Classes of Int-Soft Filters in Residuated Lattices

Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 595160, 12 pages http://dx.doi.org/10.1155/2014/595160

Research Article Classes of Int-Soft Filters in Residuated Lattices Young Bae Jun,1 Sun Shin Ahn,2 and Kyoung Ja Lee3 1

Department of Mathematics Education, Gyeongsang National University, Jinju 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; [email protected] Received 12 July 2014; Accepted 13 August 2014; Published 27 August 2014 Academic Editor: Xiaolong Xin Copyright © 2014 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 int-soft filters, int-soft G-filters, regular int-soft filters, and MV-int-soft filters in residuated lattices are introduced, and their relations, properties, and characterizations are investigated. Conditions for an int-soft filter to be an int-soft G-filter, a regular int-soft filter, or an MV-int-soft filter are provided. The extension property for an int-soft G-filter is discussed. Finally, it is shown that the notion of an MV-int-soft filter coincides with the notion of a regular int-soft filter in BL-algebras.

1. Introduction In order to deal with fuzzy and uncertain information, nonclassical logic has become a formal and useful tool. As the semantic systems of nonclassical logic systems, various logical algebras have been proposed. Residuated lattices are important algebraic structures which are basic of 𝑀𝑇𝐿algebras, 𝐵𝐿-algebras, 𝑀𝑉-algebras, G¨odel algebras, 𝑅0 algebras, lattice implication algebras, and so forth. The (fuzzy) filter theory in the logical algebras has an important role in studying these algebras and completeness of the corresponding nonclassical logics, and it is studied in [1–8]. Uncertainty is an attribute of information. As a new mathematical tool for dealing with uncertainties, Molodtsov [9] introduced the concept of soft sets. Since then several authors studied (fuzzy) algebraic structures based on soft set theory in several algebraic structures. Acar et al. [10] introduced initial concepts of soft rings. Ahn et al. [11] introduced the notion of int-soft filters of a 𝐵𝐸-algebra and investigated related properties. They also discussed characterization of an int-soft filter and solved the problem of classifying int-soft filters by their 𝛾-inclusive filter. Aktas and C ¸ agman [12] defined soft groups and derived their basic properties using Molodtsov’s definition of the soft sets. Atag¨un and Sezgin [13] introduced and studied soft subrings and soft ideals of a ring by using Molodtsov’s definition of the soft sets. Moreover, they introduced soft subfields of a field and soft submodule of a left R-module and

investigated some related properties about soft substructures of rings, fields, and modules. C ¸ a˘gman and Engino˘glu [14] constructed a uni-int decision making method which selects a set of optimum elements from the alternatives. Feng et al. [15] improved and further extended C ¸ a˘gman and Engino˘glu’s uni-int decision making method in virtue of choice value soft sets and k-satisfaction relations. C ¸ a˘gman and Engino˘glu [16] discussed fuzzy parameterized (FP) soft sets and their related properties and proposed a decision making method based on FP-soft set theory. Feng [17] considered the application of soft rough approximations in multicriteria group decision making problems. Feng et al. [18] initiated the study of soft semirings by using the soft set theory. Jun et al. applied the notion of soft sets by Molodtsov to the theory of 𝐵𝐶𝐾/𝐵𝐶𝐼algebras, 𝑑-algebras, and subtraction algebras (see [19–22]). Jun et al. [23] discussed (strong) intersection-soft filters in 𝑅0 algebras. Zhan and jun [24] investigated characterizations of (implicative, positive implicative, and fantastic) filteristic soft 𝐵𝐿-algebras by means of ∈-soft sets and 𝑞-soft sets. Recently, Feng and Li [25] explored some relationships among five different types of soft subsets and investigated free soft algebras with respect to soft product operations. They pointed out that soft sets have some nonclassical algebraic properties which are distinct from those of crisp sets or fuzzy sets. In this paper, we introduce the notions of int-soft filters, int-soft 𝐺-filters, regular int-soft filters, and 𝑀𝑉-int-soft filters in residuated lattices and investigate their relations and

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properties. We consider characterizations of int-soft filters, int-soft 𝐺-filters, regular int-soft filters, and 𝑀𝑉-int-soft filters. We provide conditions for an int-soft filter to be an int-soft 𝐺-filter, a regular int-soft filter, or an 𝑀𝑉-int-soft filter. We establish the extension property for an int-soft 𝐺filter. Finally, we show that the notion of an 𝑀𝑉-int-soft filter coincides with the notion of a regular int-soft filter in 𝐵𝐿algebras.

Definition 1 (see [1, 26, 27]). A residuated lattice is an algebra (𝐿, ∨, ∧, ⊙, → , 0, 1) of type (2, 2, 2, 2, 0, 0) such that (1) (𝐿, ∨, ∧, 0, 1) is a bounded lattice; (2) (𝐿, ⊙, 1) is a commutative monoid;

𝑥 ≤ 𝑦 󳨐⇒ 𝑥 ⊙ 𝑧 ≤ 𝑦 ⊙ 𝑧,

(14)

𝑦 󳨀→ 𝑧 ≤ 𝑥 ∨ 𝑦 󳨀→ 𝑥 ∨ 𝑧,

(15)

𝑥 󳨀→ (𝑦 ∧ 𝑧) = (𝑥 󳨀→ 𝑦) ∧ (𝑥 󳨀→ 𝑧) , (𝑥 ∨ 𝑦) 󳨀→ 𝑧 = (𝑥 󳨀→ 𝑧) ∧ (𝑦 󳨀→ 𝑧) ,

(𝑥 ≤ 𝑦 󳨀→ 𝑧 ⇐⇒ 𝑥 ⊙ 𝑦 ≤ 𝑧) .

= 𝑦 ⇐⇒ 𝑥 󳨀→ 𝑦 = 1)

(1)

(2)

Proposition 2 (see [1, 2, 6, 7, 26, 27]). In a residuated lattice 𝐿, the following properties are valid: 𝑥 󳨀→ 1 = 1,

𝑥 󳨀→ 𝑥 = 1,

0 󳨀→ 𝑥 = 1,

𝑥 󳨀→ (𝑦 󳨀→ 𝑥) = 1.

(3)

𝑦 ≤ (𝑦 󳨀→ 𝑥) 󳨀→ 𝑥,

(4)

𝑥 ≤ 𝑦 󳨀→ 𝑧 ⇐⇒ 𝑦 ≤ 𝑥 󳨀→ 𝑧,

(5)

𝑥 󳨀→ (𝑦 󳨀→ 𝑧) = (𝑥 ⊙ 𝑦) 󳨀→ 𝑧 = 𝑦 󳨀→ (𝑥 󳨀→ 𝑧) , 𝑥 ≤ 𝑦 󳨐⇒ 𝑧 󳨀→ 𝑥 ≤ 𝑧 󳨀→ 𝑦,

𝑧 󳨀→ 𝑦 ≤ (𝑦 󳨀→ 𝑥) 󳨀→ (𝑧 󳨀→ 𝑥) , (𝑥 󳨀→ 𝑦) ⊙ (𝑦 󳨀→ 𝑧) ≤ 𝑥 󳨀→ 𝑧, ¬𝑥 = ¬¬¬𝑥, ¬1 = 0,

(6)

𝑦 󳨀→ 𝑧 ≤ 𝑥 󳨀→ 𝑧, (7)

𝑧 󳨀→ 𝑦 ≤ (𝑥 󳨀→ 𝑧) 󳨀→ (𝑥 󳨀→ 𝑦) ,

𝑥 ≤ ¬¬𝑥, ¬0 = 1,

(∀𝑥, 𝑦 ∈ 𝐿)

(𝑥, 𝑦 ∈ 𝐹 󳨐⇒ 𝑥 ⊙ 𝑦 ∈ 𝐹) ,

(∀𝑥, 𝑦 ∈ 𝐿)

(𝑥 ∈ 𝐹, 𝑥 ≤ 𝑦 󳨐⇒ 𝑦 ∈ 𝐹) .

(18)

(∀𝑥 ∈ 𝐹) (∀𝑦 ∈ 𝐿)

(𝑥 󳨀→ 𝑦 ∈ 𝐹 󳨐⇒ 𝑦 ∈ 𝐹) .

(19) (20)

A soft set theory is introduced by Molodtsov [9], and C ¸ a˘gman and Engino˘glu [14] provided new definitions and various results on soft set theory. In what follows, 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. ̃ 𝐴) over 𝑈 is defined Definition 5 (see [9, 14]). A soft set (𝑓, to be the set of ordered pairs

and ¬𝑥 = 𝑥 → 0 for all 𝑥 ∈ 𝐿.

1 󳨀→ 𝑥 = 𝑥,

(17)

Definition 3 (see [5]). A nonempty subset 𝐹 of a residuated lattice 𝐿 is called a filter of 𝐿 if it satisfies the conditions

1 ∈ 𝐹,

In a residuated lattice 𝐿, the ordering ≤ and negation ¬ are defined as follows: (𝑥 ≤ 𝑦 ⇐⇒ 𝑥 ∧ 𝑦 = 𝑥 ⇐⇒ 𝑥 ∨ 𝑦

(16)

Proposition 4 (see [5]). A nonempty subset 𝐹 of a residuated lattice 𝐿 is a filter of 𝐿 if and only if it satisfies

(3) ⊙ and → form an adjoint pair; that is,

(∀𝑥, 𝑦 ∈ 𝐿)

(13)

𝑥 ≤ 𝑦 󳨀→ (𝑥 ⊙ 𝑦) .

