Implicative Int-Soft Filters of 𝑅₀-Algebras

Abstract

The notion of implicative int-soft filters is introduced, and related properties are investigated. A relation between an int-soft filter and an implicative int-soft filter is discussed, and conditions for an int-soft filter to be an implicative int-soft filter are provided. Characterizations of an implicative int-soft filter are considered, and a new implicative int-soft filter from an old one is displayed. The extension property of an implicative int-soft filter is established.

1. Introduction

To solve complicated problems in economics, engineering, and the 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]. The filter theory in 𝑅₀-algebras is discussed in [7]. In [8], Jun et al. applied the notion of intersection-soft sets to the filter theory in 𝑅₀-algebras. They introduced the concept of strong int-soft filters in 𝑅₀-algebras and investigated related properties. They established characterizations of a strong int-soft filter and provided a condition for an int-soft filter to be strong. They also constructed an extension property of a strong int-soft filter.

In this paper, we introduce a new notion which is called an implicative int-soft filter and investigate related properties. We discuss a relation between an int-soft filter and an implicative int-soft filter. We provide conditions for an int-soft filter to be an implicative int-soft filter. We consider characterizations of an implicative int-soft filter and construct a new implicative int-soft filter from an old one. We establish the extension property of an implicative int-soft filter.

2. Preliminaries

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 and 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 [9] 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 the following:(i)1∈𝐹, (ii)(∀𝑥∈𝐹)(∀𝑦∈𝐿)(𝑥→𝑦∈𝐹⇒𝑦∈𝐹).

Definition 4 (see [10]). A subset 𝐹 of 𝐿 is called an implicative filter of 𝐿 if it satisfies the following:(i)1∈𝐹, (ii)(∀𝑥,𝑦,𝑧∈𝐿)(𝑥→(𝑦→𝑧)∈𝐹,𝑥→𝑦∈𝐹⇒𝑥→𝑧∈𝐹).

Note that every implicative filter is a filter. The following is a characterization of filters.

Soft set theory was introduced by Molodtsov [1] and Çağman and Enginğlu [11].

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)) denote 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, 11]). A soft set of 𝐸 over 𝑈 (a soft set of 𝐸 for short) is anyfunction𝑓_𝐴∶𝐸→𝒫(𝑈),suchthat𝑓_𝐴(𝑥)=∅if𝑥∉𝐴,for𝐴∈𝒫(𝐸), (16) or, equivalently, any set ℱ_𝐴∶=𝑥,𝑓_𝐴(𝑥)∣𝑥∈𝐸,𝑓_𝐴(𝑥)(𝑈),𝑓∈𝒫_𝐴(𝑥)=∅if𝑥∉𝐴 (17) for 𝐴∈𝒫(𝐸).

3. Implicative Int-Soft Filters

In what follows, 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 6 (see [8]). 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, (18) 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.

Lemma 7 (see [8]). 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: (∀𝑥∈𝐿)𝑓_𝐿(𝑥)⊆𝑓_𝐿(1),∀𝑥,𝑦∈𝐿𝑓_𝐿𝑥⟶𝑦∩𝑓_𝐿(𝑥)⊆𝑓_𝐿𝑦. (19)(20)

Definition 8. Let ℱ_𝐿∈𝑆(𝑈,𝐿). Then ℱ_𝐿 is called an implicative int-soft filter of 𝐿 if and only if it satisfies (19) and ∀𝑥,𝑦,𝑧∈𝐿𝑓_𝐿(𝑥⟶𝑧)⊇𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦. (21)

Example 9. Let 𝐿={0,𝑎,𝑏,𝑐,𝑑,1} be a set with the order <𝑎<𝑏<𝑐<𝑑<1, 0 and the following Cayley tables:

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,𝑑,𝛾₂,1,𝛾₂, (22) where 𝛾₁ and 𝛾₂ are subsets of 𝑈 with 𝛾₁⊊𝛾₂. Then ℱ_𝐿 is an implicative int-soft filter of 𝐿.

We provide a relation between an int-soft filter and an implicative int-soft filter.

Theorem 10. Every implicative int-soft filter is an int-soft filter.

