χ2, ψ2, ς) = (Θ3,Λ3, ς), Hence, i˜̃g (χ1, ψ1, ς) ˜̃∩ i˜̃g(χ2, ψ2, ς) = {(eϱ3, {ω4}, {ω1, ω2, ω3}), (ϱ4, {ω4}, {ω1, ω2, ω3})}. Also, i˜̃g ((χ1, ψ1, ς) ˜̃∩(χ2, ψ2, ς)) = i˜̃g{(ϱ3, {ω4}, {ω1, ω2, ω3}), (ϱ4, {ω4}, {ω1, ω2, ω3})} = (Φ, ˜̃ Ω, ς). Therefore, i˜̃g(χ1, ψ1, ς) ˜̃∩ i˜̃g (χ2, ψ2, ς) ̸= i˜̃g ((χ1, ψ1, ς) ˜̃∩ (χ2, ψ2, ς)). H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 657 Now, to show the converse of (vi). Let (χ1, ψ1, ς) = {(ϱ3, {ω1, ω3}, {ω2}), (ϱ4, {ω1, ω3}, {ω2})} and (χ2, ψ2, ς) = {(e3, {ω4, ω5}, {ω1, ω2, ω3}), (ϱ4, {ω4, ω5}, {ω1, ω2, ω3})}. Then i˜̃g(χ1, ψ1, ς) = (Φ, ˜̃ Ω, ς) and i˜̃g(χ2, ψ2, ς) = (Φ, ˜̃ Ω, ς). Hence, i˜̃g(χ1, ψ1, ς) ˜̃∪ i˜̃g(χ2, ψ2, ς) = (Φ, ˜̃ Ω, ς). But (χ1, ψ1, ς) ˜̃∪(χ2, ψ2, ς) = {(ϱ3, {ω1, ω3, ω4, ω5}, {ω2}), (ϱ4, {ω1, ω3, ω4, ω5}, {ω2})}. While i˜̃g ((χ1, ψ1, ς) ˜̃∪ (χ2, ψ2, ς)) = (Θ4,Λ4, ς). So, i˜̃g ((χ1, ψ1, ς) ˜̃∪ (χ2, ψ2, ς)) ̸= i˜̃g(χ1, ψ1, ς) ˜̃∪ i˜̃g(χ2, ψ2, ς). Definition 20. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then (Θ,Λ, ς) is said to be bipolar soft ˜̃g-closed if its bipolar soft complement (Θ,Λ, ς)c is bipolar soft˜̃g-open. Theorem 6. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S, then ¬g̃ = {(Λ,¬ς) : (Θ,Λ, ς) ˜̃∈ ˜̃g} is SGT in terms of soft g̃-closed. Proof. Similar to Theorem 1. Theorem 7. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S . Then the following properties hold (i) ( ˜̃ Ω,Φ, ς) is bipolar soft ˜̃g-closed. (ii) Arbitrary bipolar soft intersections of the bipolar soft ˜̃g-closed sets are bipolar soft˜̃g-closed. Proof. (i) Since the complement of the absolute bipolar soft set ( ˜̃ Ω,Φ, ς) is the relative null bipolar soft set (Φ, ˜̃ Ω, ς), and (Φ, ˜̃ Ω, ς)˜̃∈ ˜̃g. Thus, (˜̃Ω,Φ, ς) is bipolar soft ˜̃g-closed. (ii) Let {(Θi,Λi, ς)}i∈I be a given collection of bipolar soft ˜̃g-closed sets. To show˜̃⋂ i∈I (Θi,Λi, ς) is bipolar soft ˜̃g-closed. Now ( ˜̃⋂ i∈I ((Θi,Λi, ς)) c = ˜̃⋃ i∈I (Θi,Λi, ς) c. Since (Θi,Λi, ς) is a bipolar soft ˜̃g-closed for each i ∈ I. So (Θi,Λi, ς) c is bipo- lar soft ˜̃g-open sets and hence ˜̃⋃ i∈I (Θi,Λi, ς) c is bipolar soft ˜̃g-open. Therefore, ( ˜̃⋂ i∈I (Θi,Λi, ς)) c is also bipolar soft ˜̃g-open. This means that, ˜̃⋂ i∈I (Θi,Λi, ς) is bipolar soft ˜̃g-closed. Definition 21. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then the bipolar soft ˜̃g-closure of (Θ,Λ, ς), denoted by c˜̃g (Θ,Λ, ς), is the bipolar soft intersection of all H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 658 bipolar soft ˜̃g-closed sets containing (Θ,Λ, ς). In other words, c˜̃g (Θ,Λ, ς) is the smallest bipolar soft ˜̃g-closed set containing (Θ,Λ, ς), so, we can write as c˜̃g (Θ,Λ, ς) = ˜̃⋂ {(χ, ψ, ς) : (χ, ψ, ς) is bipolar soft ˜̃g-closed; (χ, ψ, ς) ˜̃⊇ (Θ,Λ, ς)}. Theorem 8. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ1,Λ1, ς), (Θ2,Λ2, ς) ˜̃∈ BSS(Ω). Then (i) (Θ1,Λ1, ς) ˜̃⊆ c˜̃g (Θ1,Λ1, ς). (ii) (Θ1,Λ1, ς) is bipolar soft ˜̃g-closed if and only if c˜̃g (Θ1,Λ1, ς) = (Θ1,Λ1, ς). (iii) c˜̃g (c˜̃g (Θ1,Λ1, ς)) = c˜̃g (Θ1,Λ1, ς). (iv) If (Θ1,Λ1, ς) ˜̃⊆(Θ2,Λ2, ς), then c˜̃g (Θ1,Λ1, ς) ˜̃⊆ c˜̃g (Θ2,Λ2, ς). (v) c˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊆ c˜̃g (Θ1,Λ1, ς) ˜̃∩ c˜̃g (Θ2,Λ2, ς). (vi) c˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς) ˜̃⊇ c˜̃g (Θ1,Λ1, ς) ˜̃∪ c˜̃g (Θ2,Λ2, ς). (vii) c˜̃g(˜̃Ω,Φ, ς) = ( ˜̃ Ω,Φ, ς). Proof. (i) Since c˜̃g (Θ1,Λ1, ς) = ˜̃⋂{(Θi,Λi, ς) : (Θi,Λi, ς) c ˜̃∈ ˜̃g, (Θ1,Λ1, ς) ˜̃⊆ (Θi,Λi, ς), i ∈ I}. Then Θ1(ϱ) ⊆ Θi(ϱ) and Λi(¬ϱ) ⊆ Λ1(¬ϱ) for all i ∈ I. So, Θ1(ϱ) ⊆ ⋂ i∈I Λi(ϱ) and ⋃ i∈I Λi(¬ϱ) ⊆ Λ1(¬ϱ). Thus, (Θ1,Λ1, ς) ˜̃⊆ c˜̃g (Θ1,Λ1, ς). (ii) Let (Θ1,Λ1, ς) be a bipolar soft ˜̃g-closed set. Then (Θ1,Λ1, ς), is the smallest bipolar soft ˜̃g-closed set containing (Θ1,Λ1, ς). From (i), we have (Θ1,Λ1, ς) ˜̃⊆ c˜̃g (Θ1,Λ1, ς). Therefore, c˜̃g (Θ1,Λ1, ς) = (Θ1,Λ1, ς). (iii) Since c˜̃g (Θ1,Λ1, ς) is a bipolar soft ˜̃g-closed set. Thus by (ii), it is equal to its bipolar soft ˜̃g-closure. Therefore c˜̃g (c˜̃g (Θ1,Λ1, ς)) = c˜̃g (Θ1,Λ1, ς). (iv) Suppose (Θ1,Λ1, ς) ˜̃⊆ (Θ2,Λ2, ς). From (i), (Θ2,Λ2, ς) ˜̃⊆ c˜̃g (Θ2,Λ2, ς). Thus, (Θ1,Λ1, ς) ˜̃⊆ c˜̃g (Θ2,Λ2, ς). Now, c˜̃g (Θ2,Λ2, ς) is a bipolar soft ˜̃g-closed set contain- ing (Θ1,Λ1, ς). So it is containing its bipolar soft ˜̃g-closure, and since c˜̃g (Θ1,Λ1, ς) is the smallest bipolar soft ˜̃g-closed set containing (Θ1,Λ1, ς). Therefore, c˜̃g (Θ1,Λ1, ς)˜̃⊆ c˜̃g (Θ2,Λ2, ς). H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 659 (v) Since (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς) ˜̃⊆ (Θ1,Λ1, ς) and (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς) ˜̃⊆(Θ2,Λ2, ς). So by (iv), we get c˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊆ c˜̃g (Θ1,Λ1, ς) and c˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊆ c˜̃g (Θ2,Λ2, ς). Hence, c˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊆ c˜̃g (Θ1,Λ1, ς) ˜̃∩ c˜̃g (Θ2,Λ2, ς). (vi) Since (Θ1,Λ1, ς) ˜̃⊆ (Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς) and (Θ2,Λ2, ς) ˜̃⊆ (Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς). Then by (iv), we have c˜̃g (Θ1,Λ1, ς) ˜̃⊆ c˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)) and c˜̃g (Θ2,Λ2, ς)˜̃⊆ c˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)). Thus, c˜̃g (Θ1,Λ1, ς) ˜̃∪ c˜̃g (Θ2,Λ2, ς) ˜̃⊆ c˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)). (vii) The proof is trivial. In the next example, we will show that the equality of parts (v) and (vi) in Theorem 8 does not hold. Example 5. Let Ω = {ω1, ω2, ω3, ω4, ω5}, A = {ϱ3, ϱ4} and˜̃g = {(Φ, ˜̃Ω, ς), (Θ1,Λ1, ς), (Θ2,Λ2, ς), (Θ3,Λ3, ς), (Θ4,Λ4, ς), (Θ5,Λ5, ς), (Θ6,Λ6, ς)} where (Θ1,Λ1, ς) = {(ϱ3, {ω2, ω4}, {ω1, ω3, ω5}), (ϱ4, {ω2, ω4}, {ω1, ω3, ω5})}, (Θ2,Λ2, ς) = {(ϱ3, {ω4, ω5}, {ω2, ω3}), (ϱ4, {ω4, ω5}, {ω2, ω3})}, (Θ3,Λ3, ς) = {(ϱ3, {ω1, ω5}, {ω2, ω4}), (ϱ4, {ω1, ω5}, {ω2, ω4})}, (Θ4,Λ4, ς) = {(ϱ3, {ω2, ω4, ω5}, {ω3}), (ϱ4, {ω2, ω4, ω5}, {ω3}), (Θ5,Λ5, ς) = {(ϱ3, {ω1, ω2, ω4, ω5}, ϕ), (ϱ4, {ω1, ω2, ω4, ω5}, ϕ)} and (Θ6,Λ6, ς) = {(ϱ3, {ω1, ω4, ω5}, {ω2}), (ϱ4, {ω1, ω4, ω5}, {ω2})}. Then ˜̃g c = {( ˜̃Ω,Φ, ς), (Θ1,Λ1, ς) c, (Θ2,Λ2, ς) c, (Θ3,Λ3, ς) c, (Θ4,Λ4, ς) c, (Θ5,Λ5, ς) c, (Θ6,Λ6, ς) c}, where (Θ1,Λ1, ς) c = {(ϱ3, {ω1, ω3, ω5}, {ω2, ω4}), (ϱ4, {ω1, ω3, ω5}, {ω2, ω4})}, H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 660 (Θ2,Λ2, ς) c = {(ϱ3, {ω2, ω3}, {ω4, ω5}), (ϱ4, {ω2, ω3}, {ω4, ω5})}, (Θ3,Λ3, ς) c = {(ϱ3, {ω2, ω4}, {ω1, ω5}), (ϱ4, {ω2, ω4}, {ω1, ω5})} (Θ4,Λ4, ς) c = {(ϱ3, {ω3}, {ω2, ω4, ω5}), (ϱ4, {ω3}, , {ω2, ω4, ω5})}, (Θ5,Λ5, ς) c = {(ϱ3, ϕ, {ω1, ω2, ω4, ω5}), (ϱ4, ϕ, {ω1, ω2, ω4, ω5})} and (Θ6,Λ6, ς) c = {(ϱ3, {ω2}, {ω1, ω4, ω5}), (ϱ4{ω2}, {ω1, ω4, ω5})}. To show the converse of (v). Let (χ1, ψ1, ς) = {(ϱ3, {ω2, ω4, ω5}, {ω1, ω3}), (ϱ4, {ω2, ω4, ω5}, {ω1, ω3})} and (χ2, ψ2, ς) = {(ϱ3, {ω1, ω2, ω4}, {ω5}), (ϱ4, {ω1, ω2, ω4}, {ω5})}. So, c˜̃g (χ1, ψ1, ς) = ( ˜̃ Ω,Φ, ς) and c˜̃g (χ2, ψ2, ς) = ( ˜̃ Ω,Φ, ς). Hence, c˜̃g (χ1, ψ1, ς) ˜̃∩ c˜̃g (χ2, ψ2, ς) = ( ˜̃ Ω,Φ, ς). Also, c˜̃g ((χ1, ψ1, ς) ˜̃∩(χ2, ψ2, ς)) = c˜̃g ({(ϱ3, {ω2, ω4}, {ω1, ω3, ω5}), (ϱ4, {ω2, ω4}, {ω1, ω3, ω5})}) = (Θ3,Λ3, ς) c. Thus, c˜̃g (χ1, ψ1, ς) ˜̃∩ c˜̃g (χ2, ψ2, ς) ̸= c˜̃g ((χ1, ψ1, ς) ˜̃∩ (χ2, ψ2, ς)). Now, to show the converse of (vi). Let (χ1, ψ1, ς) = {(ϱ3, {ω1, ω5}, {ω2, ω4}), (ϱ4, {ω1, ω5}, {ω2, ω4})}, and (χ2, ψ2, ς) = {(ϱ3, {ω2, ω3}, {ω1, ω4, ω5}), (ϱ4, {ω2, ω3}, {ω1, ω4, ω5})}. Then c˜̃g (χ1, ψ1, ς) = (Θ1,Λ1, ς) c and c˜̃g (χ2, ψ2, ς) = (Θ2,Λ2, ς) c. Thus, c˜̃g (χ1, ψ1, ς) ˜̃∪ c˜̃g (χ2, ψ2, ς) = (Θ1,Λ1, ς) c ˜̃∪ (Θ2,Λ2, ς) c = {(ϱ3, {ω1, ω2, ω3, ω5}, {ω4}), (ϱ4, {ω1, ω2, ω3, ω5}, {ω4})}. But (χ1, ψ1, ς) ˜̃∪ (χ2, ψ2, ς) = {(ϱ3, {ω1, ω2, ω3, ω5}, {ω4}), (ϱ4, {ω1, ω2, ω3, ω5}, {ω4})}. Hence, c˜̃g ((χ1, ψ1, ς) ˜̃∪ (χ2, ψ2, ς)) = ( ˜̃ Ω,Φ, ς). Therefore, c˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς) ̸= c˜̃g (Θ1,Λ1, ς) ˜̃∪ c˜̃g (Θ2,Λ2, ς). H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 661 Proposition 2. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then i˜̃g (Θ,Λ, ς)˜̃⊆ (Θ,Λ, ς) ˜̃⊆ c˜̃g (Θ,Λ, ς). Theorem 9. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς), (χ, ψ, ς) ˜̃∈ BSS(Ω). Then (i) c˜̃g (Θ,Λ, ς)c = (i˜̃g (Θ,Λ, ς))c. (ii) i˜̃g (Θ,Λ, ς)c = (c˜̃g (Θ,Λ, ς))c. (iii) i˜̃g (Θ,Λ, ς) = (c˜̃g (Θ,Λ, ς)c)c. (iv) c˜̃g (Θ,Λ, ς) = (i˜̃g (Θ,Λ, ς)c)c. (v) i˜̃g ((Θ,Λ, ς) ˜̃\ (χ, ψ, ς)) ˜̃⊆ i˜̃g (Θ,Λ, ς) ˜̃\ i˜̃g (χ, ψ, ς). Proof. It is enough to prove only parts (i) and (v) since the proof of other parts are similar. (i) Since (i˜̃g (Θ,Λ, ς))c = ( ˜̃⋃{(Θi,Λi, ς) : (Θi,Λi, ς) ˜̃∈ ˜̃g, (Θi,Λi, ς) ˜̃⊆ (Θ,Λ, ς), i ∈ I})c = ˜̃⋂ {(Θi,Λi, ς) c: (Θi,Λi, ς) ˜̃∈ ˜̃g, (Θ,Λ, ς)c ˜̃⊆ (Θi,Λi, ς) c, i ∈ I} = c˜̃g(Θ,Λ, ς)c. (v) Since i˜̃g ((Θ,Λ, ς)˜̃\(χ, ψ, ς)) = i˜̃g((Θ,Λ, ς)˜̃∩ (χ, ψ, ς)c)˜̃⊆i˜̃g (Θ,Λ, ς)˜̃∩ i˜̃g (χ, ψ, ς)c (by Theorem 5(v)) = i˜̃g (Θ,Λ, ς) ˜̃∩(c˜̃g (χ, ψ, ς))c˜̃⊆ i˜̃g (Θ,Λ, ς) ˜̃∩(i˜̃g (χ, ψ, ς))c = i˜̃g (Θ,Λ, ς)˜̃\i˜̃g (χ, ψ, ς). Definition 22. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then the bipolar soft ˜̃g-boundary of (Θ,Λ, ς), denoted by b˜̃g (Θ,Λ, ς), is defined as b˜̃g (Θ,Λ, ς) = c˜̃g (Θ,Λ, ς) ˜̃∩ c˜̃g (Θ,Λ, ς)c. Proposition 3. It is clear that b˜̃g (Θ,Λ, ς) = b˜̃g (Θ,Λ, ς)c. Theorem 10. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then (i) b˜̃g (Θ,Λ, ς) ˜̃⊆ c˜̃g (Θ,Λ, ς). (ii) (Θ,Λ, ς) ˜̃∪ b˜̃g (Θ,Λ, ς) ˜̃⊆ c˜̃g (Θ,Λ, ς). H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 662 (iii) i˜̃g (Θ,Λ, ς) ˜̃⊆ (Θ,Λ, ς) ˜̃\ b˜̃g (Θ,Λ, ς). (iv) b˜̃g(i˜̃g (Θ,Λ, ς)) ˜̃⊆ b˜̃g(Θ,Λ, ς). (v) b˜̃g(c˜̃g (Θ,Λ, ς)) ˜̃⊆ b˜̃g(Θ,Λ, ς). Proof. (i) Since b˜̃g (Θ,Λ, ς) = c˜̃g (Θ,Λ, ς) ˜̃∩ c˜̃g (Θ,Λ, ς)c. Then b˜̃g (Θ,Λ, ς) ˜̃⊆ c˜̃g (Θ,Λ, ς). (ii) (Θ,Λ, ς)˜̃∪ b˜̃g (Θ,Λ, ς) = (Θ,Λ, ς)˜̃∪ (c˜̃g (Θ,Λ, ς)˜̃∩c˜̃g (Θ,Λ, ς)c) = ((Θ,Λ, ς)˜̃∪ c˜̃g (Θ,Λ, ς))˜̃∩((Θ,Λ, ς)˜̃∪ c˜̃g (Θ,Λ, ς)c) = c˜̃g (Θ,Λ, ς)˜̃∩((Θ,Λ, ς)˜̃∪ c˜̃g (Θ,Λ, ς)c)˜̃⊆c˜̃g (Θ,Λ, ς). (iii) (Θ,Λ, ς) ˜̃\b˜̃g (Θ,Λ, ς) = (Θ,Λ, ς)˜̃∩(b˜̃g (Θ,Λ, ς))c = (Θ,Λ, ς)˜̃∩(c˜̃g (Θ,Λ, ς)˜̃∩ c˜̃g (Θ,Λ, ς)c)c = (Θ,Λ, ς)˜̃∩(i˜̃g (Θ,Λ, ς)c ˜̃∪ i˜̃g (Θ,Λ, ς)) = ((Θ,Λ, ς)˜̃∩i˜̃g (Θ,Λ, ς)c)˜̃∪ ((Θ,Λ, ς)˜̃∩i˜̃g (Θ,Λ, ς)) = (Φ,Λ, ς)˜̃∪i˜̃g (Θ,Λ, ς)˜̃⊇i˜̃g (Θ,Λ, ς). (iv) b˜̃g (i˜̃g (Θ,Λ, ς)) = c˜̃g(i˜̃g(Θ,Λ, ς))˜̃∩c˜̃g(i˜̃g(Θ,Λ, ς))c = c˜̃g(i˜̃g(Θ,Λ, ς))˜̃∩c˜̃g(c˜̃g(Θ,Λ, ς)c)˜̃⊆c˜̃g(Θ,Λ, ς)˜̃∩c˜̃g(Θ,Λ, ς)c = b˜̃g (Θ,Λ, ς). (v) b˜̃g (c˜̃g (Θ,Λ, ς)) = c˜̃g(c˜̃g(Θ,Λ, ς))˜̃∩c˜̃g(c˜̃g(Θ,Λ, ς))c H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 663 ˜̃⊆(c˜̃g(Θ,Λ, ς))˜̃∩c˜̃g(Θ,Λ, ς)c = b˜̃g (Θ,Λ, ς). The following example shows that the equality of (ii), (iii), (iv) and (v) in Theorem 10 does not hold in general. Example 6. Let Ω = {ω1, ω2, ω3, ω4}, ς = {ϱ1, ϱ2} and˜̃g = {(Φ, ˜̃Ω, ς), (Θ1,Λ1, ς), (Θ2,Λ2, ς), (Θ3,Λ3, ς)}, where (Θ1,Λ1, ς) = {(ϱ1, {ω3}, {ω1}), (ϱ2, {ω3}, {ω1, ω2})}, (Θ2,Λ2, ς) = {(ϱ1, ϕ, {ω2, ω3}), (ϱ2, {ω1}, {ω3})} and (Θ3,Λ3, ς) = {(ϱ1, {ω3}, ϕ), (ϱ2, {ω1, ω3}, ϕ)}. Let (Θ,Λ, ς) = {(ϱ1, {ω1}, {ω3}), (ϱ2, {ω1}, {ω3})}. Now, c˜̃g (Θ,Λ, ς) = (Θ1,Λ1, ς) c, c˜̃g (Θ,Λ, ς)c = (Θ2,Λ2, ς) c and i˜̃g (Θ,Λ, ς) = (Θ2,Λ2, ς). Thus, b˜̃g (Θ,Λ, ς) = (Θ3,Λ3, ς) c. Hence b˜̃g (Θ,Λ, ς) ˜̃∪ (Θ,Λ, ς) = {(ϱ1, {ω1}, {ω3}), (ϱ2, {ω1}, {ω3})} = (Θ,Λ, ς). Therefore, c˜̃g (Θ,Λ, ς) ̸= b˜̃g (Θ,Λ, ς) ˜̃∪ (Θ,Λ, ς). Also, (Θ,Λ, ς) ˜̃\b˜̃g (Θ,Λ, ς) = {(ϱ1, ϕ, {ω3}), (ϱ2, {ω1}, {ω3})}. Thus, i˜̃g (Θ,Λ, ς) ̸= (Θ,Λ, ς) ˜̃\b˜̃g (Θ,Λ, ς). Now, if we take (χ, ψ, ς) = {(ϱ1, ϕ, {ω2}), (ϱ2, ϕ, {ω3})} is a bipolar soft set. Then i˜̃g (χ, ψ, ς) = (Φ, ˜̃ Ω, ς). Hence b˜̃g (i˜̃g (χ, ψ, ς)) = {(ϱ1, ϕ, {ω3}), (ϱ2, ϕ, {ω1, ω3})} = (Θ3,Λ3, ς) c and b˜̃g (χ, ψ, ς) = ( ˜̃ Ω,Φ, ς). Thus, b˜̃g (i˜̃g (χ, ψ, ς)) ̸= b˜̃g (χ, ψ, ς). Also, c˜̃g (χ, ψ, ς) = ( ˜̃ Ω,Φ, ς). Hence, b˜̃g (c˜̃g (χ, ψ, ς)) = (Θ3,Λ3, ς) c. Therefore, b˜̃g (c˜̃g (χ, ψ, ς)) ̸= b˜̃g (χ, ψ, ς). Theorem 11. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then b˜̃g (Θ,Λ, ς) ˜̃∩ i˜̃g (Θ,Λ, ς) = (Φ,Λ, ς). Proof. We start by b˜̃g (Θ,Λ, ς)˜̃∩i˜̃g (Θ,Λ, ς) = (c˜̃g (Θ,Λ, ς)˜̃\i˜̃g (Θ,Λ, ς))˜̃∩i˜̃g (Θ,Λ, ς) H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 664 = c˜̃g (Θ,Λ, ς)˜̃∩(i˜̃g (Θ,Λ, ς)c ˜̃∩i˜̃g (Θ,Λ, ς) = c˜̃g (Θ,Λ, ς)˜̃∩(Φ,Λ, ς) = (Φ,Λ, ς). Theorem 12. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then the following hold: (i) If (Θ,Λ, ς) ˜̃∈ ˜̃g, then (Θ,Λ, ς) ˜̃∩ b˜̃g (Θ,Λ, ς) = (Φ,Λ, ς). (ii) If (Θ,Λ, ς) is bipolar soft ˜̃g-closed, then b˜̃g (Θ,Λ, ς) ˜̃⊆ (Θ,Λ, ς). Proof. (i) Suppose (Θ,Λ, ς) ˜̃∈ ˜̃g. Then by Theorem 5 (ii), we have (Θ,Λ, ς) = i˜̃g (Θ,Λ, ς). Since i˜̃g (Θ,Λ, ς) ˜̃⊆ (b˜̃g (Θ,Λ, ς))c. That means (Θ,Λ, ς) ˜̃⊆ (b˜̃g (Θ,Λ, ς))c. Therefore, (Θ,Λ, ς) ˜̃∩ b˜̃g (Θ,Λ, ς) = (Φ,Λ, ς). (ii) By