Intersection-Soft Filters in 𝑅₀-Algebras

Abstract

The notions of (strong) intersection-soft filters in 𝑅₀-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.

𝑅₀-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 𝑅₀-algebras is discussed in [7].

In this paper, we apply the notion of intersection-soft sets to the filter theory in 𝑅₀-algebras. We introduced the concept of (strong) intersection-soft filters in 𝑅₀-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 𝑅₀-Algebras

Definition 1 (see [5]). Let 𝐿 be a bounded distributive lattice with order-reversing involution ¬ and a binary operation →. Then (𝐿,∧,∨,¬,→) is called an 𝑅₀-algebra if it satisfies the following axioms: (R1) 𝑥→𝑦=¬𝑦→¬𝑥, (R2) 1→𝑥=𝑥, (R3) (𝑦→𝑧)∧((𝑥→𝑦)→(𝑥→𝑧))=𝑦→𝑧, (R4) 𝑥→(𝑦→𝑧)=𝑦→(𝑥→𝑧), (R5) 𝑥→(𝑦∨𝑧)=(𝑥→𝑦)∨(𝑥→𝑧), (R6) (𝑥→𝑦)∨((𝑥→𝑦)→(¬𝑥∨𝑦))=1.
Let 𝐿 be an 𝑅₀-algebra. For any 𝑥,𝑦∈𝐿, we define 𝑥⊙𝑦=¬(𝑥→¬𝑦) and 𝑥⊕𝑦=¬𝑥→𝑦. It is proven that ⊙ and ⊕ are commutative, associative, and 𝑥⊕𝑦=¬(¬𝑥⊙¬𝑦), and (𝐿,∧,∨,⊙,→,0,1) is a residuated lattice. In the following, let 𝑥^𝑛 denote 𝑥⊙𝑥⊙⋯⊙𝑥 where 𝑥 appears 𝑛 times for 𝑛∈ℕ.
We refer the reader to the book [8] for further information regarding 𝑅₀-algebras.

Lemma 2 (see [7]). Let 𝐿 be an 𝑅₀-algebra. Then the following properties hold: ∀𝑥,𝑦∈𝐿𝑥≤𝑦⟺𝑥⟶𝑦=1,∀𝑥,𝑦∈𝐿𝑥≤𝑦⟶𝑥,(∀𝑥∈𝐿)(¬𝑥=𝑥⟶0),∀𝑥,𝑦∈𝐿𝑥⟶𝑦∨𝑦⟶𝑥=1,×,∀𝑥,𝑦∈𝐿𝑥≤𝑦⇒𝑦⟶𝑧≤𝑥⟶𝑧,𝑧⟶𝑥≤𝑧⟶𝑦∀𝑥,𝑦∈𝐿𝑥⟶𝑦⟶𝑦⟶𝑦=𝑥⟶𝑦,×∧,∀𝑥,𝑦∈𝐿𝑥∨𝑦=𝑥⟶𝑦⟶𝑦𝑦⟶𝑥⟶𝑥(∀𝑥∈𝐿)(𝑥⊙¬𝑥=0,𝑥⊕¬𝑥=1),∀𝑥,𝑦∈𝐿𝑥⊙𝑦≤𝑥∧𝑦,𝑥⊙x⟶𝑦≤𝑥∧𝑦,∀𝑥,𝑦,𝑧∈𝐿𝑥⊙𝑦⟶𝑧=𝑥⟶𝑦⟶𝑧,∀𝑥,𝑦∈𝐿𝑥≤𝑦⟶𝑥⊙𝑦,∀𝑥,𝑦,𝑧∈𝐿𝑥⊙𝑦≤𝑧⟺𝑥≤𝑦⟶𝑧,∀𝑥,𝑦,𝑧∈𝐿𝑥≤𝑦⇒𝑥⊙𝑧≤𝑦⊙𝑧,∀𝑥,𝑦,𝑧∈𝐿𝑥⟶𝑦≤𝑦⟶𝑧⟶(𝑥⟶𝑧),∀𝑥,𝑦,𝑧∈𝐿𝑥⟶𝑦⊙𝑦⟶𝑧≤𝑥⟶𝑧. (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)

Definition 3 (see [7]). A nonempty subset 𝐹 of 𝐿 is called a filter of 𝐿 if it satisfies (i)1∈𝐹, (ii)(∀𝑥∈𝐹)(∀𝑦∈𝐿)(𝑥→𝑦∈𝐹⇒𝑦∈𝐹).

Lemma 4 (see [7]). Let 𝐹 be a nonempty subset of 𝐿. Then 𝐹 is a filter of 𝐿 if and only if it satisfies (1)(∀𝑥∈𝐹)(∀𝑦∈𝐿)(𝑥≤𝑦⇒𝑦∈𝐹), (2)(∀𝑥,𝑦∈𝐹)(𝑥⊙𝑦∈𝐹).

2.2. Basic Results on Soft Set Theory

Soft set theory was introduced by Molodtsov [1] and Çağman and Enginoğlu [9].