Proof. Let ℱ_𝐿 be an implicative int-soft filter of 𝐿. If we take 𝑥=1 in (21), then 𝑓_𝐿(𝑧)=𝑓_𝐿(1⟶𝑧)⊇𝑓_𝐿1⟶𝑦⟶𝑧∩𝑓_𝐿1⟶𝑦=𝑓_𝐿𝑦⟶𝑧∩𝑓_𝐿𝑦 (23) for all 𝑦,𝑧∈𝐿. Therefore, ℱ_𝐿 is an int-soft filter of 𝐿.

The following example shows that the converse of Theorem 10 is not true in general.

Example 11. Let 𝐿={0,𝑎,𝑏,𝑐,1} be a set with the order 0<𝑎<𝑏<𝑐<1 and the following Cayley tables:

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, (24) where 𝛾₁ and 𝛾₂ are subsets of 𝑈 with 𝛾₁⊊𝛾₂. Then ℱ_𝐿 is an int-soft filter of 𝐿 (see [8]). But it is not an implicative int-soft filter of 𝐿 since 𝑓_𝐿(𝑏⟶𝑎)=𝛾₁⊋𝛾₂=𝑓_𝐿(𝑏⟶(𝑏⟶𝑎))∩𝑓_𝐿(𝑏⟶𝑏). (25)

We provide conditions for an int-soft filter to be an implicative int-soft filter.

Theorem 12. An int-soft filter ℱ_𝐿 of 𝐿 is implicative if and only if it satisfies 𝑓_𝐿(𝑥⟶𝑧)⊇𝑓_𝐿𝑥⟶¬𝑧⟶𝑦∩𝑓_𝐿𝑦⟶𝑧 (26) for all 𝑥,𝑦,𝑧∈𝐿.

Proof. Let ℱ_𝐿 be an int-soft filter of 𝐿 that satisfies condition (26). Using (R1), (R4), and (26), we have 𝑓_𝐿(𝑥⟶𝑧)=𝑓_𝐿(¬𝑧⟶¬𝑥)⊇𝑓_𝐿¬𝑧⟶¬¬𝑥⟶¬𝑦∩𝑓_𝐿¬𝑦⟶¬𝑥=𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦 (27) for all 𝑥,𝑦,𝑧∈𝐿. Thus, ℱ_𝐿 is an implicative int-soft filter of 𝐿.
Conversely, suppose that ℱ_𝐿 is an implicative int-soft filter of 𝐿. Then 𝑓_𝐿(𝑥⟶𝑧)=𝑓_𝐿(¬𝑧⟶¬𝑥)⊇𝑓_𝐿¬𝑧⟶¬𝑦⟶¬𝑥∩𝑓_𝐿¬𝑧⟶¬𝑦=𝑓_𝐿𝑥⟶¬𝑧⟶𝑦∩𝑓_𝐿𝑦⟶𝑧 (28) for all 𝑥,𝑦,𝑧∈𝐿 by (R1), (R4), and (21).

Lemma 13 (see [8]). Every int-soft filter ℱ_𝐿 is order preserving; that is, ∀𝑥,𝑦∈𝐿𝑥≤𝑦⟹𝑓_𝐿(𝑥)⊆𝑓_𝐿𝑦,∀𝑥,𝑦,𝑧∈𝐿𝑓_𝐿𝑥⟶¬𝑧⟶𝑦∩𝑓_𝐿𝑦⟶𝑧⊆𝑓_𝐿(𝑥⟶(¬𝑧⟶𝑧)),∀𝑥,𝑦∈𝐿𝑓_𝐿𝑥⊙𝑦=𝑓_𝐿(𝑥)∩𝑓_𝐿𝑦=𝑓_𝐿𝑥∧𝑦. (29)(30)(31)

Proposition 14. Every implicative int-soft filter ℱ_𝐿 of 𝐿 satisfies the following assertions: ∀𝑥,𝑦∈𝐿𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿𝑥⟶¬𝑦⟶𝑦,∀𝑥,𝑦,𝑧∈𝐿𝑓_𝐿𝑥⟶𝑦⊇𝑓_𝐿𝑧⟶𝑥⟶¬𝑦⟶𝑦∩𝑓_𝐿(𝑧)∀𝑥,𝑦∈𝐿𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿𝑥⟶𝑥⟶𝑦,∀𝑥,𝑦∈𝐿𝑓_𝐿𝑥⟶𝑦⊇𝑓_𝐿𝑧⟶𝑥⟶𝑥⟶𝑦∩𝑓_𝐿(𝑧),(∀𝑥∈𝐿)(∀𝑛∈ℕ)𝑓_𝐿(𝑛𝑥)=𝑓_𝐿(𝑥),(∀𝑥∈𝐿)𝑓_𝐿(𝑥∨¬𝑥)⊇𝑓_𝐿(1). (32)(33)(34)(35)(36)(37)

