Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 950897, 7 pages http://dx.doi.org/10.1155/2013/950897 Research Article Intersection-Soft Filters in π 0-Algebras Young Bae Jun,1 Sun Shin Ahn,2 and Kyoung Ja Lee3 1 Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea 3 Department of Mathematics Education, Hannam University, Daejeon 306-791, Republic of Korea 2 Correspondence should be addressed to Sun Shin Ahn; sunshine@dongguk.edu Received 26 January 2013; Revised 1 March 2013; Accepted 6 March 2013 Academic Editor: Yong Zhou Copyright © 2013 Young Bae Jun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The notions of (strong) intersection-soft filters in π 0 -algebras are introduced, and related properties are investigated. Characterizations of a (strong) intersection-soft filter are established, and a new intersection-soft filter from old one is constructed. A condition for an intersection-soft filter to be strong is given, and an extension property of a strong intersection-soft filter is established. 1. Introduction To solve complicated problem in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. 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 [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. π 0 -algebras, which are different from BL-algebras, have been introduced by Wang [5] in order to get an algebraic proof of the completeness theorem of a formal deductive system [6]. The filter theory in π 0 -algebras is discussed in [7]. In this paper, we apply the notion of intersection-soft sets to the filter theory in π 0 -algebras. We introduced the concept of (strong) intersection-soft filters in π 0 -algebras and investigate related properties. We establish characterizations of a (strong) intersection-soft filter and make a new intersection-soft filter from old one. We provide a condition for an intersection-soft filter to be strong and construct an extension property of a strong intersection-soft filter. 2. Preliminaries 2.1. Basic Results on π 0 -Algebras Definition 1 (see [5]). Let πΏ be a bounded distributive lattice with order-reversing involution ¬ and a binary operation → . Then (πΏ, ∧, ∨, ¬, → ) is called an π 0 -algebra if it satisfies the following axioms: (R1) π₯ → π¦ = ¬π¦ → ¬π₯, (R2) 1 → π₯ = π₯, (R3) (π¦ → π§) ∧ ((π₯ → π¦) → (π₯ → π§)) = π¦ → π§, (R4) π₯ → (π¦ → π§) = π¦ → (π₯ → π§), (R5) π₯ → (π¦ ∨ π§) = (π₯ → π¦) ∨ (π₯ → π§), (R6) (π₯ → π¦) ∨ ((π₯ → π¦) → (¬π₯ ∨ π¦)) = 1. 2 Discrete Dynamics in Nature and Society Let πΏ be an π 0 -algebra. For any π₯, π¦ ∈ πΏ, we define π₯βπ¦ = ¬(π₯ → ¬π¦) and π₯ ⊕ π¦ = ¬π₯ → π¦. It is proven that β and ⊕ are commutative, associative, and π₯ ⊕ π¦ = ¬(¬π₯ β ¬π¦), and (πΏ, ∧, ∨, β, → , 0, 1) is a residuated lattice. In the following, let π₯π denote π₯ β π₯ β ⋅ ⋅ ⋅ β π₯ where π₯ appears π times for π ∈ N. We refer the reader to the book [8] for further information regarding π 0 -algebras. Lemma 2 (see [7]). Let πΏ be an π 0 -algebra. Then the following properties hold: (∀π₯, π¦ ∈ πΏ) (π₯ ≤ π¦ ⇐⇒ π₯ σ³¨→ π¦ = 1) , (1) (∀π₯, π¦ ∈ πΏ) (π₯ ≤ π¦ σ³¨→ π₯) , (2) (∀π₯ ∈ πΏ) (¬π₯ = π₯ σ³¨→ 0) , (3) (∀π₯, π¦ ∈ πΏ) ((π₯ σ³¨→ π¦) ∨ (π¦ σ³¨→ π₯) = 1) , (4) (∀π₯, π¦ ∈ πΏ) × (π₯ ≤ π¦ σ³¨⇒ π¦ σ³¨→ π§ ≤ π₯ σ³¨→ π§, π§ σ³¨→ π₯ ≤ π§ σ³¨→ π¦) , (5) (∀π₯, π¦ ∈ πΏ) (((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π¦ = π₯ σ³¨→ π¦) , (6) (∀π₯, π¦ ∈ πΏ) × (π₯ ∨ π¦ = ((π₯ σ³¨→ π¦) σ³¨→ π¦) ∧ ((π¦ σ³¨→ π₯) σ³¨→ π₯)) , (7) (∀π₯ ∈ πΏ) (π₯ β ¬π₯ = 0, π₯ ⊕ ¬π₯ = 1) , universe. Let P(π) (resp., P(πΈ)) denotes the power set of π (resp., πΈ). By analogy with fuzzy set theory, the notion of soft set is defined as follows. Definition 5 (see [1, 9]). A soft set of πΈ over π (a soft → set of πΈ for short) is any function ππ΄ : πΈ P(π), such that ππ΄(π₯) = 0 if π₯ ∉ π΄, for π΄ ∈ P(πΈ), or, equivalently, any set Fπ΄ := {(π₯, ππ΄ (π₯)) | π₯ ∈ πΈ, ππ΄ (π₯) ∈ P (π) , ππ΄ (π₯) = 0 if π₯ ∉ π΄} , for π΄ ∈ P(πΈ). Definition 6 (see [9]). Let Fπ΄ and Fπ΅ be soft sets of πΈ. We Μ Fπ΅ , if say that Fπ΄ is a soft subset of Fπ΅ , denoted by Fπ΄ ⊆ ππ΄(π₯) ⊆ ππ΅ (π₯) for all π₯ ∈ πΈ. 3. Intersection-Soft Filters In what follows, we denote by π(π, πΏ) the set of all soft sets of πΏ over π where πΏ is an π 0 -algebra unless otherwise specified. Definition 7. A soft set FπΏ ∈ π(π, πΏ) is called an int-soft filter of πΏ if it satisfies πΎ πΎ (∀πΎ ∈ P (π)) (FπΏ =ΜΈ 0 σ³¨⇒ FπΏ is a filter of πΏ) , (17) πΎ (8) (∀π₯, π¦ ∈ πΏ) (π₯ β π¦ ≤ π₯ ∧ π¦, π₯ β (x σ³¨→ π¦) ≤ π₯ ∧ π¦) , (9) (∀π₯, π¦, π§ ∈ πΏ) ((π₯ β π¦) σ³¨→ π§ = π₯ σ³¨→ (π¦ σ³¨→ π§)) , (10) (∀π₯, π¦ ∈ πΏ) (π₯ ≤ π¦ σ³¨→ (π₯ β π¦)) , (11) (∀π₯, π¦, π§ ∈ πΏ) (π₯ β π¦ ≤ π§ ⇐⇒ π₯ ≤ π¦ σ³¨→ π§) , (12) (∀π₯, π¦, π§ ∈ πΏ) (π₯ ≤ π¦ σ³¨⇒ π₯ β π§ ≤ π¦ β π§) , (13) where FπΏ = {π₯ ∈ πΏ | πΎ ⊆ ππΏ (π₯)} which is called the πΎinclusive set of FπΏ . πΎ If FπΏ is an int-soft filter of πΏ, every πΎ-inclusive set FπΏ is called an inclusive filter of πΏ. Example 8. Let πΏ = {0, π, π, π, 1} be a set with the order 0 < π < π < π < 1, and the following Cayley tables: π₯ 0 π π π 1 (∀π₯, π¦, π§ ∈ πΏ) (π₯ σ³¨→ π¦ ≤ (π¦ σ³¨→ π§) σ³¨→ (π₯ σ³¨→ π§)) , (14) (∀π₯, π¦, π§ ∈ πΏ) ((π₯ σ³¨→ π¦) β (π¦ σ³¨→ π§) ≤ π₯ σ³¨→ π§) . (16) (15) Definition 3 (see [7]). A nonempty subset πΉ of πΏ is called a filter of πΏ if it satisfies (i) 1 ∈ πΉ, (ii) (∀π₯ ∈ πΉ)(∀π¦ ∈ πΏ)(π₯ → π¦ ∈ πΉ ⇒ π¦ ∈ πΉ). Lemma 4 (see [7]). Let πΉ be a nonempty subset of πΏ. Then πΉ is a filter of πΏ if and only if it satisfies (1) (∀π₯ ∈ πΉ)(∀π¦ ∈ πΏ)(π₯ ≤ π¦ ⇒ π¦ ∈ πΉ), (2) (∀π₯, π¦ ∈ πΉ)(π₯ β π¦ ∈ πΉ). 