In what follows, let 𝑈 be an initial universe set, and let 𝐸 be a set of parameters. We say that the pair (𝑈,𝐸) is a soft universe. Let 𝒫(𝑈) (resp., 𝒫(𝐸)) denotes the power set of 𝑈 (resp., 𝐸).

By analogy with fuzzy set theory, the notion of soft set is defined as follows.

Definition 5 (see [1, 9]). A soft set of 𝐸 over 𝑈 (a soft set of 𝐸 for short) is any function 𝑓_𝐴∶𝐸→𝒫(𝑈),suchthat𝑓_𝐴(𝑥)=∅if𝑥∉𝐴,for𝐴∈𝒫(𝐸), or, equivalently, any set ℱ_𝐴∶=𝑥,𝑓_𝐴(𝑥)∣𝑥∈𝐸,𝑓_𝐴(𝑥)(𝑈),𝑓∈𝒫_𝐴(𝑥)=∅if𝑥∉𝐴, (16) for 𝐴∈𝒫(𝐸).

Definition 6 (see [9]). Let ℱ_𝐴 and ℱ_𝐵 be soft sets of 𝐸. We say that ℱ_𝐴 is a soft subset of ℱ_𝐵, denoted by ℱ⊆_𝐴^∼ℱ_𝐵, if 𝑓_𝐴(𝑥)⊆𝑓_𝐵(𝑥) for all 𝑥∈𝐸.

3. Intersection-Soft Filters

In what follows, we denote by 𝑆(𝑈,𝐿) the set of all soft sets of 𝐿 over 𝑈 where 𝐿 is an 𝑅₀-algebra unless otherwise specified.

Definition 7. A soft set ℱ_𝐿∈𝑆(𝑈,𝐿) is called an int-soft filter of 𝐿 if it satisfies ∀𝛾∈𝒫(𝑈)ℱ^𝛾_𝐿≠∅⇒ℱ^𝛾_𝐿isafilterof𝐿, (17) where ℱ^𝛾_𝐿={𝑥∈𝐿∣𝛾⊆𝑓_𝐿(𝑥)} which is called the 𝛾-inclusive set of ℱ_𝐿.
If ℱ_𝐿 is an int-soft filter of 𝐿, every 𝛾-inclusive set ℱ^𝛾_𝐿 is called an inclusive filter of 𝐿.

Example 8. Let 𝐿={0,𝑎,𝑏,𝑐,1} be a set with the order 0<𝑎<𝑏<𝑐<1, and the following Cayley tables: 𝑥¬𝑥01𝑎𝑐𝑏𝑏𝑐𝑎10⟶0𝑎𝑏𝑐10𝑎11111𝑏𝑐1111𝑐𝑏𝑏1111𝑎𝑎𝑏110𝑎𝑏𝑐1 (18) Then (𝐿,∧,∨,¬,→) is an 𝑅₀-algebra (see [10]) where 𝑥∧𝑦=min{𝑥,𝑦} and 𝑥∨𝑦=max{𝑥,𝑦}. Let ℱ_𝐿∈𝑆(𝑈,𝐿) be given as follows: ℱ_𝐿=0,𝛾₁,𝑎,𝛾₁,𝑏,𝛾₁,𝑐,𝛾₂,1,𝛾₂, (19) where 𝛾₁ and 𝛾₂ are subsets of 𝑈 with 𝛾₁⊊𝛾₂. Then ℱ_𝐿 is an int-soft filter of 𝐿.

We provide characterizations of an int-soft filter.

Theorem 9. Let ℱ_𝐿∈𝑆(𝑈,𝐿). Then ℱ_𝐿 is an int-soft filter of 𝐿 if and only if the following assertions are valid: (1)(∀𝑥∈𝐿)(𝑓_𝐿(𝑥)⊆𝑓_𝐿(1)), (2)(∀𝑥,𝑦∈𝐿)(𝑓_𝐿(𝑥→𝑦)∩𝑓_𝐿(𝑥)⊆𝑓_𝐿(𝑦)).

Proof. Assume that ℱ_𝐿 is an int-soft filter of 𝐿. For any 𝑥∈𝐿, let 𝑓_𝐿(𝑥)=𝛾. Then 𝑥∈ℱ^𝛾_𝐿. Since ℱ^𝛾_𝐿 is a filter of 𝐿, we have 1∈ℱ^𝛾_𝐿 and so 𝑓_𝐿(𝑥)=𝛾⊆𝑓_𝐿(1). For any 𝑥,𝑦∈𝐿, let 𝑓_𝐿(𝑥→𝑦)∩𝑓_𝐿(𝑥)=𝛾. Then 𝑥→𝑦∈ℱ^𝛾_𝐿 and 𝑥∈ℱ^𝛾_𝐿. Since ℱ^𝛾_𝐿 is a filter of 𝐿, it follows that 𝑦∈ℱ^𝛾_𝐿. Hence 𝑓_𝐿(𝑥→𝑦)∩𝑓_𝐿(𝑥)=𝛾⊆𝑓_𝐿(𝑦).
Conversely, suppose that ℱ_𝐿 satisfies two conditions (1) and (2). Let 𝛾∈𝒫(𝑈) such that ℱ^𝛾_𝐿≠∅. Then there exists 𝑎∈ℱ^𝛾_𝐿, and so 𝛾⊆𝑓_𝐿(𝑎). It follows from (1) that 𝛾⊆𝑓_𝐿(𝑎)⊆𝑓_𝐿(1). Thus 1∈ℱ^𝛾_𝐿. Let 𝑥,𝑦∈𝐿 such that 𝑥→𝑦∈ℱ^𝛾_𝐿 and 𝑥∈ℱ^𝛾_𝐿. Then 𝛾⊆𝑓_𝐿(𝑥→𝑦) and 𝛾⊆𝑓_𝐿(𝑥). It follows from (2) that 𝛾⊆𝑓_𝐿𝑥⟶𝑦∩𝑓_𝐿(𝑥)⊆𝑓_𝐿𝑦, (20) that is, 𝑦∈ℱ^𝛾_𝐿. Thus ℱ^𝛾_𝐿(≠∅) is a filter of 𝐿, and hence ℱ_𝐿 is an int-soft filter of 𝐿.

Proposition 10. Let ℱ_𝐿∈𝑆(𝑈,𝐿) be an int-soft filter of 𝐿. Then the following properties are valid. (1)ℱ_𝐿 is order preserving, that is, ∀𝑥,𝑦∈𝐿𝑥≤𝑦⇒𝑓_𝐿(𝑥)⊆𝑓_𝐿𝑦. (21)(2)(∀𝑥,𝑦∈𝐿)(𝑓_𝐿(𝑥→𝑦)=𝑓_𝐿(1)⇒𝑓_𝐿(𝑥)⊆𝑓_𝐿(𝑦)). (3)(∀𝑥,𝑦∈𝐿)(f_𝐿(𝑥⊙𝑦)=𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦)=𝑓_𝐿(𝑥∧𝑦)). (4)(∀𝑥∈𝐿)(∀𝑛∈ℕ)(𝑓_𝐿(𝑥^𝑛)=𝑓_𝐿(𝑥)). (5)(∀𝑥∈𝐿)(𝑓_𝐿(0)=𝑓_𝐿(𝑥)∩𝑓_𝐿(¬𝑥)). (6)(∀𝑥,𝑦∈𝐿)(𝑓_𝐿(𝑥→𝑦)∩𝑓_𝐿(𝑦→𝑧)⊆𝑓_𝐿(𝑥→𝑧)). (7)(∀𝑥,𝑦,𝑧∈𝐿)(𝑥⊙𝑦≤𝑧⇒𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦)⊆𝑓_𝐿(𝑧)). (8)(∀𝑥,𝑦∈𝐿)(𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑥→𝑦)=𝑓_𝐿(𝑦)∩𝑓_𝐿(𝑦→𝑥)=𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦)). (9)(∀𝑥,𝑦∈𝐿)(𝑓_𝐿(𝑥⊙(𝑥→𝑦))=𝑓_𝐿(𝑦⊙(𝑦→𝑥))=𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦)). (10)(∀𝑥,𝑦,𝑧∈𝐿)(𝑓_𝐿(𝑥→(¬𝑧→𝑦))∩𝑓_𝐿(𝑦→𝑧)⊆𝑓_𝐿(𝑥→(¬𝑧→𝑧))). (11)(∀𝑥,𝑦,𝑧∈𝐿)(𝑓_𝐿(𝑥→(𝑦→𝑧))∩𝑓_𝐿(𝑥→𝑦)⊆𝑓_𝐿(𝑥→(𝑥→𝑧))).