Theorem 10 (i), we have b˜̃g (Θ,Λ, ς) ˜̃⊆ c˜̃g (Θ,Λ, ς). Since (Θ,Λ, ς) is a bipolar soft ˜̃g-closed set, then b˜̃g (Θ,Λ, ς) ˜̃⊆(Θ,Λ, ς). The following example shows that the converse of Theorem 12 does not hold in general. Example 7. Take the bipolar soft set (Θ,Λ, ς) as in Example 6. Then b˜̃g (Θ,Λ, ς) ˜̃∩ (Θ,Λ, ς) = {(ϱ1, ϕ, {ω3}), (ϱ2, ϕ, {ω1, ω3})}, and b˜̃g (Θ,Λ, ς) ˜̃⊆ (Θ,Λ, ς). While (Θ,Λ, ς) ˜̃ /∈ ˜̃g and (Θ,Λ, ς) is not bipolar soft ˜̃g-closed. Theorem 13. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω) If (Θ,Λ, ς) is both bipolar soft ˜̃g-open and bipolar soft ˜̃g-closed. Then b˜̃g (Θ,Λ, ς) = (Φ,Λ, ς). Proof. Assume that (Θ,Λ, ς) is bipolar soft ˜̃g-open and bipolar soft ˜̃g-closed. Thus b˜̃g (Θ,Λ, ς) = c˜̃g (Θ,Λ, ς)˜̃∩c˜̃g (Θ,Λ, ς)c = c˜̃g (Θ,Λ, ς)˜̃∩(i˜̃g (Θ,Λ, ς))c = (Θ,Λ, ς)˜̃∩(Θ,Λ, ς)c = (Φ,Λ, ς). The following example shows that the converse of Theorem 13 does not hold in general. H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 665 Example 8. Take the bipolar soft set (Θ,Λ, ς) as in Example 6. Then b˜̃g (Θ,Λ, ς) = (Φ,Λ, ς), but (Θ,Λ, ς) is neither bipolar soft ˜̃g-open nor bipolar soft ˜̃g-closed. Definition 23. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then the bipolar soft ˜̃g-exterior of (Θ,Λ, ς) denoted by e˜̃g (Θ,Λ, ς) , is the bipolar soft ˜̃g-interior of the bipolar soft ˜̃g-complement of (Θ,Λ, ς). In the other words, e˜̃g (Θ,Λ, ς) = i˜̃g (Θ,Λ, ς)c. Proposition 4. The following statements are true for any (Θ,Λ, ς) ˜̃∈ BSS(Ω): (i) e˜̃g (Θ,Λ, ς) = (c˜̃g (Θ,Λ, ς))c. (ii) e˜̃g (Θ,Λ, ς)c = i˜̃g((Θ,Λ, ς)c)c = i˜̃g(Θ,Λ, ς). (iii) e˜̃g (Θ,Λ, ς) ˜̃∩ (Θ,Λ, ς) = (Φ,Λ, ς). (iv) e˜̃g (Θ,Λ, ς) is the largest bipolar soft ˜̃g-open set contained in (Θ,Λ, ς)c and hence e˜̃g (Θ,Λ, ς) ˜̃∈ ˜̃g. Proposition 5. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then e˜̃g ((Θ,Λ, ς) = ˜̃⋃{(χ, ψ, ς) : (χ, ψ, ς) ˜̃∈ ˜̃g, (χ, ψ, ς) ˜̃⊆ (Θ,Λ, ς)c}. Proof. It is obvious. Theorem 14. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ1,Λ1, ς), (Θ2,Λ2, ς) ˜̃∈ BSS(Ω). Then (i) e˜̃g(Φ, ˜̃Ω, ς) ˜̃⊆ ( ˜̃ Ω,Φ, ς) and e˜̃g( ˜̃Ω,Φ, ς) = (Φ, ˜̃ Ω, ς). (ii) e˜̃g(Θ1,Λ1, ς) ˜̃⊆ (Θ1,Λ1, ς) c. (iii) e˜̃g(e˜̃g(Θ1,Λ1, ς)) c = e˜̃g(Θ1,Λ1, ς). (iv) If (Θ1,Λ1, ς) ˜̃⊆ (Θ2,Λ2, ς), then e˜̃g (Θ2,Λ2, ς) ˜̃⊆ e˜̃g (Θ1,Λ1, ς). (v) i˜̃g(Θ1,Λ1, ς) ˜̃⊆ e˜̃g(e˜̃g(Θ1,Λ1, ς)). (vi) e˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)) ˜̃⊆ e˜̃g (Θ1,Λ1, ς) ˜̃∩ e˜̃g (Θ2,Λ2, ς). (vii) e˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊇ e˜̃g (Θ1,Λ1, ς) ˜̃∪ e˜̃g (Θ2,Λ2, ς). Proof. H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 666 (i) e˜̃g(Φ, ˜̃Ω, ς) = i˜̃g(Φ, ˜̃Ω, ς)c = i˜̃g( ˜̃Ω,Φ, ς) ˜̃⊆ ( ˜̃ Ω,Φ, ς) and e˜̃g( ˜̃Ω,Φ, ς) = i˜̃g(Φ, ˜̃Ω, ς) = (Φ, ˜̃ Ω, ς). (ii) Since i˜̃g(Θ1,Λ1, ς) c ˜̃⊆ (Θ1,Λ1, ς) c. Then e˜̃g(Θ1,Λ1, ς) ˜̃⊆ (Θ1,Λ1, ς) c. (iii) e˜̃g (e˜̃g (Θ1,Λ1, ς)) c = i˜̃g(e˜̃g (Θ1,Λ1, ς)) = i˜̃g (i˜̃g (Θ1,Λ1, ς) c) = i˜̃g (Θ1,Λ1, ς) c = e˜̃g (Θ1,Λ1, ς). (iv) Obvious. (v) Follows directly from (ii) and (iv). (vi) e˜̃g ((Θ1,Λ1, ς) ˜̃∪(Θ2,Λ2, ς)) = i˜̃g ((Θ1,Λ1, ς) ˜̃∪(Θ2,Λ2, ς)) c = i˜̃g ((Θ1,Λ1, ς) c ˜̃∩ i˜̃g (Θ2,Λ2, ς) c)˜̃⊆i˜̃g (Θ1,Λ1, ς) c ˜̃∩ i˜̃g (Θ2,Λ2, ς)) c (by Theorem 5(v)) = e˜̃g (Θ1,Λ1, ς) ˜̃∩ e˜̃g (Θ2,Λ2, ς). Thus, e˜̃g ((Θ1,Λ1, ς) ˜̃∪ (Θ2,Λ2, ς)) ˜̃⊆ e˜̃g (Θ1,Λ1, ς) ˜̃∩ e˜̃g (Θ2,Λ2, ς). (vii) e˜̃g (Θ1,Λ1, ς) ˜̃∪ e˜̃g (Θ2,Λ2, ς) = i˜̃g (Θ1,Λ1, ς) c ˜̃∪ i˜̃g (Θ2,Λ2, ς) c ˜̃⊆i˜̃g ((Θ1,Λ1, ς) c ˜̃∪ (Θ2,Λ2, ς) c) (by Theorem 5(vi)) = i˜̃g (Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς) c = e˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)). Thus, e˜̃g ((Θ1,Λ1, ς) ˜̃∩ (Θ2,Λ2, ς)) ˜̃⊇ e˜̃g (Θ1,Λ1, ς) ˜̃∪ e˜̃g (Θ2,Λ2, ς). Theorem 15. Let (Ω, ˜̃g, ς,¬ς) be a BSGT S and (Θ,Λ, ς) ˜̃∈ BSS(Ω). Then (i) (b˜̃g (Θ,Λ, ς))c = i˜̃g (Θ,Λ, ς) ˜̃∪ e˜̃g (Θ,Λ, ς). (ii) b˜̃g (Θ,Λ, ς) ˜̃∪ i˜̃g (Θ,Λ, ς) ˜̃∪ e˜̃g (Θ,Λ, ς) ˜̃⊆ ( ˜̃ Ω,Φ, ς). Proof. (i) Since (b˜̃g (Θ,Λ, ς))c = i˜̃g (Θ,Λ, ς) ˜̃∪ i˜̃g (Θ,Λ, ς)c = i˜̃g (Θ,Λ, ς) ˜̃∪ e˜̃g (Θ,Λ, ς). H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 667 (ii) From (i), we have (b˜̃g (Θ,Λ, ς))c = i˜̃g (Θ,Λ, ς) ˜̃∪ e˜̃g (Θ,Λ, ς). Therefore, b˜̃g (Θ,Λ, ς) ˜̃∪ i˜̃g (Θ,Λ, ς) ˜̃∪ e˜̃g (Θ,Λ, ς) ˜̃⊆ ( ˜̃ Ω,Φ, ς). The following example shows that the converse of Theorem 15 does not hold in general. Example 9. Take the bipolar soft set (Θ,Λ, ς) as in Example 6. Then i˜̃g (Θ,Λ, ς) = {(ϱ1, ϕ, {ω2, ω3}), (ϱ2, {ω1}, {ω3})}, b˜̃g (Θ,Λ, ς) = {(ϱ1, ϕ, {ω3}), (ϱ2, ϕ, {ω1, ω3})} and e˜̃g (Θ,Λ, ς) = {(ϱ1, {ω3}, {ω1}), (ϱ2, {ω3}, {ω1, ω2})}. Thus, b˜̃g (Θ,Λ, ς) ˜̃∪ i˜̃g (Θ,Λ, ς) ˜̃∪ e˜̃g (Θ,Λ, ς) ̸= ( ˜̃ Ω,Φ, ς). 4. An Application on BSGT S The present section gives the application of BSGT Ss and investigates some of its properties. Definition 24. Let ς = {ϱ1, ϱ2, . . . , ϱn} be a parameters set, Ω = {ω1, ω2, . . . , ωn} be an initial universe and (Θ,Λ, ς) be a BSS over Ω. Then the score of an object by κi, is computed as κi = pi − ni where pi represents the set of positive description (ϱi) which is available for those who are applying for a job