2. Preliminaries

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

𝑥 ⊙ 𝑦 ≤ 𝑥 ∧ 𝑦,

(8)

̃ 𝐴) := {(𝑥, 𝑓̃ (𝑥)) : 𝑥 ∈ 𝐸, 𝑓̃ (𝑥) ∈ P (𝑈)} , (𝑓, ̃ = 0 if 𝑥 ∉ 𝐴. where 𝑓̃ : 𝐸 → P(𝑈) such that 𝑓(𝑥)

3. Int-Soft Filters In what follows, we take a residuated lattice 𝐿 as a set of parameters. ̃ 𝐿) over 𝑈 is called an int-soft filter Definition 6. A soft set (𝑓, of 𝐿 if it satisfies (∀𝑥, 𝑦 ∈ 𝐿)

(𝑥 ≤ 𝑦 󳨐⇒ 𝑓̃ (𝑥) ⊆ 𝑓̃ (𝑦)) ,

(22)

(∀𝑥, 𝑦 ∈ 𝐿)

(𝑓̃ (𝑥) ∩ 𝑓̃ (𝑦) ⊆ 𝑓̃ (𝑥 ⊙ 𝑦)) .

(23)

̃ 𝐿) of 𝐿 satisfies Proposition 7. Every int-soft filter (𝑓, (𝑓̃ (𝑥) ⊆ 𝑓̃ (1)) ,

(24)

(𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑦)) .

(25)

(∀𝑥 ∈ 𝐿) (9) (10)

¬𝑥 ∧ ¬𝑦 = ¬ (𝑥 ∨ 𝑦) ,

(11)

𝑥 ∨ ¬𝑥 = 1 󳨐⇒ 𝑥 ∧ ¬𝑥 = 0,

(12)

(21)

(∀𝑥, 𝑦 ∈ 𝐿)

̃ ⊆ 𝑓(1) ̃ by (22). Proof. Let 𝑥, 𝑦 ∈ 𝐿. Since 𝑥 ≤ 1, we have 𝑓(𝑥) Since 𝑥 ⊙ (𝑥 → 𝑦) ≤ 𝑦, it follows from (23) and (22) that 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑥 ⊙ (𝑥 󳨀→ 𝑦)) ⊆ 𝑓̃ (𝑦) . This completes the proof.

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̃ 𝐿) be a soft set over 𝑈 that satisfies two Lemma 8. Let (𝑓, conditions (24) and (25). Then one has (𝑥 ≤ 𝑦 󳨀→ 𝑧 󳨐⇒ 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑦) ⊆ 𝑓̃ (𝑧)) , (27)

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

̃ 𝐿) be a soft set over 𝑈 satisfying (24) Conversely, let (𝑓, and (31). Taking 𝑥 := 1 in (31) and using (3), we get 𝑓̃ (𝑧) = 𝑓̃ (1 󳨀→ 𝑧) ⊇ 𝑓̃ (1 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (𝑦) = 𝑓̃ (𝑦 󳨀→ 𝑧) ∩ 𝑓̃ (𝑦)

(33)

(𝑥 ⊙ 𝑦 ≤ 𝑧 󳨐⇒ 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑦) ⊆ 𝑓̃ (𝑧)) . (28)

̃ 𝐿) is an int-soft filter of 𝐿 by for all 𝑦, 𝑧 ∈ 𝐿. Thus (𝑓, Theorem 9.

Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝐿 be such that 𝑥 ≤ 𝑦 → 𝑧. Then 𝑥 → (𝑦 → 𝑧) = 1, and so

̃ 𝐿) over 𝑈 satisfies the Lemma 12. Every int-soft filter (𝑓, following condition:

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

𝑓̃ (𝑥) ∩ 𝑓̃ (𝑦) = (𝑓̃ (𝑥) ∩ 𝑓̃ (1)) ∩ 𝑓̃ (𝑦)

(∀𝑎, 𝑥 ∈ 𝐿)

= (𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧))) ∩ 𝑓̃ (𝑦)

Proof. If we take 𝑦 := (𝑎 → 𝑥) → 𝑥 and 𝑥 := 𝑎 in (25), then

(29)

= 𝑓̃ (𝑎) ∩ 𝑓̃ ((𝑎 󳨀→ 𝑥) 󳨀→ (𝑎 󳨀→ 𝑥))

Since 𝑥 ≤ 𝑦 → 𝑧 ⇔ 𝑥 ⊙ 𝑦 ≤ 𝑧, (28) is from (27).

= 𝑓̃ (𝑎) ∩ 𝑓̃ (1) = 𝑓̃ (𝑎) .

̃ 𝐿) over 𝑈 is an int-soft filter of 𝐿 if Theorem 9. A soft set (𝑓, and only if it satisfies two conditions (24) and (25). Proof. The necessity is from Proposition 7. ̃ 𝐿) be a soft set over 𝑈 that satisfies (24) Conversely, let (𝑓, and (25). If 𝑥 ≤ 𝑦, then 𝑥 → 𝑦 = 1 and so 𝑓̃ (𝑥) = 𝑓̃ (𝑥) ∩ 𝑓̃ (1) = 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑦) . (30) Since 𝑥 ⊙ 𝑦 ≤ 𝑥 ⊙ 𝑦 for all 𝑥, 𝑦 ∈ 𝐿, it follows from (28) that ̃ ∩ 𝑓(𝑦) ̃ ⊆ 𝑓(𝑥 ̃ ⊙ 𝑦) for all 𝑥, 𝑦 ∈ 𝐿. Therefore (𝑓, ̃ 𝐿) is an 𝑓(𝑥) int-soft filter of 𝐿. ̃ 𝐿) over 𝑈 is an int-soft filter of 𝐿 if Theorem 10. A soft set (𝑓, and only if it satisfies condition (27). Proof. The necessity is from Lemma 8 and Theorem 9. ̃ 𝐿) be a soft set over 𝑈 satisfying (27). Conversely, let (𝑓, Since 𝑥 ≤ 𝑥 → 1 and 𝑥 → 𝑦 ≤ 𝑥 → 𝑦 for all 𝑥, 𝑦 ∈ 𝐿, ̃ ̃ ∩ 𝑓(𝑥) ̃ ̃ and it follows from (27) that 𝑓(𝑥) = 𝑓(𝑥) ⊆ 𝑓(1) ̃ ∩ 𝑓(𝑥 ̃ → 𝑦) ⊆ 𝑓(𝑦) ̃ for all 𝑥, 𝑦 ∈ 𝐿. Hence (𝑓, ̃ 𝐿) is an 𝑓(𝑥) int-soft filter of 𝐿 by Theorem 9. ̃ 𝐿) over 𝑈 is an int-soft filter of 𝐿 if Theorem 11. A soft set (𝑓, ̃ and only if (𝑓, 𝐿) satisfies condition (24) and (𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (𝑦) ⊆ 𝑓̃ (𝑥 󳨀→ 𝑧)) .

(31)

̃ 𝐿) is an int-soft filter of 𝐿. Then Proof. Assume that (𝑓, condition (24) is valid. Using (6) and (25), we have 𝑓̃ (𝑥 󳨀→ 𝑧) ⊇ 𝑓̃ (𝑦) ∩ 𝑓̃ (𝑦 󳨀→ (𝑥 󳨀→ 𝑧)) = 𝑓̃ (𝑦) ∩ 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) for all 𝑥, 𝑦, 𝑧 ∈ 𝐿.

(34)

𝑓̃ ((𝑎 󳨀→ 𝑥) 󳨀→ 𝑥) ⊇ 𝑓̃ (𝑎) ∩ 𝑓̃ (𝑎 󳨀→ ((𝑎 󳨀→ 𝑥) 󳨀→ 𝑥))

⊆ 𝑓̃ (𝑦) ∩ 𝑓̃ (𝑦 󳨀→ 𝑧) ⊆ 𝑓̃ (𝑧) .

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

(𝑓̃ (𝑎) ⊆ 𝑓̃ ((𝑎 󳨀→ 𝑥) 󳨀→ 𝑥)) .

(32)

(35) This completes the proof. ̃ 𝐿) over 𝑈 is an int-soft filter of 𝐿 if Theorem 13. A soft set (𝑓, and only if it satisfies the following conditions: (∀𝑥, 𝑦 ∈ 𝐿) (∀𝑥, 𝑎, 𝑏 ∈ 𝐿)

(𝑓̃ (𝑥) ⊆ 𝑓̃ (𝑦 󳨀→ 𝑥)) ,

(36)

(𝑓̃ (𝑎) ∩ 𝑓̃ (𝑏) ⊆ 𝑓̃ ((𝑎 󳨀→ (𝑏 󳨀→ 𝑥)) 󳨀→ 𝑥)) .

(37)

̃ 𝐿) is an int-soft filter of 𝐿. Using (3), Proof. Assume that (𝑓, (24), and (25), we have 𝑓̃ (𝑦 󳨀→ 𝑥) ⊇ 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑥)) = 𝑓̃ (𝑥) ∩ 𝑓̃ (1) = 𝑓̃ (𝑥)

(38)

for all 𝑥, 𝑦 ∈ 𝐿. Using (31) and (34), we get 𝑓̃ ((𝑎 󳨀→ (𝑏 󳨀→ 𝑥)) 󳨀→ 𝑥) ⊇ 𝑓̃ ((𝑎 󳨀→ (𝑏 󳨀→ 𝑥)) 󳨀→ (𝑏 󳨀→ 𝑥)) ∩ 𝑓̃ (𝑏)

(39)

⊇ 𝑓̃ (𝑎) ∩ 𝑓̃ (𝑏) for all 𝑎, 𝑏, 𝑥 ∈ 𝐿. ̃ 𝐿) be a soft set over 𝑈 satisfying two Conversely, let (𝑓, ̃ ⊆ conditions (36) and (37). If we take 𝑦 := 𝑥 in (36), then 𝑓(𝑥) ̃ ̃ 𝑓(𝑥 → 𝑥) = 𝑓(1) for all 𝑥 ∈ 𝐿. Using (37) induces 𝑓̃ (𝑦) = 𝑓̃ (1 󳨀→ 𝑦) = 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ 𝑦)

(40)

⊇ 𝑓̃ (𝑥 󳨀→ 𝑦) ∩ 𝑓̃ (𝑥) ̃ 𝐿) is an int-soft filter of 𝐿 by for all 𝑥, 𝑦 ∈ 𝐿. Therefore (𝑓, Theorem 9.