Proof. (1) Let 𝑥,𝑦∈𝐿 such that 𝑥≤𝑦. Then 𝑥→𝑦=1, and so 𝑓_𝐿(𝑥)=𝑓_𝐿(𝑥)∩𝑓_𝐿(1)=𝑓_𝐿(𝑥)∩𝑓_𝐿𝑥⟶𝑦⊆𝑓_𝐿𝑦 (22) by (1) and (2) of Theorem 9.
(2) Let 𝑥,𝑦∈𝐿 such that 𝑓_𝐿(𝑥→𝑦)=𝑓_𝐿(1). Then 𝑓_𝐿(𝑥)=𝑓_𝐿(𝑥)∩𝑓_𝐿(1)=𝑓_𝐿(𝑥)∩𝑓_𝐿𝑥⟶𝑦⊆𝑓_𝐿𝑦 (23) by (1) and (2) of Theorem 9.
(3) Since 𝑥⊙𝑦≤𝑥∧𝑦 for all 𝑥,𝑦∈𝐿, it follows from (1) that 𝑓_𝐿(𝑥⊙𝑦)⊆𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦). Using (11) and (1), we have 𝑓_𝐿(𝑥)⊆𝑓_𝐿(𝑦→(𝑥⊙𝑦)). It follows from Theorem 9(2) that 𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦)⊆𝑓_𝐿(𝑦→(𝑥⊙𝑦))∩𝑓_𝐿(𝑦)⊆𝑓_𝐿(𝑥⊙𝑦). Therefore 𝑓_𝐿(𝑥⊙𝑦)=𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦). Since 𝑦≤𝑥→𝑦 and 𝑥⊙(𝑥→𝑦)≤𝑥∧𝑦 for all 𝑥,𝑦∈𝐿, we have 𝑓_𝐿(𝑦)⊆𝑓_𝐿(𝑥→𝑦) and 𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦)⊆𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑥→𝑦)=𝑓_𝐿(𝑥⊙(𝑥→𝑦))⊆𝑓_𝐿(𝑥∧𝑦)⊆𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦) by (1). Hence 𝑓_𝐿(𝑥∧𝑦)=𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦) for all 𝑥,𝑦∈𝐿.
(4) It follows from (3).
(5) Note that 𝑥⊙¬𝑥= for all 𝑥∈𝐿. Using (3), we have 𝑓_𝐿(0)=𝑓_𝐿(𝑥⊙¬𝑥)=𝑓_𝐿(𝑥)∩𝑓_𝐿(¬𝑥) (24) for all 𝑥,𝑦∈𝐿.
(6) Combining (15), (1), and (3), we have the desired result.
(7) It follows from (1) and (3).
(8) Since 𝑦≤𝑥→𝑦 for all 𝑥,𝑦∈𝐿, it follows from (1) that 𝑓_𝐿(𝑥)∩𝑓_𝐿𝑦⊆𝑓_𝐿(𝑥)∩𝑓_𝐿𝑥⟶𝑦. (25) Since 𝑥⊙(𝑥→𝑦)≤𝑥∧𝑦 for all 𝑥,𝑦∈𝐿, we have 𝑓_𝐿(𝑥)∩𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿𝑥⊙𝑥⟶𝑦⊆𝑓_𝐿𝑥∧𝑦=𝑓_𝐿(𝑥)∩𝑓_𝐿𝑦 (26) by (3) and (1). Hence 𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦)=𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑥→𝑦). Similarly, 𝑓_𝐿(𝑦)  ∩  𝑓_𝐿(𝑦→𝑥)  =  𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦) for all 𝑥,𝑦∈𝐿.
(9) Using (3), we have 𝑓_𝐿𝑥⊙𝑥⟶𝑦=𝑓_𝐿(𝑥)∩𝑓_𝐿,𝑓𝑥⟶𝑦_𝐿𝑦⊙𝑦⟶𝑥=𝑓_𝐿𝑦∩𝑓_𝐿𝑦⟶𝑥, (27) for all 𝑥,𝑦∈𝐿. It follows from (8) that 𝑓_𝐿(𝑥⊙(𝑥→𝑦))=𝑓_𝐿(𝑦⊙(𝑦→𝑥))=𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦).
(10) Note that ⊙=)⊙≤()()𝑥⟶¬𝑧⟶𝑦𝑦⟶𝑧(𝑥⊙¬𝑧⟶𝑦𝑦⟶𝑧𝑥⊙¬𝑧⟶𝑧=𝑥⟶¬𝑧⟶𝑧 (28) for all 𝑥,𝑦,𝑧∈𝐿. Using (1) and (3), we have 𝑓_𝐿𝑥⟶¬𝑧⟶𝑦∩𝑓_𝐿𝑦⟶𝑧=𝑓_𝐿𝑥⟶¬𝑧⟶𝑦⊙𝑦⟶𝑧⊆𝑓_𝐿(𝑥⟶(¬𝑧⟶𝑧)) (29) for all 𝑥,𝑦,𝑧∈𝐿.
(11) Note that (𝑥→(𝑦→𝑧))⊙(𝑥→𝑦)=(𝑦→(𝑥→𝑧))⊙(𝑥→𝑦)≤𝑥→(𝑥→𝑧) for all 𝑥,𝑦,𝑧∈𝐿. It follows from (1) and (3) that 𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦⊆𝑓_𝐿(𝑥⟶(𝑥⟶𝑧)) (30) for all 𝑥,𝑦,𝑧∈𝐿.

Theorem 11. Let ℱ_𝐿∈𝑆(𝑈,𝐿). Then ℱ_𝐿 is an int-soft filter of 𝐿 if and only if the following assertions are valid: (1)ℱ_𝐿 is order preserving, (2)(∀𝑥,𝑦∈𝐿)(𝑓_𝐿(𝑥⊙𝑦)=𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑦)).

Proof. The necessity follows from (1) and (3) of Proposition 10.
Conversely, suppose that ℱ_𝐿 satisfies two conditions (1) and (2). Let 𝑥,𝑦∈𝐿. Since 𝑥≤1, we have 𝑓_𝐿(𝑥)⊆𝑓_𝐿(1) by (1). Note that 𝑥⊙(𝑥→𝑦)≤𝑦. It follows from (2) and (1) that 𝑓_𝐿(𝑥)∩𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿𝑥⊙𝑥⟶𝑦⊆𝑓_𝐿𝑦. (31) Therefore ℱ_𝐿 is an int-soft filter of 𝐿.