Proof. If we put 𝑧=𝑦 in (26), then 𝑓_𝐿𝑥⟶𝑦⊇𝑓_𝐿𝑥⟶¬𝑦⟶𝑦∩𝑓_𝐿𝑦⟶𝑦=𝑓_𝐿𝑥⟶¬𝑦⟶𝑦∩𝑓_𝐿(1)=𝑓_𝐿𝑥⟶¬𝑦⟶𝑦 (38) for all 𝑥,𝑦∈𝐿. Since 𝑥→𝑦≤¬𝑦→(𝑥→𝑦) for all 𝑥,𝑦∈𝐿, it follows from (29) and (R4) that 𝑓_𝐿𝑥⟶𝑦⊆𝑓_𝐿¬𝑦⟶𝑥⟶𝑦=𝑓_𝐿𝑥⟶¬𝑦⟶𝑦 (39) for all 𝑥,𝑦∈𝐿. Consequently, we get 𝑓_𝐿(𝑥→𝑦)=𝑓_𝐿(𝑥→(¬𝑦→𝑦)) for all 𝑥,𝑦∈𝐿. Equation (33) follows from (32) and (20).
If we put 𝑧=𝑦 and 𝑦=𝑥 in (21), then 𝑓_𝐿𝑥⟶𝑦⊇𝑓_𝐿𝑥⟶𝑥⟶𝑦∩𝑓_𝐿(𝑥⟶𝑥)=𝑓_𝐿𝑥⟶𝑥⟶𝑦∩𝑓_𝐿(1)=𝑓_𝐿𝑥⟶𝑥⟶𝑦 (40) for all 𝑥,𝑦∈𝐿. Since 𝑥→𝑦≤𝑥→(𝑥→𝑦) for all 𝑥,𝑦∈𝐿, (29) implies that 𝑓_𝐿𝑥⟶𝑦⊆𝑓_𝐿𝑥⟶𝑥⟶𝑦. (41)
Combining (40) and (41), we have 𝑓_𝐿(𝑥→𝑦)=𝑓_𝐿(𝑥→(𝑥→𝑦)) for all 𝑥,𝑦∈𝐿.
Using (20) and (34), we have 𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿𝑥⟶𝑥⟶𝑦⊇𝑓_𝐿𝑧⟶𝑥⟶𝑥⟶𝑦∩𝑓_𝐿(𝑧) (42) for all 𝑥,𝑦,𝑧∈𝐿. The proof of (36) is by induction on 𝑛. For 𝑛=2, if we use (34), then 𝑓_𝐿(2𝑥)=𝑓_𝐿(𝑥⊕𝑥)=𝑓_𝐿(¬𝑥⟶𝑥)=𝑓_𝐿(¬𝑥⟶(¬𝑥⟶0))=𝑓_𝐿(¬𝑥⟶0)=𝑓_𝐿(𝑥) (43)
for all 𝑥∈𝐿. Suppose (36) holds for 𝑛=𝑘; that is, 𝑓_𝐿(𝑘𝑥)=𝑓_𝐿(𝑥) for all 𝑥∈𝐿. It follows from (34) that 𝑓_𝐿((𝑘+1)𝑥)=𝑓_𝐿(𝑥⊕𝑘𝑥)=𝑓_𝐿(¬𝑥⟶𝑘𝑥)=𝑓_𝐿(¬𝑥⟶(𝑥⊕(𝑘−1)𝑥))=𝑓_𝐿(¬𝑥⟶(¬𝑥⟶(𝑘−1)𝑥))=𝑓_𝐿(¬𝑥⟶(𝑘−1)𝑥)=𝑓_𝐿(𝑥⊕(𝑘−1)𝑥)=𝑓_𝐿(𝑘𝑥)=𝑓_𝐿(𝑥) (44)
for all 𝑥∈𝐿. Therefore, (36) is valid. Note that ))⟶))))(1=((¬𝑥⟶𝑥⟶𝑥((¬𝑥⟶𝑥⟶𝑥=¬𝑥⟶(((¬𝑥⟶𝑥⟶𝑥⟶¬¬𝑥⟶𝑥)) (45) and ¬𝑥→((¬𝑥→𝑥)→𝑥)=1 for all 𝑥∈𝐿. It follows from (21) and (R1) that 𝑓_𝐿(1)=𝑓_𝐿(¬𝑥⟶(((¬𝑥⟶𝑥)⟶𝑥)⟶¬(¬𝑥⟶𝑥)))∩𝑓_𝐿(¬𝑥⟶((¬𝑥⟶𝑥)⟶𝑥))⊆𝑓_𝐿(¬𝑥⟶¬(¬𝑥⟶𝑥))=𝑓_𝐿((¬𝑥⟶𝑥)⟶𝑥) (46)
for all 𝑥∈𝐿. Similarly 𝑓_𝐿(1)⊆𝑓_𝐿((𝑥→¬𝑥)→¬𝑥) for all 𝑥∈𝐿. It follows from (31) that 𝑓_𝐿(1)⊆𝑓_𝐿((𝑥⟶¬𝑥)⟶¬𝑥)∩𝑓_𝐿((¬𝑥⟶𝑥)⟶𝑥)=𝑓_𝐿(((𝑥⟶¬𝑥)⟶¬𝑥)∧((¬𝑥⟶𝑥)⟶𝑥))=𝑓_𝐿(𝑥∨¬𝑥) (47) for all 𝑥∈𝐿.