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̃ 𝐿) over 𝑈 is an int-soft filter of 𝐿 if Theorem 14. A soft set (𝑓, and only if the set 𝑓̃𝜏 := {𝑥 ∈ 𝐿 | 𝜏 ⊆ 𝑓̃ (𝑥)}

(41)

𝑓̃ (𝑎) ⊆ 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑦)

is a filter of 𝐿 for all 𝜏 ∈ P(𝑈) with 𝑓̃𝜏 ≠ 0. ̃ 𝐿) is an int-soft filter of 𝐿. Let 𝑥, 𝑦 ∈ 𝐿 Proof. Assume that (𝑓, and 𝜏 ∈ P(𝑈) be such that 𝑥 ∈ 𝑓̃𝜏 and 𝑥 → 𝑦 ∈ 𝑓̃𝜏 . Then ̃ and 𝜏 ⊆ 𝑓(𝑥 ̃ → 𝑦). It follows from (24) and (25) 𝜏 ⊆ 𝑓(𝑥) ̃ ̃ ̃ ̃ ∩ 𝑓(𝑥 ̃ → 𝑦) ⊇ 𝜏 that 𝑓(1) ⊇ 𝑓(𝑥) ⊇ 𝜏 and 𝑓(𝑦) ⊇ 𝑓(𝑥) and so that 1 ∈ 𝑓̃𝜏 and 𝑦 ∈ 𝑓̃𝜏 . Hence 𝑓̃𝜏 is a filter of 𝐿 by Proposition 4. Conversely, suppose that 𝑓̃𝜏 is a filter of 𝐿 for all 𝜏 ∈ P(𝑈) ̃ = 𝛿. Then 𝑥 ∈ 𝑓̃ and 𝑓̃ with 𝑓̃𝜏 ≠ 0. For any 𝑥 ∈ 𝐿, let 𝑓(𝑥) 𝛿 𝛿 ̃ ̃ = 𝛿 ⊆ 𝑓(1). ̃ is a filter of 𝐿. Hence 1 ∈ 𝑓𝛿 and so 𝑓(𝑥) For any ̃ ̃ → 𝑦) = 𝛿 𝑥, 𝑦 ∈ 𝐿, let 𝑓(𝑥) = 𝛿𝑥 and 𝑓(𝑥 𝑥 → 𝑦 . If we take ̃ 𝛿 = 𝛿𝑥 ∩ 𝛿𝑥 → 𝑦 , then 𝑥 ∈ 𝑓𝛿 and 𝑥 → 𝑦 ∈ 𝑓̃𝛿 which imply that 𝑦 ∈ 𝑓̃𝛿 . Thus 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) = 𝛿𝑥 ∩ 𝛿𝑥 → 𝑦 = 𝛿 ⊆ 𝑓̃ (𝑦) .

(42)

̃ 𝐿) is an int-soft filter of 𝐿 by Theorem 9. Therefore (𝑓, ̃ 𝐿) over 𝑈, let (𝑓̃∗ , 𝐿) be a soft Theorem 15. For a soft set (𝑓, set over 𝑈, where 𝑓̃∗ : 𝐿 󳨀→ P (𝑈) ,

𝑓̃ (𝑥) 𝑖𝑓 𝑥 ∈ 𝑓̃𝜏 , 𝑥 󳨃󳨀→ { 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,

(43)

̃ 𝐿) is an int-soft filter of 𝐿, then so where 𝜏 ∈ P(𝑈) \ {0}. If (𝑓, ∗ ̃ is (𝑓 , 𝐿). ̃ 𝐿) is an int-soft filter of 𝐿. Then 𝑓̃ is Proof. Suppose that (𝑓, 𝜏 a filter of 𝐿 for all 𝜏 ∈ P(𝑈) with 𝑓̃𝜏 ≠ 0 by Theorem 14. Thus ̃ ⊇ 𝑓(𝑥) ̃ ⊇ 𝑓̃∗ (𝑥) for all 𝑥 ∈ 𝐿. Let 1 ∈ 𝑓̃𝜏 , and so 𝑓̃∗ (1) = 𝑓(1) 𝑥, 𝑦 ∈ 𝐿. If 𝑥 ∈ 𝑓̃𝜏 and 𝑥 → 𝑦 ∈ 𝑓̃𝜏 , then 𝑦 ∈ 𝑓̃𝜏 . Hence 𝑓̃∗ (𝑥) ∩ 𝑓̃∗ (𝑥 󳨀→ 𝑦) = 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑦) = 𝑓̃∗ (𝑦) .

(44)

If 𝑥 ∉ 𝑓̃𝜏 or 𝑥 → 𝑦 ∉ 𝑓̃𝜏 , then 𝑓̃∗ (𝑥) = 0 or 𝑓̃∗ (𝑥 → 𝑦) = 0. Thus 𝑓̃∗ (𝑥) ∩ 𝑓̃∗ (𝑥 󳨀→ 𝑦) = 0 ⊆ 𝑓̃∗ (𝑦) .

(45)

̃ 𝐿) is an int-soft filter of 𝐿, then the set Theorem 16. If (𝑓,

is a filter of 𝐿 for every 𝑎 ∈ 𝐿.

(46)

(47)

so that 𝑦 ∈ 𝐿 𝑎 . Hence 𝐿 𝑎 is a filter of 𝐿 by Proposition 4. ̃ 𝐿) be a soft set over 𝑈. Then Theorem 17. Let 𝑎 ∈ 𝐿 and let (𝑓, ̃ 𝐿) satisfies the following (1) if 𝐿 𝑎 is a filter of 𝐿, then (𝑓, condition: (∀𝑥, 𝑦 ∈ 𝐿)

(𝑓̃ (𝑎) ⊆ 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) 󳨐⇒ 𝑓̃ (𝑎) ⊆ 𝑓̃ (𝑦)) ; (48)

̃ 𝐿) satisfies (24) and (48), then 𝐿 is a filter of 𝐿. (2) if (𝑓, 𝑎 Proof. (1) Assume that 𝐿 𝑎 is a filter of 𝐿. Let 𝑥, 𝑦 ∈ 𝐿 be such that 𝑓̃ (𝑎) ⊆ 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) .

(49)

Then 𝑥 → 𝑦 ∈ 𝐿 𝑎 and 𝑥 ∈ 𝐿 𝑎 . Using (20), we have 𝑦 ∈ 𝐿 𝑎 ̃ ⊆ 𝑓(𝑦). ̃ and so 𝑓(𝑎) ̃ 𝐿) satisfies (24) and (48). From (24) (2) Suppose that (𝑓, it follows that 1 ∈ 𝐿 𝑎 . Let 𝑥, 𝑦 ∈ 𝐿 be such that 𝑥 ∈ 𝐿 𝑎 and ̃ ̃ and 𝑓(𝑎) ̃ ̃ → 𝑦), 𝑥 → 𝑦 ∈ 𝐿 𝑎 . Then 𝑓(𝑎) ⊆ 𝑓(𝑥) ⊆ 𝑓(𝑥 ̃ ̃ ̃ ̃ which imply that 𝑓(𝑎) ⊆ 𝑓(𝑥) ∩ 𝑓(𝑥 → 𝑦). Thus 𝑓(𝑎) ⊆ ̃ 𝑓(𝑦) by (48), and so 𝑦 ∈ 𝐿 𝑎 . Therefore 𝐿 𝑎 is a filter of 𝐿 by Proposition 4.

4. Int-Soft 𝐺-Filters Definition 18 (see [28]). A nonempty subset 𝐹 of 𝐿 is called a 𝐺-filter of 𝐿 if it is a filter of 𝐿 that satisfies the following condition: (∀𝑥, 𝑦 ∈ 𝐿)

((𝑥 ⊙ 𝑥) 󳨀→ 𝑦 ∈ 𝐹 󳨐⇒ 𝑥 󳨀→ 𝑦 ∈ 𝐹) . (50)

̃ 𝐿) over 𝑈 is called an int-soft 𝐺Definition 19. A soft set (𝑓, filter of 𝐿 if it is an int-soft filter of 𝐿 that satisfies (∀𝑥, 𝑦 ∈ 𝐿)

(𝑓̃ ((𝑥 ⊙ 𝑥) 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑥 󳨀→ 𝑦)) .

(51)

Note that condition (51) is equivalent to the following condition: (∀𝑥, 𝑦 ∈ 𝐿)

Therefore (𝑓̃∗ , 𝐿) is an int-soft filter of 𝐿.