Proposition 12. Let 𝑏∈𝐿 such that ¬𝑏=𝑏. If ℱ_𝐿∈𝑆(𝑈,𝐿) is an int-soft filter of 𝐿, then 𝑓_𝐿(𝑏)=𝑓_𝐿(𝑥) for all 𝑥∈{𝑎∈𝐿∣0≤𝑎≤𝑏}.

Proof. Suppose that there exists 𝑦∈{𝑎∈𝐿∣0≤𝑎≤𝑏} such that 𝑓_𝐿(𝑏)≠𝑓_𝐿(𝑦). Then 𝑓_𝐿(𝑦)⊊𝑓_𝐿(𝑏), and so 𝑏∈ℱ^𝛾_𝐿 and 𝑦∉ℱ^𝛾_𝐿 where 𝛾=𝑓_𝐿(𝑏). Since ℱ^𝛾_𝐿 is a filter of 𝐿, we have 0=𝑏⊙𝑏∈ℱ^𝛾_𝐿. This shows that ℱ^𝛾_𝐿=𝐿, and it is a contradiction. Hence 𝑓_𝐿(𝑏)=𝑓_𝐿(𝑥) for all 𝑥∈{𝑎∈𝐿∣0≤𝑎≤𝑏}.

Theorem 13. Let ℱ_𝐿∈𝑆(𝑈,𝐿). Then ℱ_𝐿 is an int-soft filter of 𝐿 if and only if the following assertion is valid: ×𝑓∀𝑥,𝑦,𝑧∈𝐿_𝐿𝑦∩𝑓_𝐿𝑥⟶𝑦⟶𝑧⊆𝑓_𝐿(𝑥⟶𝑧). (32)

Proof. Assume that ℱ_𝐿 is an int-soft filter of 𝐿, and let 𝑥,𝑦,𝑧∈𝐿. Since 𝑦≤𝑥→𝑦, it follows from Proposition 10(1) and Theorem 9(2) that 𝑓_𝐿𝑦∩𝑓_𝐿𝑥⟶𝑦⟶𝑧⊆𝑓_𝐿𝑥⟶𝑦∩𝑓_𝐿𝑥⟶𝑦⟶𝑧⊆𝑓_𝐿(𝑧)⊆𝑓_𝐿(𝑥⟶𝑧). (33)
Conversely, suppose that ℱ_𝐿 satisfies the inclusion (32). Let 𝑥,𝑦∈𝐿. Then 𝑓_𝐿(𝑥)=𝑓_𝐿(𝑥)∩𝑓_𝐿((0⟶𝑥)⟶𝑥)⊆𝑓_𝐿(0⟶𝑥)=𝑓_𝐿(1),𝑓_𝐿(𝑥)∩𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿(𝑥)∩𝑓_𝐿(1⟶𝑥)⟶𝑦⊆𝑓_𝐿1⟶𝑦=𝑓_𝐿𝑦 (34) by (32). Therefore ℱ_𝐿 is an int-soft filter of 𝐿 by Theorem 9.

Theorem 14. Let ℱ_𝐿∈𝑆(𝑈,𝐿). Then ℱ_𝐿 is an int-soft filter of 𝐿 if and only if the following assertion is valid: ∀𝑥,𝑦,𝑧∈𝐿𝑥≤𝑦⟶𝑧⇒𝑓_𝐿(𝑥)∩𝑓_𝐿𝑦⊆𝑓_𝐿(𝑧). (35)

Proof. Suppose that ℱ_𝐿 is an int-soft filter of 𝐿. Let 𝑥,𝑦,𝑧∈𝐿 such that 𝑥≤𝑦→𝑧. Then 𝑓_𝐿(𝑥)⊆𝑓_𝐿(𝑦→𝑧) by Proposition 10(1), and so 𝑓_𝐿(𝑥)∩𝑓_𝐿𝑦⊆𝑓_𝐿𝑦⟶𝑧∩𝑓_𝐿𝑦⊆𝑓_𝐿(𝑧) (36) by Theorem 9 (2).
Conversely, assume that ℱ_𝐿 satisfies the condition (35). Let 𝑥,𝑦∈𝐿. Since 𝑥≤1=𝑥→1, we have 𝑓_𝐿(𝑥)⊆𝑓_𝐿(1) by (35). Note that 𝑥→𝑦≤𝑥→𝑦. It follows from (35) that 𝑓_𝐿(𝑥→𝑦)∩𝑓_𝐿(𝑥)⊆𝑓_𝐿(𝑦). Therefore ℱ_𝐿 is an int-soft filter of 𝐿 by Theorem 9.