Theorem 15. Let ℱ_𝐿 be an int-soft filter of 𝐿. If ℱ_𝐿 satisfies condition (33), then ℱ_𝐿 is an implicative int-soft filter of 𝐿.

Proof. Let ℱ_𝐿 be an int-soft filter of 𝐿 which satisfies condition (33). If we take 𝑧=1 and 𝑦=𝑧 in (33) and use (30), then 𝑓_𝐿(𝑥⟶𝑧)⊇𝑓_𝐿(1⟶(𝑥⟶(¬𝑧⟶𝑧)))∩𝑓_𝐿(1)=𝑓_𝐿(𝑥⟶(¬𝑧⟶𝑧))⊇𝑓_𝐿𝑥⟶¬𝑧⟶𝑦∩𝑓_𝐿𝑦⟶𝑧 (48) for all 𝑥,𝑦,𝑧∈𝐿. It follows from Theorem 12 that ℱ_𝐿 is an implicative int-soft filter of 𝐿.

Corollary 16. Let ℱ_𝐿 be an int-soft filter of 𝐿. If ℱ_𝐿 satisfies condition (32), then ℱ_𝐿 is an implicative int-soft filter of 𝐿.

Theorem 17. If an int-soft filter ℱ_𝐿 of 𝐿 satisfies condition (36), then it is implicative.

Proof. Since 𝑥→𝑦≤𝑥→(¬𝑦→𝑦) for all 𝑥,𝑦∈𝐿, it follows from (29) that 𝑓_𝐿𝑥⟶𝑦⊆𝑓_𝐿𝑥⟶¬𝑦⟶𝑦 (49) for all 𝑥,𝑦∈𝐿. Now condition (36) implies that 𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿2𝑥⟶𝑦=𝑓_𝐿¬𝑥⟶𝑦⟶𝑥⟶𝑦=𝑓_𝐿𝑥⟶¬𝑥⟶𝑦⟶𝑦⊇𝑓_𝐿𝑥⟶¬𝑦⟶𝑦 (50)
for all 𝑥,𝑦∈𝐿. Combining (49) and (50) induces 𝑓_𝐿(𝑥→𝑦)=𝑓_𝐿(𝑥→(¬𝑦→𝑦)) for all 𝑥,𝑦∈𝐿. Therefore, ℱ_𝐿 is an implicative int-soft filter of 𝐿 by Corollary 16.

Theorem 18. If an int-soft filter ℱ_𝐿 of 𝐿 satisfies condition (37), then it is implicative.