𝐿 𝑎 := {𝑥 ∈ 𝐿 | 𝑓̃ (𝑎) ⊆ 𝑓̃ (𝑥)}

̃ ⊇ 𝑓(𝑎) ̃ for all 𝑎 ∈ 𝐿, we have 1 ∈ 𝐿 . Let Proof. Since 𝑓(1) 𝑎 ̃ ⊇ 𝑥, 𝑦 ∈ 𝐿 be such that 𝑥 ∈ 𝐿 𝑎 and 𝑥 → 𝑦 ∈ 𝐿 𝑎 . Then 𝑓(𝑥) ̃ and 𝑓(𝑥 ̃ → 𝑦) ⊇ 𝑓(𝑎). ̃ ̃ 𝐿) is an int-soft filter 𝑓(𝑎) Since (𝑓, of 𝐿, it follows from (25) that

(𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ⊆ 𝑓̃ (𝑥 󳨀→ 𝑦)) . (52)

̃ 𝐿) of 𝐿 satisfies the followLemma 20. Every int-soft filter (𝑓, ing condition: (∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

(𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑧))) .

(53)

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5 ̃ 𝐿) is an int-soft filter of 𝐿 by Theorem 9. Let Hence (𝑓, 𝑥, 𝑦, 𝑧 ∈ 𝐿. Since

Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝐿. Using (6) and (8), we have 𝑥 󳨀→ (𝑦 󳨀→ 𝑧) = 𝑦 󳨀→ (𝑥 󳨀→ 𝑧) ≤ (𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ (𝑥 󳨀→ 𝑧)) .

(54)

It follows from Theorem 10 that 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑧)) . (55)

𝑥 󳨀→ (𝑦 󳨀→ 𝑧) ≤ (𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ (𝑥 󳨀→ 𝑧))

(62)

̃ → (𝑦 → 𝑧)) ⊆ 𝑓((𝑥 ̃ by (6) and (8), we have 𝑓(𝑥 → 𝑦) → (𝑥 → (𝑥 → 𝑧))) by (22). It follows from (22), (24), (25), (8), and (60) that 𝑓̃ (𝑥 󳨀→ 𝑦) ∩ 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧))

This completes the proof.

⊆ 𝑓̃ (𝑥 󳨀→ 𝑦) ∩ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ (𝑥 󳨀→ 𝑧)))

̃ 𝐿) be a soft set over 𝑈. Then (𝑓, ̃ 𝐿) is an Theorem 21. Let (𝑓, int-soft 𝐺-filter of 𝐿 if and only if it is an int-soft filter of 𝐿 that satisfies the following condition:

⊆ 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑧))

(𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦)

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

⊆ 𝑓̃ (𝑥 󳨀→ 𝑧)) .

(56)

̃ 𝐿) is an int-soft 𝐺-filter of 𝐿. Then Proof. Assume that (𝑓, ̃ (𝑓, 𝐿) is an int-soft filter of 𝐿. Note that 𝑥 ≤ 1 = (𝑥 → 𝑦) → (𝑥 → 𝑦), and thus 𝑥 → 𝑦 ≤ 𝑥 → (𝑥 → 𝑦) for all 𝑥, 𝑦 ∈ 𝐿. ̃ → 𝑦) ⊆ 𝑓(𝑥 ̃ → (𝑥 → 𝑦)). It follows from (22) that 𝑓(𝑥 Combining this and (52), we have 𝑓̃ (𝑥 󳨀→ 𝑦) = 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦))

(57)

for all 𝑥, 𝑦 ∈ 𝐿. Using (53) and (57), we have 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑥 󳨀→ 𝑧)

(58)

for all 𝑥, 𝑦, 𝑧 ∈ 𝐿. ̃ 𝐿) be an int-soft filter of 𝐿 that satisfies Conversely, let (𝑓, condition (56). If we put 𝑦 = 𝑥 and 𝑧 = 𝑦 in (56) and use (3) and (24), then 𝑓̃ (𝑥 󳨀→ 𝑦) ⊇ 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ∩ 𝑓̃ (𝑥 󳨀→ 𝑥) = 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ∩ 𝑓̃ (1)

(59)

= 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ̃ 𝐿) is an int-soft 𝐺-filter of 𝐿. for all 𝑥, 𝑦 ∈ 𝐿. Therefore (𝑓, ̃ 𝐿) be a soft set over 𝑈 that satisfies Theorem 22. Let (𝑓, condition (24) and (∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

(𝑓̃ (𝑥) ∩ 𝑓̃ ((𝑦 󳨀→ 𝑧) 󳨀→ (𝑥 󳨀→ 𝑦)) ⊆ 𝑓̃ (𝑦)) . (60)

̃ 𝐿) is an int-soft 𝐺-filter of 𝐿. Then (𝑓,

⊆ 𝑓̃ (((𝑥 󳨀→ 𝑧) 󳨀→ 𝑧) 󳨀→ (𝑥 󳨀→ 𝑧)) = 𝑓̃ (((𝑥 󳨀→ 𝑧) 󳨀→ 𝑧) 󳨀→ (1 󳨀→ (𝑥 󳨀→ 𝑧))) ⊆ 𝑓̃ (𝑥 󳨀→ 𝑧) . (63) ̃ 𝐿) is an int-soft 𝐺-filter of 𝐿 by Theorem 21. Therefore (𝑓, The following example shows that any int-soft 𝐺-filter may not satisfy condition (60). Example 23. Let 𝐿 := [0, 1] (unit interval). For any 𝑎, 𝑏 ∈ 𝐿, define 𝑎 ∨ 𝑏 = max {𝑎, 𝑏} , 𝑎 ∧ 𝑏 = min {𝑎, 𝑏} , 1 if 𝑎 ≤ 𝑏, 𝑎 󳨀→ 𝑏 = { 𝑏 otherwise,

𝑎 ⊙ 𝑏 = min {𝑎, 𝑏} .

(64)

̃ 𝐿) be Then (𝐿, ∨, ∧, ⊙, → , 0, 1) is a residuated lattice. Let (𝑓, a soft set over 𝑈 defined by {𝜏 if 𝑥 ∈ [ 1 , 1] , (65) 𝑥 󳨃󳨀→ { 2 {0 otherwise, ̃ 𝐿) is an int-soft 𝐺-filter of 𝐿. where 𝜏 ∈ P(𝑈) \ {0}. Then (𝑓, But it does not satisfy condition (60). For example, 1 1 2 1 2 𝑓̃ ( ) ∩ 𝑓̃ (( 󳨀→ ) 󳨀→ ( 󳨀→ )) 3 3 4 3 3 (66) 2 1 ̃ ̃ ̃ = 𝑓 ( ) ∩ 𝑓 (1) = 𝜏 ⊈ 0 = 𝑓 ( ) . 3 3 ̃ 𝐿) of 𝐿, condition Proposition 24. For an int-soft filter (𝑓, (60) is equivalent to the following condition: 𝑓̃ : 𝐿 󳨀→ P (𝑈) ,

(∀𝑥, 𝑦 ∈ 𝐿)

(𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑥) ⊆ 𝑓̃ (𝑥)) .

(67)

Proof. Assume that condition (60) is valid. It follows from (24) and (3) that 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑥) = 𝑓̃ (1) ∩ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑥)

Proof. If we take 𝑧 := 1 in (60) and use (3), then 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦) = 𝑓̃ (𝑥) ∩ 𝑓̃ (1 󳨀→ (𝑥 󳨀→ 𝑦))

= 𝑓̃ (1) ∩ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (1 󳨀→ 𝑥))

= 𝑓̃ (𝑥) ∩ 𝑓̃ ((𝑦 󳨀→ 1) 󳨀→ (𝑥 󳨀→ 𝑦))

⊆ 𝑓̃ (𝑥)

⊆ 𝑓̃ (𝑦) .

(68) (61)

for all 𝑥, 𝑦 ∈ 𝐿.

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Conversely, suppose that condition (67) is valid. It follows from (6) and (25) that 𝑓̃ (𝑥) ∩ 𝑓̃ ((𝑦 󳨀→ 𝑧) 󳨀→ (𝑥 󳨀→ 𝑦)) = 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ ((𝑦 󳨀→ 𝑧) 󳨀→ 𝑦))

̃ 𝐿) be an int-soft 𝐺-filter of 𝐿 that satisfies Proof. Let (𝑓, condition (70). For any 𝑥, 𝑦, 𝑧 ∈ 𝐿, we have 𝑓̃ (𝑧) ∩ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑧 󳨀→ 𝑥))

(69)

⊆ 𝑓̃ ((𝑦 󳨀→ 𝑧) 󳨀→ 𝑦) ⊆ 𝑓̃ (𝑦)

= 𝑓̃ (𝑧) ∩ 𝑓̃ (𝑧 󳨀→ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑥)) ⊆ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑥)

for all 𝑥, 𝑦 ∈ 𝐿.

⊆ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦))

Combining Theorem 22 and Proposition 24, we have the following result.

⊆ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦)

Theorem 25. Every int-soft filter satisfying condition (67) is an int-soft 𝐺-filter. ̃ 𝐿) of 𝐿 with condition Proposition 26. Every int-soft filter (𝑓, (60) satisfies the following condition: (∀𝑥, 𝑦 ∈ 𝐿)

⊆ 𝑓̃ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) by (6), (25), (22), (8), (52), and (70). Since (𝑥 → 𝑦) → 𝑥 ≤ ̃ 𝑦 → 𝑥 ≤ 𝑧 → (𝑦 → 𝑥), it follows from (22) that 𝑓((𝑥 → ̃ 𝑦) → 𝑥) ⊆ 𝑓(𝑧 → (𝑦 → 𝑥)) and so from (25) that 𝑓̃ (𝑧) ∩ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑧 󳨀→ 𝑥))

(𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) ⊆ 𝑓̃ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥)) .