Proposition 15. Every int-soft filter ℱ_𝐿 of 𝐿 satisfies. ×𝑓∀𝑥,𝑦,𝑧∈𝐿_𝐿𝑥⟶𝑦⟶𝑧⊆𝑓_𝐿𝑥⟶𝑦⟶𝑧. (37)

Proof. Let 𝑥,𝑦,𝑧∈𝐿. Since 1=𝑦→(𝑥→𝑦)≤((𝑥→𝑦)→𝑧)→(𝑦→𝑧), we have 𝑓_𝐿(1)⊆𝑓_𝐿𝑥⟶𝑦⟶𝑧⟶𝑦⟶𝑧 (38) by Proposition 10  (1). It follows from (2) and Theorem 9 that 𝑓_𝐿𝑥⟶𝑦⟶𝑧=𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿(1)⊆𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿×𝑥⟶𝑦⟶𝑧⟶𝑦⟶𝑧⊆𝑓_𝐿𝑦⟶𝑧⊆𝑓_𝐿𝑥⟶𝑦⟶𝑧. (39) This completes the proof.

The following example shows that the converse of Proposition 15 may not be true in general.

Example 16. Let 𝐿={0,𝑎,𝑏,𝑐,𝑑,1} be a set with the order 0<𝑎<𝑏<𝑐<𝑑<1, and the following Cayley tables: 𝑥¬𝑥01𝑎𝑑𝑏𝑐𝑐𝑏𝑑𝑎10⟶0𝑎𝑏𝑐𝑑10𝑎111111𝑏𝑑11111𝑐𝑐𝑐1111𝑑𝑏𝑏𝑏1111𝑎𝑎𝑏𝑐110𝑎𝑏𝑐𝑑1 (40) Then (𝐿,∧,∨,¬,→) is an 𝑅₀-algebra (see [10]) where 𝑥∧𝑦=min{𝑥,𝑦} and 𝑥∨𝑦=max{𝑥,𝑦}. Let ℱ_𝐿∈𝑆(𝑈,𝐿) be given as follows: ℱ_𝐿=0,𝛾₁,𝑎,𝛾₂,𝑏,𝛾₂,𝑐,𝛾₂,𝑑,𝛾₁,1,𝛾₁, (41) where 𝛾₁ and 𝛾₂ are subsets of 𝑈 with 𝛾₁⊊𝛾₂. Then ℱ_𝐿 satisfies the condition (37), but ℱ_𝐿 is not an int-soft filter of 𝐿 since 𝑓_𝐿(𝑎)⊈𝑓_𝐿(1).

Proposition 17. For an int-soft filter ℱ_𝐿 of 𝐿, the following are equivalent: ∀𝑥,𝑦∈𝐿𝑓_𝐿𝑦⟶𝑦⟶𝑥⊆𝑓_𝐿𝑦⟶𝑥,∀𝑥,𝑦,𝑧∈𝐿𝑓_𝐿𝑧⟶𝑦⟶𝑥⊆𝑓_𝐿𝑧⟶𝑦⟶(𝑧⟶𝑥). (42)(43)

Proof. Assume that (42) is valid, and let 𝑥,𝑦,𝑧∈𝐿. Using (R4), (5), and (14), we have 𝑧⟶𝑦⟶𝑥≤𝑧⟶𝑧⟶𝑧⟶𝑦⟶𝑥. (44) It follows from Proposition 10 (1), (42), and (R4) that 𝑓_𝐿𝑧⟶𝑦⟶𝑥⊆𝑓_𝐿𝑧⟶𝑧⟶𝑧⟶𝑦⟶𝑥⊆𝑓_𝐿𝑧⟶𝑧⟶𝑦⟶𝑥=𝑓_𝐿𝑧⟶𝑦⟶(𝑧⟶𝑥). (45)
Conversely, suppose that (43) holds. If we use 𝑧 instead of 𝑦 in (43), then 𝑓_𝐿(𝑧⟶(𝑧⟶𝑥))⊆𝑓_𝐿((𝑧⟶𝑧)⟶(𝑧⟶𝑥))=𝑓_𝐿(1⟶(𝑧⟶𝑥))=𝑓_𝐿(𝑧⟶𝑥), (46) which proves (42).

We make a new int-soft filter from old one.

Theorem 18. Let ℱ_𝐿∈𝑆(𝑈,𝐿). For a subset 𝛾 of 𝑈, define a soft set ℱ^∗_𝐿 of 𝐿 by 𝑓^∗_𝐿(𝑈)𝑓∶𝐿⟶𝒫,𝑥↦^𝐿(𝑥)𝑖𝑓𝑥∈ℱ^𝛾_L,∅𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (47) If ℱ_𝐿 is an int-soft filter of 𝐿, then so is ℱ^∗_𝐿.