2.2. Basic Results on Soft Set Theory. Soft set theory was introduced by Molodtsov [1] and CΜ§agΜman and EnginogΜlu [9]. In what follows, let π be an initial universe set, and let πΈ be a set of parameters. We say that the pair (π, πΈ) is a soft σ³¨→ 0 π π π 1 0 1 π π π 0 ¬π₯ 1 π π π 0 π 1 1 π π π π 1 1 1 π π π 1 1 1 1 π 1 1 1 1 1 1 (18) Then (πΏ, ∧, ∨, ¬, → ) is an π 0 -algebra (see [10]) where π₯ ∧ π¦ = min{π₯, π¦} and π₯ ∨ π¦ = max{π₯, π¦}. Let FπΏ ∈ π(π, πΏ) be given as follows: FπΏ = {(0, πΎ1 ) , (π, πΎ1 ) , (π, πΎ1 ) , (π, πΎ2 ) , (1, πΎ2 )} , (19) where πΎ1 and πΎ2 are subsets of π with πΎ1 β πΎ2 . Then FπΏ is an int-soft filter of πΏ. We provide characterizations of an int-soft filter. Discrete Dynamics in Nature and Society 3 Theorem 9. Let FπΏ ∈ π(π, πΏ). Then FπΏ is an int-soft filter of πΏ if and only if the following assertions are valid: (1) (∀π₯ ∈ πΏ)(ππΏ (π₯) ⊆ ππΏ (1)), (2) (∀π₯, π¦ ∈ πΏ)(ππΏ (π₯ → π¦) ∩ ππΏ (π₯) ⊆ ππΏ (π¦)). Proof. Assume that FπΏ is an int-soft filter of πΏ. For any π₯ ∈ πΏ, πΎ πΎ let ππΏ (π₯) = πΎ. Then π₯ ∈ FπΏ . Since FπΏ is a filter of πΏ, we have πΎ 1 ∈ FπΏ and so ππΏ (π₯) = πΎ ⊆ ππΏ (1). For any π₯, π¦ ∈ πΏ, let πΎ πΎ ππΏ (π₯ → π¦) ∩ ππΏ (π₯) = πΎ. Then π₯ → π¦ ∈ FπΏ and π₯ ∈ FπΏ . πΎ πΎ Since FπΏ is a filter of πΏ, it follows that π¦ ∈ FπΏ . Hence ππΏ (π₯ → π¦) ∩ ππΏ (π₯) = πΎ ⊆ ππΏ (π¦). Conversely, suppose that FπΏ satisfies two conditions (1) πΎ and (2). Let πΎ ∈ P(π) such that FπΏ =ΜΈ 0. Then there exists πΎ π ∈ FπΏ , and so πΎ ⊆ ππΏ (π). It follows from (1) that πΎ ⊆ ππΏ (π) ⊆ πΎ πΎ ππΏ (1). Thus 1 ∈ FπΏ . Let π₯, π¦ ∈ πΏ such that π₯ → π¦ ∈ FπΏ and πΎ π₯ ∈ FπΏ . Then πΎ ⊆ ππΏ (π₯ → π¦) and πΎ ⊆ ππΏ (π₯). It follows from (2) that πΎ ⊆ ππΏ (π₯ σ³¨→ π¦) ∩ ππΏ (π₯) ⊆ ππΏ (π¦) , (20) πΎ πΎ FπΏ . Thus FπΏ ( =ΜΈ 0) is a filter of πΏ, and hence FπΏ that is, π¦ ∈ an int-soft filter of πΏ. is Proposition 10. Let FπΏ ∈ π(π, πΏ) be an int-soft filter of πΏ. Then the following properties are valid. (3) Since π₯ β π¦ ≤ π₯ ∧ π¦ for all π₯, π¦ ∈ πΏ, it follows from (1) that ππΏ (π₯ β π¦) ⊆ ππΏ (π₯) ∩ ππΏ (π¦). Using (11) and (1), we have ππΏ (π₯) ⊆ ππΏ (π¦ → (π₯ β π¦)). It follows from Theorem 9 (2) that ππΏ (π₯) ∩ ππΏ (π¦) ⊆ ππΏ (π¦ → (π₯ β π¦)) ∩ ππΏ (π¦) ⊆ ππΏ (π₯ β π¦). Therefore ππΏ (π₯ β π¦) = ππΏ (π₯) ∩ ππΏ (π¦). Since π¦ ≤ π₯ → π¦ and π₯ β (π₯ → π¦) ≤ π₯ ∧ π¦ for all π₯, π¦ ∈ πΏ, we have ππΏ (π¦) ⊆ ππΏ (π₯ → π¦) and ππΏ (π₯) ∩ ππΏ (π¦) ⊆ ππΏ (π₯) ∩ ππΏ (π₯ → π¦) = ππΏ (π₯ β (π₯ → π¦)) ⊆ ππΏ (π₯ ∧ π¦) ⊆ ππΏ (π₯) ∩ ππΏ (π¦) by (1). Hence ππΏ (π₯ ∧ π¦) = ππΏ (π₯) ∩ ππΏ (π¦) for all π₯, π¦ ∈ πΏ. (4) It follows from (3). (5) Note that π₯ β ¬π₯ = for all π₯ ∈ πΏ. Using (3), we have ππΏ (0) = ππΏ (π₯ β ¬π₯) = ππΏ (π₯) ∩ ππΏ (¬π₯) for all π₯, π¦ ∈ πΏ. (6) Combining (15), (1), and (3), we have the desired result. (7) It follows from (1) and (3). (8) Since π¦ ≤ π₯ → π¦ for all π₯, π¦ ∈ πΏ, it follows from (1) that ππΏ (π₯) ∩ ππΏ (π¦) ⊆ ππΏ (π₯) ∩ ππΏ (π₯ σ³¨→ π¦) . ππΏ (π₯) ∩ ππΏ (π₯ σ³¨→ π¦) = ππΏ (π₯ β (π₯ σ³¨→ π¦)) ⊆ ππΏ (π₯ ∧ π¦) = ππΏ (π₯) ∩ ππΏ (π¦) (26) (21) (2) (∀π₯, π¦ ∈ πΏ)(ππΏ (π₯ → π¦) = ππΏ (1) ⇒ ππΏ (π₯) ⊆ ππΏ (π¦)). (3) (∀π₯, π¦ ∈ πΏ)(fπΏ (π₯ β π¦) = ππΏ (π₯) ∩ ππΏ (π¦) = ππΏ (π₯ ∧ π¦)). (4) (∀π₯ ∈ πΏ)(∀π ∈ N)(ππΏ (π₯π ) = ππΏ (π₯)). (5) (∀π₯ ∈ πΏ)(ππΏ (0) = ππΏ (π₯) ∩ ππΏ (¬π₯)). (6) (∀π₯, π¦ ∈ πΏ)(ππΏ (π₯ → π¦) ∩ ππΏ (π¦ → π§) ⊆ ππΏ (π₯ → π§)). (7) (∀π₯, π¦, π§ ∈ πΏ)(π₯ β π¦ ≤ π§ ⇒ ππΏ (π₯) ∩ ππΏ (π¦) ⊆ ππΏ (π§)). (8) (∀π₯, π¦ ∈ πΏ)(ππΏ (π₯) ∩ ππΏ (π₯ → π¦) = ππΏ (π¦) ∩ ππΏ (π¦ → π₯) = ππΏ (π₯) ∩ ππΏ (π¦)). (9) (∀π₯, π¦ ∈ πΏ)(ππΏ (π₯ β (π₯ → π¦)) = ππΏ (π¦ β (π¦ → π₯)) = ππΏ (π₯) ∩ ππΏ (π¦)). (10) (∀π₯, π¦, π§ ∈ πΏ)(ππΏ (π₯ → (¬π§ → π¦)) ∩ ππΏ (π¦ → π§) ⊆ ππΏ (π₯ → (¬π§ → π§))). (11) (∀π₯, π¦, π§ ∈ πΏ)(ππΏ (π₯ → (π¦ → π§)) ∩ ππΏ (π₯ → π¦) ⊆ ππΏ (π₯ → (π₯ → π§))). Proof. (1) Let π₯, π¦ ∈ πΏ such that π₯ ≤ π¦. Then π₯ → π¦ = 1, and so ππΏ (π₯) = ππΏ (π₯) ∩ ππΏ (1) = ππΏ (π₯) ∩ ππΏ (π₯ σ³¨→ π¦) ⊆ ππΏ (π¦) (22) by (1) and (2) of Theorem 9. (2) Let π₯, π¦ ∈ πΏ such that ππΏ (π₯ → π¦) = ππΏ (1). Then ππΏ (π₯) = ππΏ (π₯) ∩ ππΏ (1) = ππΏ (π₯) ∩ ππΏ (π₯ σ³¨→ π¦) ⊆ ππΏ (π¦) (23) by (1) and (2) of Theorem 9. (25) Since π₯ β (π₯ → π¦) ≤ π₯ ∧ π¦ for all π₯, π¦ ∈ πΏ, we have (1) FπΏ is order preserving, that is, (∀π₯, π¦ ∈ πΏ) (π₯ ≤ π¦ σ³¨⇒ ππΏ (π₯) ⊆ ππΏ (π¦)) . (24) by (3) and (1). Hence ππΏ (π₯) ∩ ππΏ (π¦) = ππΏ (π₯) ∩ ππΏ (π₯ → π¦). Similarly, ππΏ (π¦) ∩ ππΏ (π¦ → π₯) = ππΏ (π₯) ∩ ππΏ (π¦) for all π₯, π¦ ∈ πΏ. (9) Using (3), we have ππΏ (π₯ β (π₯ σ³¨→ π¦)) = ππΏ (π₯) ∩ ππΏ (π₯ σ³¨→ π¦) , ππΏ (π¦ β (π¦ σ³¨→ π₯)) = ππΏ (π¦) ∩ ππΏ (π¦ σ³¨→ π₯) , (27) for all π₯, π¦ ∈ πΏ. It follows from (8) that ππΏ (π₯ β (π₯ → π¦)) = ππΏ (π¦ β (π¦ → π₯)) = ππΏ (π₯) ∩ ππΏ (π¦). (10) Note that (π₯ σ³¨→ (¬π§ σ³¨→ π¦)) β (π¦ σ³¨→ π§) = ((π₯ β ¬π§) σ³¨→ π¦) β (π¦ σ³¨→ π§) (28) ≤ (π₯ β ¬π§) σ³¨→ π§ = π₯ σ³¨→ (¬π§ σ³¨→ π§) for all π₯, π¦, π§ ∈ πΏ. Using (1) and (3), we have ππΏ (π₯ σ³¨→ (¬π§ σ³¨→ π¦)) ∩ ππΏ (π¦ σ³¨→ π§) = ππΏ ((π₯ σ³¨→ (¬π§ σ³¨→ π¦)) β (π¦ σ³¨→ π§)) (29) ⊆ ππΏ (π₯ σ³¨→ (¬π§ σ³¨→ π§)) for all π₯, π¦, π§ ∈ πΏ. (11) Note that (π₯ → (π¦ → π§)) β (π₯ → π¦) = (π¦ → (π₯ → π§)) β (π₯ → π¦) ≤ π₯ → (π₯ → π§) for all π₯, π¦, π§ ∈ πΏ. It follows from (1) and (3) that ππΏ (π₯ σ³¨→ (π¦ σ³¨→ π§)) ∩ ππΏ (π₯ σ³¨→ π¦) ⊆ ππΏ (π₯ σ³¨→ (π₯ σ³¨→ π§)) (30) for all π₯, π¦, π§ ∈ πΏ. 4 Discrete Dynamics in Nature and Society Theorem 11. Let FπΏ ∈ π(π, πΏ). Then FπΏ is an int-soft filter of πΏ if and only if the following assertions are valid: (∀π₯, π¦, π§ ∈ πΏ) (π₯ ≤ π¦ σ³¨→ π§ σ³¨⇒ ππΏ (π₯) ∩ ππΏ (π¦) ⊆ ππΏ (π§)) . (35) (1) FπΏ is order preserving, (2) (∀π₯, π¦ ∈ πΏ)(ππΏ (π₯ β π¦) = ππΏ (π₯) ∩ ππΏ (π¦)). Proof. The necessity follows from (1) and (3) of Proposition 10. Conversely, suppose that FπΏ satisfies two conditions (1) and (2). Let π₯, π¦ ∈ πΏ. Since π₯ ≤ 1, we have ππΏ (π₯) ⊆ ππΏ (1) by (1). Note that π₯ β (π₯ → π¦) ≤ π¦. It follows from (2) and (1) that ππΏ (π₯) ∩ ππΏ (π₯ σ³¨→ π¦) = ππΏ (π₯ β (π₯ σ³¨→ π¦)) ⊆ ππΏ (π¦) . (31) Therefore FπΏ is an int-soft filter of πΏ. Proposition 12. Let π ∈ πΏ such that ¬π = π. If FπΏ ∈ π(π, πΏ) is an int-soft filter of πΏ, then ππΏ (π) = ππΏ (π₯) for all π₯ ∈ {π ∈ πΏ | 0 ≤ π ≤ π}. Proof. Suppose that there exists π¦ ∈ {π ∈ πΏ | 0 ≤ π ≤ π} such πΎ that ππΏ (π) =ΜΈ ππΏ (π¦). Then ππΏ (π¦) β ππΏ (π), and so π ∈ FπΏ and πΎ πΎ π¦ ∉ FπΏ where πΎ = ππΏ (π). Since FπΏ is a filter of πΏ, we have 0 = πΎ πΎ π β π ∈ FπΏ . This shows that FπΏ = πΏ, and it is a contradiction. Hence ππΏ (π) = ππΏ (π₯) for all π₯ ∈ {π ∈ πΏ | 0 ≤ π ≤ π}. Theorem 13. Let FπΏ ∈ π(π, πΏ). Then FπΏ is an int-soft filter of πΏ if and only if the following assertion is valid: (∀π₯, π¦, π§ ∈ πΏ) × (ππΏ (π¦) ∩ ππΏ ((π₯ σ³¨→ π¦) σ³¨→ π§) ⊆ ππΏ (π₯ σ³¨→ π§)) . Proof. Suppose that FπΏ is an int-soft filter of πΏ. Let π₯, π¦, π§ ∈ πΏ such that π₯ ≤ π¦ → π§. Then ππΏ (π₯) ⊆ ππΏ (π¦ → π§) by Proposition 10 (1), and so ππΏ (π₯) ∩ ππΏ (π¦) ⊆ ππΏ (π¦ σ³¨→ π§) ∩ ππΏ (π¦) ⊆ ππΏ (π§) (36) by Theorem 9 (2). Conversely, assume that FπΏ satisfies the condition (35). Let π₯, π¦ ∈ πΏ. Since π₯ ≤ 1 = π₯ → 1, we have ππΏ (π₯) ⊆ ππΏ (1) by (35). Note that π₯ → π¦ ≤ π₯ → π¦. It follows from (35) that ππΏ (π₯ → π¦) ∩ ππΏ (π₯) ⊆ ππΏ (π¦). Therefore FπΏ is an int-soft filter of πΏ by Theorem 9. Proposition 15. Every int-soft filter FπΏ of πΏ satisfies. (∀π₯, π¦, π§ ∈ πΏ) × (ππΏ ((π₯ σ³¨→ π¦) σ³¨→ π§) ⊆ ππΏ (π₯ σ³¨→ (π¦ σ³¨→ π§))) . (37) Proof. Let π₯, π¦, π§ ∈ πΏ. Since 1 = π¦ → (π₯ → π¦) ≤ ((π₯ → π¦) → π§) → (π¦ → π§), we have ππΏ (1) ⊆ ππΏ (((π₯ σ³¨→ π¦) σ³¨→ π§) σ³¨→ (π¦ σ³¨→ π§)) (38) by Proposition 10 (1). It follows from (2) and Theorem 9 that ππΏ ((π₯ σ³¨→ π¦) σ³¨→ π§) = ππΏ ((π₯ σ³¨→ π¦) σ³¨→ π§) ∩ ππΏ (1) (32) Proof. Assume that FπΏ is an int-soft filter of πΏ, and let π₯, π¦, π§ ∈ πΏ. Since π¦ ≤ π₯ → π¦, it follows from Proposition 10 (1) and Theorem 9 (2) that ππΏ (π¦) ∩ ππΏ ((π₯ σ³¨→ π¦) σ³¨→ π§) ⊆ ππΏ (π₯ σ³¨→ π¦) ∩ ππΏ ((π₯ σ³¨→ π¦) σ³¨→ π§) Theorem 14. Let FπΏ ∈ π(π, πΏ). Then FπΏ is an int-soft filter of πΏ if and only if the following assertion is valid: (33) ⊆ ππΏ ((π₯ σ³¨→ π¦) σ³¨→ π§) ∩ ππΏ (39) × (((π₯ σ³¨→ π¦) σ³¨→ π§) σ³¨→ (π¦ σ³¨→ π§)) ⊆ ππΏ (π¦ σ³¨→ π§) ⊆ ππΏ (π₯ σ³¨→ (π¦ σ³¨→ π§)) . This completes the proof. The following example shows that the converse of Proposition 15 may not be true in general. Example 16. Let πΏ = {0, π, π, π, π, 1} be a set with the order 0 < π < π < π < π < 1, and the following Cayley tables: ⊆ ππΏ (π§) ⊆ ππΏ (π₯ σ³¨→ π§) . Conversely, suppose that FπΏ satisfies the inclusion (32). Let π₯, π¦ ∈ πΏ. Then π₯ 0 π π π π 1 ππΏ (π₯) = ππΏ (π₯) ∩ ππΏ ((0 σ³¨→ π₯) σ³¨→ π₯) ⊆ ππΏ (0 σ³¨→ π₯) = ππΏ (1) , ππΏ (π₯) ∩ ππΏ (π₯ σ³¨→ π¦) = ππΏ (π₯) ∩ ππΏ ((1 σ³¨→ π₯) σ³¨→ π¦) ⊆ ππΏ (1 σ³¨→ π¦) = ππΏ (π¦) (34) by (32). Therefore FπΏ is an int-soft filter of πΏ by Theorem 9. σ³¨→ 0 π π π π 1 0 1 π π π π 0 π 1 1 π π π π ¬π₯ 1 π π π π 0 π 1 1 1 π π π π 1 1 1 1 π π π 1 1 1 1 1 π 1 1 1 1 1 1 1 (40) Discrete Dynamics in Nature and Society 5 Then (πΏ, ∧, ∨, ¬, → ) is an π 0 -algebra (see [10]) where π₯ ∧ π¦ = min{π₯, π¦} and π₯ ∨ π¦ = max{π₯, π¦}. Let FπΏ ∈ π(π, πΏ) be given as follows: FπΏ = {(0, πΎ1 ) , (π, πΎ2 ) , (π, πΎ2 ) , (π, πΎ2 ) , (π, πΎ1 ) , (1, πΎ1 )} , (41) where πΎ1 and πΎ2 are subsets of π with πΎ1 β πΎ2 . Then FπΏ satisfies the condition (37), but FπΏ is not an int-soft filter of πΏ since ππΏ (π) ⊆ΜΈ ππΏ (1). Proposition 17. For an int-soft filter FπΏ of πΏ, the following are equivalent: (∀π₯, π¦ ∈ πΏ) (ππΏ (π¦ σ³¨→ (π¦ σ³¨→ π₯)) ⊆ ππΏ (π¦ σ³¨→ π₯)) , (42) (∀π₯, π¦, π§ ∈ πΏ) (ππΏ (π§ σ³¨→ (π¦ σ³¨→ π₯)) (43) ⊆ ππΏ ((π§ σ³¨→ π¦) σ³¨→ (π§ σ³¨→ π₯))) . Proof. Assume that (42) is valid, and let π₯, π¦, π§ ∈ πΏ. Using (R4), (5), and (14), we have π§ σ³¨→ (π¦ σ³¨→ π₯) ≤ π§ σ³¨→ (π§ σ³¨→ ((π§ σ³¨→ π¦) σ³¨→ π₯)) . (44) It follows from Proposition 10 (1), (42), and (R4) that ππΏ (π§ σ³¨→ (π¦ σ³¨→ π₯)) ⊆ ππΏ (π§ σ³¨→ (π§ σ³¨→ ((π§ σ³¨→ π¦) σ³¨→ π₯))) Theorem 19. Any filter of πΏ can be realized as an inclusive filter of some int-soft filter of πΏ. Proof. Let πΉ be a filter of πΏ. For a nonempty subset πΎ of π, let FπΏ be a soft set of πΏ defined by ππΏ : πΏ σ³¨→ P (π) , = ππΏ ((π§ σ³¨→ π¦) σ³¨→ (π§ σ³¨→ π₯)) . (45) Conversely, suppose that (43) holds. If we use π§ instead of π¦ in (43), then ππΏ (π§ σ³¨→ (π§ σ³¨→ π₯)) ⊆ ππΏ ((π§ σ³¨→ π§) σ³¨→ (π§ σ³¨→ π₯)) (46) = ππΏ (1 σ³¨→ (π§ σ³¨→ π₯)) = ππΏ (π§ σ³¨→ π₯) , Definition 20. An int-soft filter FπΏ of πΏ is said to be strong if the following assertion is valid: (∀π₯, π¦ ∈ πΏ) (ππΏ (π¦ σ³¨→ π₯) ⊆ ππΏ (((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯)) . (51) Example 21. The int-soft filter FπΏ in Example 8 is strong. (1) (∀π₯ ∈ πΏ)(ππΏ (π₯) ⊆ ππΏ (1)), (2) (∀π₯, π¦, π§ ∈ πΏ)(ππΏ (π§ → (π¦ → π₯)) ∩ ππΏ (π§) ⊆ ππΏ (((π₯ → π¦) → π¦) → π₯)). Proof. Suppose that FπΏ is a strong int-soft filter of πΏ. Obviously, (1) is valid. For every π₯, π¦, π§ ∈ πΏ, we have ⊆ ππΏ (((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) We make a new int-soft filter from old one. Theorem 18. Let FπΏ ∈ π(π, πΏ). For a subset πΎ of π, define a soft set F∗πΏ of πΏ by πΎ ππ π₯ ∈ FL , ππ‘βπππ€ππ π. (47) by Theorem 9 (2) and (51). Conversely, assume that FπΏ satisfies two conditions (1) and (2). If we take π¦ = 1 in (2), then ππΏ (π§) ∩ ππΏ (π§ → π₯) ⊆ ππΏ (π₯) for all π₯, π§ ∈ πΏ. Hence FπΏ is an int-soft filter of πΏ. Now if we put π§ = 1 in (2), then = ππΏ (1 σ³¨→ (π¦ σ³¨→ π₯)) ∩ ππΏ (1) πΎ Proof. Assume that FπΏ is an int-soft filter of πΏ. Then FπΏ ( =ΜΈ 0) πΎ is a filter of πΏ for all πΎ ∈ P(π). Hence 1 ∈ FπΏ , and so ππΏ∗ (1) = πΎ ∗ ππΏ (1) ⊇ ππΏ (π₯) ⊇ ππΏ (π₯) for all π₯ ∈ πΏ. Let π₯, π¦ ∈ πΏ. If π₯ ∈ FπΏ πΎ πΎ and π₯ → π¦ ∈ FπΏ , then π¦ ∈ FπΏ . Hence (π₯) ∩ ππΏ∗ (π₯ σ³¨→ π¦) = ππΏ (π₯) ∩ ππΏ (π₯ σ³¨→ π¦) ⊆ ππΏ (π¦) = ππΏ∗ (π¦) . πΎ (48) πΎ If π₯ ∉ FπΏ or π₯ → π¦ ∉ FπΏ , then ππΏ∗ (π₯) = 0 or ππΏ∗ (π₯ → π¦) = 0. Thus ππΏ∗ (π₯) ∩ ππΏ∗ (π₯ σ³¨→ π¦) = 0 ⊆ ππΏ∗ (π¦) . Therefore F∗πΏ is an int-soft filter of πΏ. (52) ππΏ (π¦ σ³¨→ π₯) = ππΏ (1 σ³¨→ (π¦ σ³¨→ π₯)) If FπΏ is an int-soft filter of πΏ, then so is F∗πΏ . ππΏ∗ (50) Obviously ππΏ (π₯) ⊆ ππΏ (1) for all π₯ ∈ πΏ. For any π₯, π¦ ∈ πΏ, if π₯ ∈ πΉ and π₯ → π¦ ∈ πΉ, then π¦ ∈ πΉ. Hence ππΏ (π₯) ∩ ππΏ (π₯ → π¦) = πΎ = ππΏ (π¦). If π₯ ∉ πΉ or π₯ → π¦ ∉ πΉ, then ππΏ (π₯) = 0 or ππΏ (π₯ → π¦) = 0. Thus ππΏ (π₯) ∩ ππΏ (π₯ → π¦) = 0 ⊆ ππΏ (π¦). πΎ Therefore FπΏ is an int-soft filter of πΏ and clearly FπΏ = πΉ. This completes the proof. ππΏ (π§ σ³¨→ (π¦ σ³¨→ π₯)) ∩ ππΏ (π§) ⊆ ππΏ (π¦ σ³¨→ π₯) which proves (42). π (π₯) π₯ σ³¨σ³¨→ { πΏ 0 if π₯ ∈ F, otherwise. Theorem 22. Let FπΏ ∈ π(π, πΏ). Then FπΏ is a strong int-soft filter of πΏ if and only if the following assertions are valid: ⊆ ππΏ (π§ σ³¨→ ((π§ σ³¨→ π¦) σ³¨→ π₯)) ππΏ∗ : πΏ σ³¨→ P (π) , πΎ π₯ σ³¨σ³¨→ { 0 (49) (53) ⊆ ππΏ (((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) by (R2) and (1). Therefore FπΏ is a strong int-soft filter of πΏ. Example 23. Let πΏ = [0, 1]. For any π, π ∈ πΏ, we define ¬π = 1 − π, π ∧ π = min {π, π} , π ∨ π = max {π, π} 1 π≤π π σ³¨→ π = { ¬π ∨ π otherwise. (54) 6 Discrete Dynamics in Nature and Society Then (πΏ, ∧, ∨, ¬, → ) is an π 0 -algebra (see [5]). Let FπΏ ∈ π(π, πΏ) be given as follows: FπΏ = {(1, πΎ2 ) , (π₯, πΎ1 ) | π₯ ∈ πΏ \ {1}} , (55) where πΎ1 and πΎ2 are subsets of π with πΎ1 β πΎ2 . Then FπΏ is an int-soft filter of πΏ. But ππΏ (1 σ³¨→ (0.3 σ³¨→ 0.8)) ∩ ππΏ (1) = πΎ2 ⊆ΜΈ πΎ1 = ππΏ (((0.8 σ³¨→ 0.3) σ³¨→ 0.3) σ³¨→ 0.8) , (56) Theorem 24. Let πΏ be an π 0 -algebra satisfying the following inequality: (57) Then every int-soft filter of πΏ is strong. Proof. Let FπΏ be an int-soft filter of πΏ. Using (5), (6), and (57), we have π¦ σ³¨→ π₯ = ((π¦ σ³¨→ π₯) σ³¨→ π₯) σ³¨→ π₯ ≤ ((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯ (58) for all π₯, π¦ ∈ πΏ. It follows from Proposition 10 (1) that ππΏ (π¦ σ³¨→ π₯) ⊆ ππΏ (((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) (59) for all π₯, π¦ ∈ πΏ. Therefore FπΏ is a strong int-soft filter of πΏ. We consider an extension property of a strong int-soft filter. Theorem 25. Let FπΏ and GπΏ be two int-soft filters of πΏ such Μ GπΏ and ππΏ (1) = ππΏ (1). If FπΏ is strong, then so is that FπΏ ⊆ GπΏ . Proof. Assume that FπΏ is a strong int-soft filter of πΏ. For any π₯, π¦ ∈ πΏ, let π = π¦ → π₯. Since FπΏ is a strong int-soft filter of πΏ, we have ππΏ (1) = ππΏ (1) = ππΏ (π¦ σ³¨→ (π σ³¨→ π₯)) ⊆ ππΏ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ (π σ³¨→ π₯)) ⊆ ππΏ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ (π σ³¨→ π₯)) , (60) by (51) and assumption, and so ππΏ (1) = ππΏ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ (π σ³¨→ π₯)) = ππΏ (π σ³¨→ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯)) . (61) Since GπΏ is an int-soft filter of πΏ, it follows that ππΏ (π) = ππΏ (π) ∩ ππΏ (1) = ππΏ (π) ∩ ππΏ × (π σ³¨→ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯)) (62) ⊆ ππΏ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) . 