and it is computed as pi = ∑n j=1 αij. Whereas, ni represents the set of negative description (¬ϱi) which is available for those who are applying for a job and it is computed as ni = ∑n j=1 βij. This means that κi is the different point between the scores of positive descriptions except the scores of negative descriptions to get the highest score for their selection to own job. Now , we can depend on the following algorithm to select a sample among those who applying for a job in vacancy jobs. Algorithm 1: The algorithm for the selection of a preferable choice is given as the following steps: Step 1. Input the BSS(Θ,Λ, ς). Step 2. Write the BSS in the tabular form. Step 3. Compute the score κi of ωi , ∀ωi ∈ Ω. Step 4. Find κS = maxκi. Step 5. If s has more than one value, then one of ωi or all of ωi could be preferable choice. Step 6. To select new κs. Go to Step 4. Example 10. We consider the problem in Example 1 to select the most suitable peo- ple who are offered by Mr. Ibrahim. According to tourism companies’ conditions, peo- ple who own specific description will be selected from ς and n selection of people in H. Y. Saleh, B. A. Asaad, R. A. Mohammed / Eur. J. Pure Appl. Math, 15 (2) (2022), 646-671 668 Duhok city. Suppose ς1 = {ϱ1 = ”Hard Working”, ϱ3 = ”Flexibility”, ϱ5 = ”Self - Confidence”, ϱ7 = ”Skillful”} and ς2 = {ϱ2 = ”Negligent”, ϱ4 = ”Rigidity”, ϱ6 = ”Shyness”, ϱ8 = ”Unskillful”}. Here, we will given ς = ς1 ∪ ς2. Now, we can use the above algorithm to select employees that are come for a job in tourism. Table 4: Tabular form the score of the bipolar soft set (Θ1,Λ1, ς) (Θ1,Λ1)(ϱi) pi ni κi ω1 1 2 −1 ω2 1 3 -2 ω3 2 2 0 ω4 2 0 2 ω5 2 1 1 ω6 1 1 0 ω7 2 0 2 ω8 1 2 -1 Table 5: Tabular form the score of the bipolar soft set (Θ2,Λ2, ς) (Θ2,Λ2)(ϱi) pi ni κi ω1 2 1 1 ω2 2 2 0 ω3 1 3 -2 ω4 2 2 0 ω5 1 2 -1 ω6 0 1 -1 ω7 0 2 -2 ω8 0 2 -2 Table 6: Tabular form the score of the bipolar soft set (Θ3,Λ3, ς) (Θ3,Λ3)(ϱi) pi ni κi ω1 2 1 1 ω2 2 2 0 ω3 2 2 0 ω4 2 0 2 ω5 2 1 1 ω6 1 1 0 ω7 2 0 2 ω8 1 2 -1 Clearly, the maximum of (Θ1,Λ1, ς), (Θ2,Λ2, ς) and (Θ3,Λ3, ς) are 2, 1, and 2 respec- tively. The optimal elements of Ω are ω4, ω1 and ω7. REFERENCES 669 5. Conclusions We have introduced bipolar soft generalized topological spaces via bipolar soft sets. The bipolar soft sets ˜̃g-interior, ˜̃g-closure, ˜̃g-exterior and ˜̃g-boundary have been investigated and some results among them are obtained. 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