Proof. Assume that ℱ_𝐿 is an int-soft filter of 𝐿. Then ℱ^𝛾_𝐿(≠∅) is a filter of 𝐿 for all 𝛾∈𝒫(𝑈). Hence 1∈ℱ^𝛾_𝐿, and so 𝑓^∗_𝐿(1)=𝑓_𝐿(1)⊇𝑓_𝐿(𝑥)⊇𝑓^∗_𝐿(𝑥) for all 𝑥∈𝐿. Let 𝑥,𝑦∈𝐿. If 𝑥∈ℱ^𝛾_𝐿 and 𝑥→𝑦∈ℱ^𝛾_𝐿, then 𝑦∈ℱ^𝛾_𝐿. Hence 𝑓^∗_𝐿(𝑥)∩𝑓^∗_𝐿𝑥⟶𝑦=𝑓_𝐿(𝑥)∩𝑓_𝐿𝑥⟶𝑦⊆𝑓_𝐿𝑦=𝑓^∗_𝐿𝑦. (48) If 𝑥∉ℱ^𝛾_𝐿 or 𝑥→𝑦∉ℱ^𝛾_𝐿, then 𝑓^∗_𝐿(𝑥)=∅ or 𝑓^∗_𝐿(𝑥→𝑦)=∅. Thus 𝑓^∗_𝐿(𝑥)∩𝑓^∗_𝐿𝑥⟶𝑦=∅⊆𝑓^∗_𝐿𝑦. (49) Therefore ℱ^∗_𝐿 is an int-soft filter of 𝐿.

Theorem 19. Any filter of 𝐿 can be realized as an inclusive filter of some int-soft filter of 𝐿.

Proof. Let 𝐹 be a filter of 𝐿. For a nonempty subset 𝛾 of 𝑈, let ℱ_𝐿 be a soft set of 𝐿 defined by 𝑓_𝐿(𝑈)𝛾∶𝐿⟶𝒫,𝑥↦if𝑥∈F,∅otherwise. (50) Obviously 𝑓_𝐿(𝑥)⊆𝑓_𝐿(1) for all 𝑥∈𝐿. For any 𝑥,𝑦∈𝐿, if 𝑥∈𝐹 and 𝑥→𝑦∈𝐹, then 𝑦∈𝐹. Hence 𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑥→𝑦)=𝛾=𝑓_𝐿(𝑦). If 𝑥∉𝐹 or 𝑥→𝑦∉𝐹, then 𝑓_𝐿(𝑥)=∅ or 𝑓_𝐿(𝑥→𝑦)=∅. Thus 𝑓_𝐿(𝑥)∩𝑓_𝐿(𝑥→𝑦)=∅⊆𝑓_𝐿(𝑦). Therefore ℱ_𝐿 is an int-soft filter of 𝐿 and clearly ℱ^𝛾_𝐿=𝐹. This completes the proof.

Definition 20. An int-soft filter ℱ_𝐿 of 𝐿 is said to be strong if the following assertion is valid: ∀𝑥,𝑦∈𝐿𝑓_𝐿𝑦⟶𝑥⊆𝑓_𝐿𝑥⟶𝑦⟶𝑦⟶𝑥. (51)

Example 21. The int-soft filter ℱ_𝐿 in Example 8 is strong.

Theorem 22. Let ℱ_𝐿∈𝑆(𝑈,𝐿). Then ℱ_𝐿 is a strong int-soft filter of 𝐿 if and only if the following assertions are valid: (1)(∀𝑥∈𝐿)(𝑓_𝐿(𝑥)⊆𝑓_𝐿(1)), (2)(∀𝑥,𝑦,𝑧∈𝐿)(𝑓_𝐿(𝑧→(𝑦→𝑥))∩𝑓_𝐿(𝑧)⊆𝑓_𝐿(((𝑥→𝑦)→𝑦)→𝑥)).

Proof. Suppose that ℱ_𝐿 is a strong int-soft filter of 𝐿. Obviously, (1) is valid. For every 𝑥,𝑦,𝑧∈𝐿, we have 𝑓_𝐿𝑧⟶𝑦⟶𝑥∩𝑓_𝐿(𝑧)⊆𝑓_𝐿𝑦⟶𝑥⊆𝑓_𝐿𝑥⟶𝑦⟶𝑦⟶𝑥 (52) by Theorem 9 (2) and (51).
Conversely, assume that ℱ_𝐿 satisfies two conditions (1) and (2). If we take 𝑦=1 in (2), then 𝑓_𝐿(𝑧)∩𝑓_𝐿(𝑧→𝑥)⊆𝑓_𝐿(𝑥) for all 𝑥,𝑧∈𝐿. Hence ℱ_𝐿 is an int-soft filter of 𝐿. Now if we put 𝑧=1 in (2), then 𝑓_𝐿𝑦⟶𝑥=𝑓_𝐿1⟶𝑦⟶𝑥=𝑓_𝐿1⟶𝑦⟶𝑥∩𝑓_𝐿(1)⊆𝑓_𝐿𝑥⟶𝑦⟶𝑦⟶𝑥 (53) by (R2) and (1). Therefore ℱ_𝐿 is a strong int-soft filter of 𝐿.