(73)

(70)

̃ 𝐿) be an int-soft filter of 𝐿 that satisfies Proof. Let (𝑓, condition (60) and let 𝑥, 𝑦 ∈ 𝐿. Since 𝑥 → ((𝑦 → 𝑥) → 𝑥) = (𝑦 → 𝑥) → (𝑥 → 𝑥) = (𝑦 → 𝑥) → 1 = 1, that is, 𝑥 ≤ (𝑦 → 𝑥) → 𝑥, we have ((𝑦 → 𝑥) → 𝑥) → 𝑦 ≤ 𝑥 → 𝑦 by (7). It follows from (8), (6), and (7) that (𝑥 󳨀→ 𝑦) 󳨀→ 𝑦

⊆ 𝑓̃ (𝑧) ∩ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑥) ⊆ 𝑓̃ (𝑧) ∩ 𝑓̃ (𝑧 󳨀→ (𝑦 󳨀→ 𝑥))

(74)

⊆ 𝑓̃ (𝑦 󳨀→ 𝑥) . ̃ ∩ 𝑓((𝑥 ̃ ̃ Therefore 𝑓(𝑧) → 𝑦) → (𝑧 → 𝑥)) ⊆ 𝑓(𝑦 → ̃ ̃ 𝑥) ∩ 𝑓((𝑦 → 𝑥) → 𝑥) ⊆ 𝑓(𝑥). Hence condition (60) is valid. ̃ 𝐿) be an int-soft filter of 𝐿. Then (𝑓, ̃ 𝐿) Theorem 28. Let (𝑓, is an int-soft 𝐺-filter of 𝐿 if and only if the following condition holds:

≤ (𝑦 󳨀→ 𝑥) 󳨀→ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑥) = (𝑥 󳨀→ 𝑦) 󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) ≤ (((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) 󳨀→ 𝑦) 󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) . (71) Using (22), (24), (3), (6), and (60), we have 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) ⊆ 𝑓̃ ((((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) 󳨀→ 𝑦) 󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥))

(∀𝑥 ∈ 𝐿)

(𝑓̃ (𝑥 󳨀→ (𝑥 ⊙ 𝑥)) = 𝑓̃ (1)) .

̃ 𝐿) is an int-soft 𝐺-filter of 𝐿. Since Proof. Suppose that (𝑓, ̃ → (𝑥 → 𝑥 → (𝑥 → (𝑥 ⊙ 𝑥)) = 1 for all 𝑥 ∈ 𝐿, we have 𝑓(𝑥 ̃ (𝑥 ⊙ 𝑥))) = 𝑓(1). It follows from (56) and (3) that 𝑓̃ (𝑥 󳨀→ (𝑥 ⊙ 𝑥)) ⊇ 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ (𝑥 ⊙ 𝑥))) ∩ 𝑓̃ (𝑥 󳨀→ 𝑥) = 𝑓̃ (1)

= 𝑓̃ (1) ∩ 𝑓̃ (1 󳨀→ ((((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) 󳨀→ 𝑦)

(75)

(76)

̃ → (𝑥 ⊙ 𝑥)) = 𝑓(1) ̃ for all 𝑥 ∈ 𝐿. and so from (24) that 𝑓(𝑥 ̃ Conversely, let (𝑓, 𝐿) be an int-soft filter of 𝐿 which satisfies condition (75) and let 𝑥, 𝑦 ∈ 𝐿. Since

󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥))) = 𝑓̃ (1) ∩ 𝑓̃ ((((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) 󳨀→ 𝑦) 󳨀→ (1 󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥)))

𝑥 󳨀→ (𝑥 󳨀→ 𝑦) = (𝑥 ⊙ 𝑥) 󳨀→ 𝑦

⊆ 𝑓̃ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) . (72) Hence condition (70) is valid. ̃ 𝐿) of 𝐿 with condiProposition 27. Every int-soft 𝐺-filter (𝑓, tion (70) satisfies condition (60).

≤ (𝑥 󳨀→ (𝑥 ⊙ 𝑥)) 󳨀→ (𝑥 󳨀→ 𝑦)

(77)

by (6) and (8), it follows from (22) that 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ⊆ 𝑓̃ ((𝑥 󳨀→ (𝑥 ⊙ 𝑥)) 󳨀→ (𝑥 󳨀→ 𝑦)) . (78)

The Scientific World Journal

7 ∩ 𝑓̃ (𝑥 󳨀→ ((𝑦 󳨀→ 𝑧) 󳨀→ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑧)))

Hence, we have

= 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧))

𝑓̃ (𝑥 󳨀→ 𝑦) ⊇ 𝑓̃ ((𝑥 󳨀→ (𝑥 ⊙ 𝑥)) 󳨀→ (𝑥 󳨀→ 𝑦)) ∩ 𝑓̃ (𝑥 󳨀→ (𝑥 ⊙ 𝑥))

∩ 𝑓̃ ((𝑦 󳨀→ 𝑧) 󳨀→ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ 𝑧)))

⊇ 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ∩ 𝑓̃ (𝑥 󳨀→ (𝑥 ⊙ 𝑥))

= 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (1)

= 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ∩ 𝑓̃ (1)

= 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) (83)

= 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) (79) ̃ 𝐿) is an int-soft 𝐺by using (25), (75), and (24). Hence (𝑓, filter of 𝐿. ̃ 𝐿) of 𝐿, the following Theorem 29. For an int-soft filter (𝑓, assertions are equivalent:

̃ → 𝑦) = 𝑓(𝑥 ̃ → (𝑥 → 𝑦))). (2) (∀𝑥, 𝑦 ∈ 𝐿) (𝑓(𝑥 ̃ 𝐿) is an int-soft 𝐺-filter of Proof. (1) ⇒ (2). Suppose that (𝑓, 𝐿 and let 𝑥, 𝑦 ∈ 𝐿. Since 𝑥 → 𝑦 ≤ 𝑥 → (𝑥 → 𝑦), it follows ̃ ̃ from (22) that 𝑓(𝑥 → 𝑦) ⊆ 𝑓(𝑥 → (𝑥 → 𝑦)). Hence ̃ ̃ 𝑓(𝑥 → 𝑦) = 𝑓(𝑥 → (𝑥 → 𝑦)) by using (52). (2) ⇒ (1). Assume that (2) holds. Using Lemma 20 and (2), we have

⊆ 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑧)) = 𝑓̃ (𝑥 󳨀→ 𝑧)

(80)

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

(𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) (81)

(82)

̃ 𝐿) be an int-soft 𝐺-filter of 𝐿. Using (6), (56), Proof. Let (𝑓, (8), and (24), we have 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ 𝑧)) = 𝑓̃ (𝑥 󳨀→ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑧)) ⊇ 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧))

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

(𝑓̃ ((𝑥 ⊙ 𝑦) 󳨀→ 𝑧) = 𝑓̃ ((𝑥 ∧ 𝑦) 󳨀→ 𝑧)) . (85)

Proof. Using the divisibility and (6), we have (𝑥 ∧ 𝑦) 󳨀→ 𝑧 = (𝑥 ⊙ (𝑥 󳨀→ 𝑦)) 󳨀→ 𝑧 = (𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ 𝑧)

(86)

𝑓̃ ((𝑥 ⊙ 𝑦) 󳨀→ 𝑧) = 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) (87)

= 𝑓̃ ((𝑥 ∧ 𝑦) 󳨀→ 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝐿. Theorem 32. Let 𝐿 satisfy the divisibility; that is, 𝑥 ∧ 𝑦 = 𝑥 ⊙ ̃ 𝐿) of 𝐿 (𝑥 → 𝑦), for all 𝑥, 𝑦 ∈ 𝐿. Then every int-soft filter (𝑓, satisfying condition (85) is an int-soft 𝐺-filter of 𝐿. Proof. Using Lemma 20 and (6) and (85), we have 𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦)

(𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) = 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ 𝑧))) .

for all 𝑥, 𝑦, 𝑧 ∈ 𝐿 by using (81).

= 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ 𝑧))

̃ 𝐿) of 𝐿 satisfies the Proposition 30. Every int-soft 𝐺-filter (𝑓, following conditions:

⊆ 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ 𝑧))) ,

(84)

for all 𝑥, 𝑦, 𝑧 ∈ 𝐿. It follows from (6) and (82) that

̃ 𝐿) is an int-soft 𝐺-filter of 𝐿 by for all 𝑥, 𝑦, 𝑧 ∈ 𝐿, and so (𝑓, Theorem 21.