Proof. Using (19), (20), (37), and (R4), we have 𝑓_𝐿𝑥⟶𝑦⊇𝑓_𝐿𝑦∨¬𝑦⟶𝑥⟶𝑦∩𝑓_𝐿𝑦∨¬𝑦⊇𝑓_𝐿𝑦∨¬𝑦⟶𝑥⟶𝑦∩𝑓_𝐿(1)=𝑓_𝐿𝑦∨¬𝑦⟶𝑥⟶𝑦=𝑓_𝐿𝑦⟶𝑥⟶𝑦∩𝑓_𝐿¬𝑦⟶𝑥⟶𝑦=𝑓_𝐿𝑥⟶𝑦⟶𝑦∩𝑓_𝐿¬𝑦⟶𝑥⟶𝑦=𝑓_𝐿(𝑥⟶1)∩𝑓_𝐿¬𝑦⟶𝑥⟶𝑦=𝑓_𝐿¬𝑦⟶𝑥⟶𝑦 (51)for all 𝑥,𝑦∈𝐿. Since 𝑥→𝑦≤¬𝑦→(𝑥→𝑦) for all 𝑥,𝑦∈𝐿, it follows from (29) that 𝑓_𝐿(𝑥→𝑦)⊆𝑓_𝐿(¬𝑦→(𝑥→𝑦)) for all 𝑥,𝑦∈𝐿. Therefore, 𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿¬𝑦⟶𝑥⟶𝑦 (52)
for all 𝑥,𝑦∈𝐿, and so ℱ_𝐿 is an implicative int-soft filter of 𝐿 by Corollary 16.

Theorem 19. Let ℱ_𝐿∈𝑆(𝑈,𝐿) satisfy condition (19) and ∀𝑥,𝑦,𝑧∈𝐿𝑓_𝐿(𝑥)⊇𝑓_𝐿𝑧⟶𝑥⟶𝑦⟶𝑥∩𝑓_𝐿(𝑧),∀𝑥,𝑦,𝑧∈𝐿𝑓_𝐿(𝑥⟶𝑧)⊇𝑓_𝐿𝑥⟶𝑦∩𝑓_𝐿𝑦⟶𝑧. (53)(54) Then ℱ_𝐿 is an implicative int-soft filter of 𝐿.

Proof. Using (R4) and (6), we have ))⟶())))()((𝑥⟶𝑧⟶𝑧𝑥⟶𝑧=𝑥⟶(((𝑥⟶𝑧⟶𝑧⟶𝑧=𝑥⟶𝑥⟶𝑧 (55) for all 𝑥,𝑧∈𝐿. It follows from (R2), (R4), (19), (53), and (54) that 𝑓_𝐿(𝑥⟶𝑧)⊇𝑓_𝐿(1⟶(((𝑥⟶𝑧)⟶𝑧)⟶(𝑥⟶𝑧)))∩𝑓_𝐿(1)=𝑓_𝐿(((𝑥⟶𝑧)⟶𝑧)⟶(𝑥⟶𝑧))=𝑓_𝐿(𝑥⟶(𝑥⟶𝑧))⊇𝑓_𝐿𝑦⟶(𝑥⟶𝑧)∩𝑓_𝐿𝑥⟶𝑦=𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦 (56)
for all 𝑥,𝑦,𝑧∈𝐿. Thus, ℱ_𝐿 is an implicative int-soft filter of 𝐿.

Corollary 20. Every int-soft filter satisfying condition (53) is an implicative int-soft filter.

Proof. Let ℱ_𝐿 be an int-soft filter of 𝐿 that satisfies condition (53). Since ℱ_𝐿 satisfies two conditions (19) and (54) (see [8]), we know that ℱ_𝐿 is an implicative int-soft filter of 𝐿.

Theorem 21. A soft set ℱ_𝐿 of 𝐿 is an implicative int-soft filter of 𝐿 if and only if the nonempty 𝛾-inclusive set ℱ^𝛾_𝐿 is an implicative filter of 𝐿 for all 𝛾∈𝒫(𝑈).

The implicative filters ℱ^𝛾_𝐿 in Theorem 21 are called inclusive implicative filters of 𝐿.