Example 23. Let 𝐿=[0,1]. For any 𝑎,𝑏∈𝐿, we define ¬𝑎=1−𝑎,𝑎∧𝑏=min{𝑎,𝑏},𝑎∨𝑏=max{𝑎,𝑏}𝑎⟶𝑏=1𝑎≤𝑏¬𝑎∨𝑏otherwise. (54) Then (𝐿,∧,∨,¬,→) is an 𝑅₀-algebra (see [5]). Let ℱ_𝐿∈𝑆(𝑈,𝐿) be given as follows: ℱ_𝐿=1,𝛾₂,𝑥,𝛾₁∣𝑥∈𝐿⧵{1}, (55) where 𝛾₁ and 𝛾₂ are subsets of 𝑈 with 𝛾₁⊊𝛾₂. Then ℱ_𝐿 is an int-soft filter of 𝐿. But 𝑓_𝐿(1⟶(0.3⟶0.8))∩𝑓_𝐿(1)=𝛾₂⊆𝛾̸₁=𝑓_𝐿(((0.8⟶0.3)⟶0.3)⟶0.8), (56) and so ℱ_𝐿 is not a strong int-soft filter of 𝐿 by Theorem 22.
We provide a condition for an int-soft filter to be strong.

Theorem 24. Let 𝐿 be an 𝑅₀-algebra satisfying the following inequality: ∀𝑥,𝑦∈𝐿𝑥⟶𝑦⟶𝑦≤𝑦⟶𝑥⟶𝑥. (57) Then every int-soft filter of 𝐿 is strong.

Proof. Let ℱ_𝐿 be an int-soft filter of 𝐿. Using (5), (6), and (57), we have ≤𝑦⟶𝑥=𝑦⟶𝑥⟶𝑥⟶𝑥𝑥⟶𝑦⟶𝑦⟶𝑥 (58) for all 𝑥,𝑦∈𝐿. It follows from Proposition 10 (1) that 𝑓_𝐿𝑦⟶𝑥⊆𝑓_𝐿𝑥⟶𝑦⟶𝑦⟶𝑥 (59) for all 𝑥,𝑦∈𝐿. Therefore ℱ_𝐿 is a strong int-soft filter of 𝐿.

We consider an extension property of a strong int-soft filter.

Theorem 25. Let ℱ_𝐿 and 𝒢_𝐿 be two int-soft filters of 𝐿 such that ℱ⊆_𝐿^∼𝒢_𝐿 and 𝑓_𝐿(1)=𝑔_𝐿(1). If ℱ_𝐿 is strong, then so is 𝒢_𝐿.

Proof. Assume that ℱ_𝐿 is a strong int-soft filter of 𝐿. For any 𝑥,𝑦∈𝐿, let 𝑎=𝑦→𝑥. Since ℱ_𝐿 is a strong int-soft filter of 𝐿, we have 𝑔_𝐿(1)=𝑓_𝐿(1)=𝑓_𝐿𝑦⟶(𝑎⟶𝑥)⊆𝑓_𝐿(𝑎⟶𝑥)⟶𝑦⟶𝑦⟶(𝑎⟶𝑥)⊆𝑔_𝐿(𝑎⟶𝑥)⟶𝑦⟶𝑦⟶(𝑎⟶𝑥), (60) by (51) and assumption, and so 𝑔_𝐿(1)=𝑔_𝐿(𝑎⟶𝑥)⟶𝑦⟶𝑦⟶(𝑎⟶𝑥)=𝑔_𝐿𝑎⟶(𝑎⟶𝑥)⟶𝑦⟶𝑦⟶𝑥. (61) Since 𝒢_𝐿 is an int-soft filter of 𝐿, it follows that 𝑔_𝐿(𝑎)=𝑔_𝐿(𝑎)∩𝑔_𝐿(1)=𝑔_𝐿(𝑎)∩𝑔_𝐿×𝑎⟶(𝑎⟶𝑥)⟶𝑦⟶𝑦⟶𝑥⊆𝑔_𝐿(𝑎⟶𝑥)⟶𝑦⟶𝑦⟶𝑥. (62) Using (R4) and (14), we have ()≤()⟶≤⟶)≤()⟶.1=𝑥⟶𝑎⟶𝑥𝑎⟶𝑥⟶𝑦𝑥⟶𝑦𝑥⟶𝑦⟶𝑦(𝑎⟶𝑥⟶𝑦⟶𝑦𝑎⟶𝑥⟶𝑦⟶𝑦⟶𝑥𝑥⟶𝑦⟶𝑦⟶𝑥 (63) It follows from (62) and Theorem 14 that 𝑔_𝐿𝑦⟶𝑥=𝑔_𝐿(𝑎)⊆𝑔_𝐿(𝑎⟶𝑥)⟶𝑦⟶𝑦⟶𝑥=𝑔_𝐿(1)∩𝑔_𝐿(𝑎⟶𝑥)⟶𝑦⟶𝑦⟶𝑥⊆𝑔_𝐿𝑥⟶𝑦⟶𝑦⟶𝑥. (64) Therefore 𝒢_𝐿 is a strong int-soft filter of 𝐿.

4. Conclusion

Using the notion of int-soft sets, we have introduced the concept of (strong) int-soft filters in 𝑅₀-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.