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) = 𝑓̃ ((𝑥 󳨀→ 𝑦) 󳨀→ (𝑥 󳨀→ 𝑧))

Proposition 31. Assume that 𝐿 satisfies the divisibility; that is, ̃ 𝐿) is an int-soft 𝑥 ∧ 𝑦 = 𝑥 ⊙ (𝑥 → 𝑦), for all 𝑥, 𝑦 ∈ 𝐿. If (𝑓, 𝐺-filter of 𝐿 satisfying (82), then the following equality is true:

̃ 𝐿) is an int-soft 𝐺-filter of 𝐿; (1) (𝑓,

𝑓̃ (𝑥 󳨀→ (𝑦 󳨀→ 𝑧)) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦)

for all 𝑥, 𝑦, 𝑧 ∈ 𝐿. Thus (81) holds. Since (𝑥 → 𝑦) → (𝑥 → 𝑧) ≤ 𝑥 → (𝑦 → 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝐿, it follows from (22) ̃ ̃ → (𝑦 → 𝑧)) and so that 𝑓((𝑥 → 𝑦) → (𝑥 → 𝑧)) ⊆ 𝑓(𝑥 that

⊆ 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑧)) = 𝑓̃ ((𝑥 ⊙ 𝑥) 󳨀→ 𝑧) = 𝑓̃ ((𝑥 ∧ 𝑥) 󳨀→ 𝑧) = 𝑓̃ (𝑥 󳨀→ 𝑧) (88) ̃ 𝐿) is an int-soft 𝐺-filter of 𝐿 for all 𝑥, 𝑦, 𝑧 ∈ 𝐿. Therefore (𝑓, by Theorem 21. ̃ 𝐿) and (𝑔, ̃ 𝐿) be Theorem 33 (extension property). Let (𝑓, ̃ ̃ ̃ ̃ int-soft filters of 𝐿 such that (𝑓, 𝐿) ⊆ (𝑔, 𝐿); that is, 𝑓(𝑥) ⊆ 𝑔(𝑥) ̃ ̃ ̃ for all 𝑥 ∈ 𝐿 and 𝑓(1) = 𝑔(1). If (𝑓, 𝐿) is an int-soft 𝐺-filter of ̃ 𝐿). 𝐿, then so is (𝑔,

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̃ 𝐿) is an int-soft 𝐺-filter of 𝐿. Using (6) Proof. Assume that (𝑓, and (3), we have 𝑥 󳨀→ (𝑥 󳨀→ ((𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ 𝑦)) = (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) = 1

Then (𝐿, ∨, ∧, ⊙, → , 0, 1) is a residuated lattice (see [29]). Let ̃ 𝐿) be a soft set over 𝑈 = [0, 1] defined by (𝑓, 𝑓̃ : 𝐿 󳨀→ P (𝑈) .

(89)

(𝑥, 1] 𝑥 󳨃󳨀→ { 0

for all 𝑥, 𝑦 ∈ 𝐿. Thus 𝑔̃ (𝑥 󳨀→ ((𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ 𝑦)) (90)

= 𝑓̃ (1) = 𝑔̃ (1)

̃ 𝐿) of 𝐿, the following Theorem 37. For an int-soft filter (𝑓, assertions are equivalent. ̃ 𝐿) is regular. (1) (𝑓, ̃ ̃ → 𝑥)). (2) (∀𝑥, 𝑦 ∈ 𝐿) (𝑓(¬𝑥 → ¬𝑦) ⊆ 𝑓(𝑦

by hypotheses and (57), and so 𝑔̃ (𝑥 󳨀→ ((𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ 𝑦)) = 𝑔̃ (1)

(91)

̃ 𝐿) is an int-soft filter of 𝐿, it for all 𝑥, 𝑦 ∈ 𝐿 by (24). Since (𝑔, follows from (25), (6), and (24) that 𝑔̃ (𝑥 󳨀→ 𝑦)

̃ 𝐿) is a regular int-soft filter of 𝐿 and Proof. Assume that (𝑓, let 𝑥, 𝑦 ∈ 𝐿. Using (7) and (10), we have

(92)

¬¬𝑥 󳨀→ 𝑥 ≤ (𝑦 󳨀→ ¬¬𝑥) 󳨀→ (𝑦 󳨀→ 𝑥) ≤ (¬𝑥 󳨀→ ¬𝑦) 󳨀→ (𝑦 󳨀→ 𝑥)

∩ 𝑔̃ (𝑥 󳨀→ ((𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ 𝑦)) 𝑓̃ (¬𝑥 󳨀→ ¬𝑦)

= 𝑔̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ̃ 𝐿) is an int-soft 𝐺-filter of 𝐿. for all 𝑥, 𝑦 ∈ 𝐿. Therefore (𝑔,

= 𝑓̃ (¬𝑥 󳨀→ ¬𝑦) ∩ 𝑓̃ (1)

5. Regular and 𝑀𝑉-Int-Soft Filters

= 𝑓̃ (¬𝑥 󳨀→ ¬𝑦) ∩ 𝑓̃ (¬¬𝑥 󳨀→ 𝑥)

Zhu and Xu [29] introduced the notion of a regular filter in a residuated lattice. Definition 34 (see [29]). A filter 𝐹 of 𝐿 is said to be regular if it satisfies the following condition: (¬¬𝑥 󳨀→ 𝑥 ∈ 𝐹) .

(93)

̃ 𝐿) of 𝐿 is said to be regular Definition 35. An int-soft filter (𝑓, if it satisfies (𝑓̃ (¬¬𝑥 󳨀→ 𝑥) = 𝑓̃ (1)) .

(94)

Example 36. Let 𝐿 := [0, 1] (unit interval). For any 𝑎, 𝑏 ∈ 𝐿, define 𝑎 ∨ 𝑏 = max {𝑎, 𝑏} ,

𝑎 ∧ 𝑏 = min {𝑎, 𝑏} ,

1 if 𝑎 ≤ 𝑏, 𝑎 󳨀→ 𝑏 = { (1 − 𝑎) ∨ 𝑏 otherwise, 0 if 𝑎 + 𝑏 ≤ 1, 𝑎⊙𝑏={ 𝑎 ∧ 𝑏 otherwise.

(98)

and so from (24), (94), and (25) that

= 𝑔̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) ∩ 𝑔̃ (1)

(∀𝑥 ∈ 𝐿)

(97)

It follows from (8) and (7) that

∩ 𝑔̃ ((𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ (𝑥 󳨀→ 𝑦)) = 𝑔̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦))

̃ ̃ (3) (∀𝑥, 𝑦 ∈ 𝐿) (𝑓(¬𝑥 → 𝑦) ⊆ 𝑓(¬𝑦 → 𝑥)).

¬𝑥 󳨀→ ¬𝑦 ≤ ¬¬𝑦 󳨀→ ¬¬𝑥 ≤ 𝑦 󳨀→ ¬¬𝑥.

⊇ 𝑔̃ (𝑥 󳨀→ (𝑥 󳨀→ 𝑦))

(∀𝑥 ∈ 𝐿)

(96)

̃ 𝐿) is a regular int-soft filter of 𝐿. Then (𝑓,

⊇ 𝑓̃ (𝑥 󳨀→ ((𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ 𝑦)) = 𝑓̃ (𝑥 󳨀→ (𝑥 󳨀→ ((𝑥 󳨀→ (𝑥 󳨀→ 𝑦)) 󳨀→ 𝑦)))

if 𝑥 ∈ [0.5, 1] , otherwise,

(99)

⊆ 𝑓̃ (¬𝑥 󳨀→ ¬𝑦) ∩ 𝑓̃ ((¬𝑥 󳨀→ ¬𝑦) 󳨀→ (𝑦 󳨀→ 𝑥)) ⊆ 𝑓̃ (𝑦 󳨀→ 𝑥) ; that is, the second condition holds. Since ¬𝑥 → 𝑦 ≤ ¬𝑦 → ¬¬𝑥, we have ¬¬𝑥 󳨀→ 𝑥 ≤ (¬𝑦 󳨀→ ¬¬𝑥) 󳨀→ (¬𝑦 󳨀→ 𝑥) ≤ (¬𝑥 󳨀→ 𝑦) 󳨀→ (¬𝑦 󳨀→ 𝑥)

(100)

by (8) and (7). It follows from (24), (94), and (25) that 𝑓̃ (¬𝑥 󳨀→ 𝑦) = 𝑓̃ (¬𝑥 󳨀→ 𝑦) ∩ 𝑓̃ (1)

(95)

= 𝑓̃ (¬𝑥 󳨀→ 𝑦) ∩ 𝑓̃ (¬¬𝑥 󳨀→ 𝑥) ⊆ 𝑓̃ (¬𝑥 󳨀→ 𝑦) ∩ 𝑓̃ ((¬𝑥 󳨀→ 𝑦) 󳨀→ (¬𝑦 󳨀→ 𝑥)) ⊆ 𝑓̃ (¬𝑦 󳨀→ 𝑥) .

(101)

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9

Hence the third condition holds. Next, suppose that the second condition is valid. Condition (10) together with the second condition induces 𝑓̃ (1) = 𝑓̃ (¬𝑥 󳨀→ ¬¬¬𝑥) ⊆ 𝑓̃ (¬¬𝑥 󳨀→ 𝑥)

(102)

̃ ̃ ̃ 𝐿) is for all 𝑥 ∈ 𝐿, and so 𝑓(¬¬𝑥 → 𝑥) = 𝑓(1). Hence (𝑓, regular. Finally, assume that the third condition is valid. Since ̃ = ¬𝑥 → ¬𝑥 = 1 for all 𝑥 ∈ 𝐿, it follows from (3) that 𝑓(1) ̃ ̃ ̃ ̃ 𝑓(¬𝑥 → ¬𝑥) ⊆ 𝑓(¬¬𝑥 → 𝑥), and so 𝑓(¬¬𝑥 → 𝑥) = 𝑓(1) ̃ by (24). Therefore (𝑓, 𝐿) is regular. ̃ 𝐿) over 𝑈 is a regular int-soft filter Theorem 38. A soft set (𝑓, of 𝐿 if and only if it satisfies condition (24) and (∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

̃ 𝐿) over 𝑈 is a regular int-soft filter Theorem 39. A soft set (𝑓, of 𝐿 if and only if it satisfies condition (24) and

Lemma 40 (see [29]). Let 𝐹 be a filter of 𝐿. Then the following assertions are equivalent: (1) 𝐹 is regular; (2) (∀𝑥, 𝑦 ∈ 𝐿) (¬𝑥 → 𝑦 ∈ 𝐹 ⇒ ¬𝑦 → 𝑥 ∈ 𝐹). ̃ 𝐿) over 𝑈 is a regular int-soft filter Theorem 41. A soft set (𝑓, of 𝐿 if and only if the set