1 = π₯ σ³¨→ (π σ³¨→ π₯) ≤ ((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ (π₯ σ³¨→ π¦) ≤ ((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ (((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) ≤ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) σ³¨→ (((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) . (63) and so FπΏ is not a strong int-soft filter of πΏ by Theorem 22. We provide a condition for an int-soft filter to be strong. (∀π₯, π¦ ∈ πΏ) ((π₯ σ³¨→ π¦) σ³¨→ π¦ ≤ (π¦ σ³¨→ π₯) σ³¨→ π₯) . Using (R4) and (14), we have It follows from (62) and Theorem 14 that ππΏ (π¦ σ³¨→ π₯) = ππΏ (π) ⊆ ππΏ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) = ππΏ (1) ∩ ππΏ ((((π σ³¨→ π₯) σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) (64) ⊆ ππΏ (((π₯ σ³¨→ π¦) σ³¨→ π¦) σ³¨→ π₯) . Therefore GπΏ is a strong int-soft filter of πΏ. 4. Conclusion Using the notion of int-soft sets, we have introduced the concept of (strong) int-soft filters in π 0 -algebras and investigated related properties. We have established characterizations of a (strong) int-soft filter and made a new int-soft filter from old one. We have provided a condition for an int-soft filter to be strong and constructed an extension property of a strong int-soft filter. Work is ongoing. Some important issues for future work are (1) to develop strategies for obtaining more valuable results, (2) to apply these notions and results for studying related notions in other (soft) algebraic structures; and (3) to study the notions of implicative int-soft filters and Boolean int-soft filters. Acknowledgments The authors wish to thank the anonymous reviewer(s) for their valuable suggestions. This work (RPP-2012-021) was supported by the Fund of Research Promotion Program, Gyeongsang National University, 2012. References [1] D. Molodtsov, “Soft set theory-first results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999. [2] P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002. [3] P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562, 2003. [4] D. Chen, E. C. C. Tsang, D. S. Yeung, and X. Wang, “The parameterization reduction of soft sets and its applications,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 757–763, 2005. Discrete Dynamics in Nature and Society [5] G. J. Wang, Non-Classical Mathematical Logic and Approximate Reasoning, Science Press, Beijing, China, 2000. [6] G. J. Wang, “On the logic foundation of fuzzy reasoning,” Information Sciences, vol. 117, no. 1-2, pp. 47–88, 1999. [7] D. W. Pei and G. J. Wang, “The completeness and application of formal systems L,” Science China, vol. 32, no. 1, pp. 56–64, 2002. [8] A. Iorgulescu, Algebras of Logic as BCK Algebras, Academy of Economic Studies, Bucharest, Romania, 2008. [9] N. CΜ§agΜman and S. EnginogΜlu, “Soft set theory and uni-int decision making,” European Journal of Operational Research, vol. 207, no. 2, pp. 848–855, 2010. [10] L. Lianzhen and L. 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