𝑓̃ (𝑧) ∩ 𝑓̃ (𝑧 󳨀→ (¬𝑥 󳨀→ 𝑦)) ⊆ 𝑓̃ (¬𝑥 󳨀→ 𝑦) ⊆ 𝑓̃ (¬𝑦 󳨀→ 𝑥)

(104)

for all 𝑥, 𝑦, 𝑧 ∈ 𝐿. ̃ 𝐿) satisfies two conditions Conversely, suppose that (𝑓, (24) and (103). Let 𝑥, 𝑦 ∈ 𝐿. Since 𝑥 → 𝑦 = 𝑥 → (1 → 𝑦) = 𝑥 → (¬0 → 𝑦) and ¬¬𝑦 = 1 → ¬¬𝑦 = 1 → (¬𝑦 → 0), it follows from (3), (24), and (103) that 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ 𝑦)

⊆ 𝑓̃ (¬𝑦 󳨀→ 0) (105)

= 𝑓̃ (1) ∩ 𝑓̃ (1 󳨀→ (¬𝑦 󳨀→ 0))

is a regular filter of 𝐿 for all 𝜏 ∈ P(𝑈) with 𝑓̃𝜏 ≠ 0. ̃ 𝐿) is a regular int-soft filter of 𝐿. Let Proof. Assume that (𝑓, ̃ 𝐿) is an int-soft filter 𝜏 ∈ P(𝑈) be such that 𝑓̃𝜏 ≠ 0. Since (𝑓, ̃ of 𝐿, the set 𝑓𝜏 is a filter of 𝐿 by Theorem 14. Let 𝑥, 𝑦 ∈ 𝐿 be ̃ ̃ such that ¬𝑥 → 𝑦 ∈ 𝑓̃𝜏 . Then 𝜏 ⊆ 𝑓(¬𝑥 → 𝑦) ⊆ 𝑓(¬𝑦 → ̃ ̃ 𝑥) by Theorem 37, and so ¬𝑦 → 𝑥 ∈ 𝑓𝜏 . Hence (𝑓, 𝐿) is regular by Lemma 40. Conversely, suppose that 𝑓̃𝜏 is a regular filter of 𝐿 for all ̃ 𝐿) 𝜏 ∈ P(𝑈) with 𝑓̃𝜏 ≠ 0. Then 𝑓̃𝜏 is a filter of 𝐿, and thus (𝑓, is an int-soft filter of 𝐿 by Theorem 14. For any 𝑥, 𝑦 ∈ 𝐿, let ̃ 𝑓(¬𝑥 → 𝑦) = 𝛿. Then ¬𝑥 → 𝑦 ∈ 𝑓̃𝛿 which implies from ̃ Lemma 40 that ¬𝑦 → 𝑥 ∈ 𝑓̃𝛿 . Hence 𝑓(¬𝑥 → 𝑦) = 𝛿 ⊆ ̃ ̃ 𝑓(¬𝑦 → 𝑥), and so (𝑓, 𝐿) is regular by Theorem 37.

̃ 𝐿) be a soft set over 𝑈 defined by Proof. Let (𝑓, 𝑓̃ : 𝐿 󳨀→ P (𝑈) ,

⊆ 𝑓̃ (¬0 󳨀→ 𝑦) = 𝑓̃ (1 󳨀→ 𝑦) = 𝑓̃ (𝑦) . ̃ 𝐿) is an int-soft filter of 𝐿 by Theorem 9. If we Therefore (𝑓, take 𝑧 := 1 in (103) and use (3) and (24), then 𝑓̃ (¬𝑥 󳨀→ 𝑦) = 𝑓̃ (1 󳨀→ (¬𝑥 󳨀→ 𝑦)) = 𝑓̃ (1) ∩ 𝑓̃ (1 󳨀→ (¬𝑥 󳨀→ 𝑦))

(108)

Theorem 42. For any regular filter 𝐹 of 𝐿, there exist 𝜏 ∈ ̃ 𝐿) of 𝐿 such that P(𝑈) \ {0} and a regular int-soft filter (𝑓, ̃ 𝐹 = 𝑓𝜏 .

= 𝑓̃ (𝑥) ∩ 𝑓̃ (𝑥 󳨀→ (¬0 󳨀→ 𝑦))

= 𝑓̃ (¬¬𝑦)

𝑓̃𝜏 := {𝑥 ∈ 𝐿 | 𝜏 ⊆ 𝑓̃ (𝑥)}

(103)

̃ 𝐿) is a regular int-soft filter of 𝐿. Proof. Assume that (𝑓, Clearly condition (24) holds. Using (25) and Theorem 37(3), we get

(107)

⊆ 𝑓̃ (𝑦 󳨀→ 𝑥)) .

(𝑓̃ (𝑧) ∩ 𝑓̃ (𝑧 󳨀→ (¬𝑥 󳨀→ 𝑦)) ⊆ 𝑓̃ (¬𝑦 󳨀→ 𝑥)) .

(𝑓̃ (𝑧) ∩ 𝑓̃ (𝑧 󳨀→ (¬𝑥 󳨀→ ¬𝑦))

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

(106)

⊆ 𝑓̃ (¬𝑦 󳨀→ 𝑥) . ̃ 𝐿) is regular by Theorem 37. Hence (𝑓, By a similar way to the proof of Theorem 38, we have the following characterization of a regular int-soft filter.

𝜏 𝑥 󳨃󳨀→ { 0

if 𝑥 ∈ 𝐹, otherwise,

(109)

̃ ⊆ 𝜏 = 𝑓(1) ̃ where 𝜏 ∈ P(𝑈) \ {0}. Since 1 ∈ 𝐹, we have 𝑓(𝑥) for all 𝑥 ∈ 𝐿. Let 𝑥, 𝑦, 𝑧 ∈ 𝐿. If 𝑧 ∈ 𝐹 and 𝑧 → (¬𝑥 → 𝑦) ∈ 𝐹, then ¬𝑦 → 𝑥 ∈ 𝐹 by Proposition 4 and Lemma 40. Hence ̃ ∩ 𝑓(𝑧 ̃ → (¬𝑥 → 𝑦)) = 𝜏 = 𝑓(¬𝑦 ̃ 𝑓(𝑧) → 𝑥). Suppose that ̃ = 0 or 𝑓(¬𝑥 ̃ 𝑧 ∉ 𝐹 or 𝑧 → (¬𝑥 → 𝑦) ∉ 𝐹. Then 𝑓(𝑧) → ̃ ̃ ̃ 𝑦) = 0, and so 𝑓(𝑧)∩ 𝑓(𝑧 → (¬𝑥 → 𝑦)) = 0 ⊆ 𝑓(¬𝑦 → 𝑥). ̃ 𝐿) is a regular int-soft filter of Therefore, by Theorem 38, (𝑓, 𝐿. Obviously, 𝐹 = 𝑓̃𝜏 . Definition 43 (see [29]). A subset 𝐹 of 𝐿 is called an 𝑀𝑉-filter of 𝐿 if it is a filter of 𝐿 that satisfies (∀𝑥, 𝑦 ∈ 𝐿)

(𝑦 󳨀→ 𝑥 ∈ 𝐹 󳨐⇒ ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ 𝑥 ∈ 𝐹) .

(110)

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Lemma 44 (see [29]). A filter 𝐹 of 𝐿 is an 𝑀𝑉-filter of 𝐿 if and only if it satisfies the condition (∀𝑥, 𝑦 ∈ 𝐿)

(((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) ∈ 𝐹) .

(111)

̃ 𝐿) over 𝑈 is called an 𝑀𝑉-intDefinition 45. A soft set (𝑓, soft filter of 𝐿 if it is an int-soft filter of 𝐿 with the following additional condition: (∀𝑥, 𝑦 ∈ 𝐿)

(112)

̃ 𝐿) over 𝑈 is an 𝑀𝑉-int-soft filter Theorem 46. A soft set (𝑓, of 𝐿 if and only if it satisfies condition (24) and

= 𝑓̃ (𝑦 󳨀→ 𝑥) ∩ 𝑓̃ (1)

(113)

̃ 𝐿) is an 𝑀𝑉-int-soft filter of 𝐿. Using Proof. Assume that (𝑓, (25) and (112), we have 𝑓̃ (𝑧) ∩ 𝑓̃ (𝑧 󳨀→ (𝑦 󳨀→ 𝑥)) ⊆ 𝑓̃ (𝑦 󳨀→ 𝑥) ⊆ 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ 𝑥)

(114)

for all 𝑥, 𝑦 ∈ 𝐿. ̃ 𝐿) be a soft set over 𝑈 which satisfies Conversely, let (𝑓, two conditions (24) and (113). Taking 𝑦 := 1 in (113) and using ̃ 𝐿) is an int-soft filter of 𝐿 (3) induce condition (25). Hence (𝑓, by Theorem 9. If we take 𝑧 := 1 in (113) and use (3) and (24), ̃ 𝐿) satisfies condition (112). Therefore then we know that (𝑓, ̃ 𝐿) is an 𝑀𝑉-int-soft filter of 𝐿. (𝑓, ̃ 𝐿) be an int-soft filter of 𝐿. Then (𝑓, ̃ 𝐿) is Theorem 47. Let (𝑓, an 𝑀𝑉-int-soft filter of 𝐿 if and only if the following assertion is valid: (𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥)) = 𝑓̃ (1)) .

(115)

̃ 𝐿) is an 𝑀𝑉-int-soft filter of 𝐿. Then Proof. Assume that (𝑓, ̃ (𝑓, 𝐿) is an int-soft filter of 𝐿, and so 𝑓̃𝜏 is a filter of 𝐿 for all 𝜏 ∈ P(𝑈) with 𝑓̃𝜏 ≠ 0 by Theorem 14. In particular, 𝑓̃𝑓(1) is a ̃ ̃ filter of 𝐿. Let 𝑥, 𝑦 ∈ 𝐿 be such that 𝑦 → 𝑥 ∈ 𝑓 ̃ . Then 𝑓(1)

𝑓̃ (1) ⊆ 𝑓̃ (𝑦 󳨀→ 𝑥) ⊆ 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ 𝑥) , (116) ̃̃ is an and so ((𝑥 → 𝑦) → 𝑦) → 𝑥 ∈ 𝑓̃𝑓(1) ̃ . Therefore 𝑓𝑓(1) 𝑀𝑉-filter of 𝐿, and thus ((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥) ∈ 𝑓̃𝑓(1) ̃

∩ 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ ((𝑦 󳨀→ 𝑥) 󳨀→ 𝑥)) (118) = 𝑓̃ (𝑦 󳨀→ 𝑥) ∩ 𝑓̃ ((𝑦 󳨀→ 𝑥) 󳨀→ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ 𝑥)) ⊆ 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ 𝑥) .

(𝑓̃ (𝑧) ∩ 𝑓̃ (𝑧 󳨀→ (𝑦 󳨀→ 𝑥)) ⊆ 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ 𝑥)) .

(∀𝑥, 𝑦 ∈ 𝐿)

𝑓̃ (𝑦 󳨀→ 𝑥)

= 𝑓̃ (𝑦 󳨀→ 𝑥)

(𝑓̃ (𝑦 󳨀→ 𝑥) ⊆ 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ 𝑥)) .

(∀𝑥, 𝑦, 𝑧 ∈ 𝐿)

̃ ⊆ 𝑓(((𝑥 ̃ by Lemma 44. Hence 𝑓(1) → 𝑦) → 𝑦) → ((𝑦 → ̃ 𝑥) → 𝑥)), and this and (24) imply that 𝑓(((𝑥 → 𝑦) → ̃ 𝑦) → ((𝑦 → 𝑥) → 𝑥)) = 𝑓(1). ̃ 𝐿) be an int-soft filter of 𝐿 that satisfies Conversely, let (𝑓, condition (115). Using (24), (115), (6), and (25), we obtain

(117)

̃ 𝐿) is an 𝑀𝑉-int-soft filter of 𝐿. Therefore (𝑓, Theorem 48. Every 𝑀𝑉-int-soft filter is regular. ̃ 𝐿) be an 𝑀𝑉-int-soft filter of 𝐿. If we take 𝑦 := Proof. Let (𝑓, 0 in (112) and use (3), then 𝑓̃ (1) = 𝑓̃ (0 󳨀→ 𝑥) ⊆ 𝑓̃ (((𝑥 󳨀→ 0) 󳨀→ 0) 󳨀→ 𝑥) = 𝑓̃ (¬¬𝑥 󳨀→ 𝑥)

(119)

̃ ̃ by (22). Therefore (𝑓, ̃ 𝐿) is a and so 𝑓(¬¬𝑥 → 𝑥) = 𝑓(1) regular int-soft filter of 𝐿. The converse of Theorem 48 is not true in general as seen in the following example. Example 49. Let (𝐿, ∨, ∧, ⊙, → , 0, 1) be the residuated lattice which is given in Example 36. Let 𝐹 := (𝑐, 1] for any 𝑐 ∈ 𝐿. Note that if 𝑐 ∈ [0.5, 1] then 𝐹 is a regular filter of 𝐿. But, if 𝑐 ∈ (0.7, 1] then 𝐹 is not an 𝑀𝑉-filter of 𝐿 since 0.4 → 0.7 = 1 ∈ 𝐹, but ((0.7 → 0.4) → 0.4) → 0.7 = 0.7 ∉ 𝐹. Hence ̃ 𝐿) over 𝑈 which is given as follows, the soft set (𝑓, 𝑓̃ : 𝐿 󳨀→ P (𝑈) ,

𝑈 𝑥 󳨃󳨀→ { 0

if 𝑥 ∈ 𝐹, otherwise,

(120)

̃ is an int-soft filter of 𝐿 which is regular. But, since 𝑓(0.4 → ̃ ̃ 0.7) = 𝑓(1) = 𝑈 and 𝑓(((0.7 → 0.4) → 0.4) → 0.7) = ̃ ̃ 𝐿) is not an 𝑀𝑉-int-soft filter of 𝐿. 𝑓(0.7) = 0, therefore (𝑓, In a 𝐵𝐿-algebra, that is, a residuated lattice 𝐿 with the following two conditions: (∀𝑥, 𝑦 ∈ 𝐿) (∀𝑥, 𝑦 ∈ 𝐿)

(𝑥 ∧ 𝑦 = 𝑥 ⊙ (𝑥 󳨀→ 𝑦)) , ((𝑥 󳨀→ 𝑦) ∨ (𝑦 󳨀→ 𝑥) = 1) ,

(121)

the converse of Theorem 48 is true which is shown in the following theorem.

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Theorem 50. In a 𝐵𝐿-algebra 𝐿, the notion of an 𝑀𝑉-int-soft filter coincides with the notion of a regular int-soft filter. Proof. Based on Theorem 48, it is sufficient to show that every ̃ 𝐿) be a regular int-soft filter is an 𝑀𝑉-int-soft filter. Let (𝑓, ̃ regular int-soft filter of 𝐿 and let 𝑥, 𝑦 ∈ 𝐿. Then 𝑓(¬𝑥 → ̃ ¬𝑦) ⊆ 𝑓(𝑦 → 𝑥) by Theorem 37. Since 𝑦 → 𝑥 ≤ ¬𝑥 → ̃ → 𝑥) ⊆ 𝑓(¬𝑥 ̃ ¬𝑦, we have 𝑓(𝑦 → ¬𝑦) by (22). Hence 𝑓̃ (𝑦 󳨀→ 𝑥)

to be an int-soft 𝐺-filter, a regular int-soft filter, or an 𝑀𝑉-intsoft filter. We have established the extension property for an int-soft 𝐺-filter and have shown that the notion of an 𝑀𝑉int-soft filter coincides with the notion of a regular int-soft filter in 𝐵𝐿-algebras. Future research will focus on applying the notions/contents to other algebraic structures.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

= 𝑓̃ (¬𝑥 󳨀→ ¬𝑦) = 𝑓̃ (¬𝑥 󳨀→ (¬𝑥 󳨀→ ¬𝑦)) = 𝑓̃ (¬𝑥 󳨀→ (¬𝑦 ⊙ (¬𝑦 󳨀→ ¬𝑥)))

Acknowledgment

= 𝑓̃ (¬𝑥 󳨀→ (¬𝑦 ⊙ (𝑥 󳨀→ ¬¬𝑦)))

The authors are deeply grateful to the referees for their valuable comments and suggestions for improving the paper.

= 𝑓̃ (¬ (¬𝑦 ⊙ (𝑥 󳨀→ ¬¬𝑦)) 󳨀→ 𝑥)

References

= 𝑓̃ (((𝑥 󳨀→ ¬¬𝑦) 󳨀→ (¬𝑦 󳨀→ 0)) 󳨀→ 𝑥) = 𝑓̃ (((𝑥 󳨀→ ¬¬𝑦) 󳨀→ ¬¬𝑦) 󳨀→ 𝑥) ,

(122)

𝑓̃ (1) = 𝑓̃ (¬𝑦 󳨀→ ¬𝑦) = 𝑓̃ (¬¬𝑦 󳨀→ 𝑦) ⊆ 𝑓̃ ((𝑥 󳨀→ ¬¬𝑦) 󳨀→ (𝑥 󳨀→ 𝑦)) ⊆ 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ ¬¬𝑦) 󳨀→ ((𝑥 󳨀→ ¬¬𝑦) 󳨀→ ¬¬𝑦)) ⊆ 𝑓̃ ((((𝑥 󳨀→ ¬¬𝑦) 󳨀→ ¬¬𝑦) 󳨀→ 𝑥) 󳨀→ (((𝑥 󳨀→ 𝑦) 󳨀→ ¬¬𝑦) 󳨀→ 𝑥)) . It follows that 𝑓̃ (𝑦 󳨀→ 𝑥) = 𝑓̃ (𝑦 󳨀→ 𝑥) ∩ 𝑓̃ (1) ⊆ 𝑓̃ (((𝑥 󳨀→ ¬¬𝑦) 󳨀→ ¬¬𝑦) 󳨀→ 𝑥) ∩ 𝑓̃ ((((𝑥 󳨀→ ¬¬𝑦) 󳨀→ ¬¬𝑦) 󳨀→ 𝑥)

(123)

󳨀→ (((𝑥 󳨀→ 𝑦) 󳨀→ ¬¬𝑦) 󳨀→ 𝑥)) ⊆ 𝑓̃ ((((𝑥 󳨀→ 𝑦) 󳨀→ ¬¬𝑦) 󳨀→ 𝑥)) ⊆ 𝑓̃ (((𝑥 󳨀→ 𝑦) 󳨀→ 𝑦) 󳨀→ 𝑥) . ̃ 𝐿) is an 𝑀𝑉-int-soft filter of 𝐿. Therefore (𝑓,

6. Conclusions We have introduced the notions of int-soft filters, int-soft 𝐺-filters, regular int-soft filters, and 𝑀𝑉-int-soft filters in residuated lattices and have investigated their relations and properties. We have considered characterizations of int-soft filters, int-soft 𝐺-filters, regular int-soft filters, and 𝑀𝑉-intsoft filters. We have provided conditions for an int-soft filter

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