Proof. Assume that ℱ_𝐿 is an implicative int-soft filter of 𝐿. Let 𝛾∈𝒫(𝑈) be such that ℱ^𝛾_𝐿≠∅. Then ℱ_𝐿 is an int-soft filter of 𝐿 (see Theorem 10), and so ℱ^𝛾_𝐿 is a filter of 𝐿. Let 𝑥,𝑦,𝑧∈𝐿 be such that 𝑥→(𝑦→𝑧)∈ℱ^𝛾_𝐿 and 𝑥→𝑦∈ℱ^𝛾_𝐿. Then 𝑓_𝐿(𝑥→(𝑦→𝑧))⊇𝛾 and 𝑓_𝐿(𝑥→𝑦)⊇𝛾. It follows from (21) that 𝑓_𝐿(𝑥⟶𝑧)⊇𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦⊇𝛾, (57) and so that 𝑥→𝑧∈ℱ^𝛾_𝐿. Therefore, ℱ^𝛾_𝐿 is an implicative filter of 𝐿 for all 𝛾∈𝒫(𝑈).
Conversely, suppose that the nonempty 𝛾-inclusive set ℱ^𝛾_𝐿 is an implicative filter of 𝐿 for all 𝛾∈𝒫(𝑈). Then ℱ^𝛾_𝐿 is a filter of 𝐿, and so ℱ_𝐿 is an int-soft filter of 𝐿. For every 𝑥,𝑦,𝑧∈𝐿, let 𝛾=𝑓_𝐿(𝑥→(𝑦→𝑧))∩𝑓_𝐿(𝑥→𝑦). Then 𝑥→(𝑦→𝑧)∈ℱ^𝛾_𝐿 and 𝑥→𝑦∈ℱ^𝛾_𝐿, which imply that 𝑥→𝑧∈ℱ^𝛾_𝐿. Hence, 𝑓_𝐿(𝑥⟶𝑧)⊇𝛾=𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦. (58) Thus, ℱ_𝐿 is an implicative int-soft filter of 𝐿.

Theorem 22 (extension property). Let ℱ_𝐿 and 𝒢_𝐿 be two int-soft filters of 𝐿 such that 𝑓_𝐿(1)=𝑔_𝐿(1) and 𝑓_𝐿(𝑥)⊆𝑔_𝐿(𝑥) for all 𝑥∈𝐿. If ℱ_𝐿 is implicative, then so is 𝒢_𝐿.

Proof. Assume that ℱ_𝐿 is implicative. Then 𝑓_𝐿(𝑥∨¬𝑥)⊇𝑓_𝐿(1) by (37). It follows from the hypothesis that 𝑔_𝐿(𝑥∨¬𝑥)⊇𝑓_𝐿(𝑥∨¬𝑥)⊇𝑓_𝐿(1)=𝑔_𝐿(1) (59) for all 𝑥∈𝐿. Therefore, 𝒢_𝐿 is an implicative int-soft filter of 𝐿 by Theorem 18.

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

Theorem 23. For any ℱ_𝐿∈𝑆(𝑈,𝐿), let ℱ^∗_𝐿 be a soft set of 𝐿 over 𝑈 defined by 𝑓^∗_𝐿(𝑈)𝑓∶𝐿⟶𝒫,𝑥↦^𝐿(𝑥),𝑖𝑓𝑥∈ℱ^𝛾_𝐿,𝛿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (60) where 𝛾 and 𝛿 are subsets of 𝑈 with 𝛿⊊𝑓_𝐿(𝑥). If ℱ_𝐿 is an implicative int-soft filter of 𝐿, then so is ℱ^∗_𝐿.

Proof. Assume that ℱ_𝐿 is an implicative int-soft filter of 𝐿. Then ℱ^𝛾_𝐿 is an implicative filter of 𝐿 for all 𝛾⊆𝑈 with ℱ^𝛾_𝐿≠∅. Hence, 1∈ℱ^𝛾_𝐿, and so 𝑓^∗_𝐿(1)=𝑓_𝐿(1)⊇𝑓_𝐿(𝑥)⊇𝑓^∗_𝐿(𝑥) (61) for all 𝑥∈𝐿. Let 𝑥,𝑦,𝑧∈𝐿. If 𝑥→(𝑦→𝑧)∈ℱ^𝛾_𝐿 and 𝑥→𝑦∈ℱ^𝛾_𝐿, then 𝑥→𝑧∈ℱ^𝛾_𝐿. Hence, 𝑓^∗_𝐿(𝑥⟶𝑧)=𝑓_𝐿(𝑥⟶𝑧)⊇𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦=𝑓^∗_𝐿𝑥⟶𝑦⟶𝑧∩𝑓^∗_𝐿𝑥⟶𝑦. (62)
If 𝑥→(𝑦→𝑧)∉ℱ^𝛾_𝐿 or 𝑥→𝑦∉ℱ^𝛾_𝐿, then 𝑓^∗_𝐿(𝑥→(𝑦→𝑧))=𝛿 or 𝑓^∗_𝐿(𝑥→𝑦)=𝛿. Thus, 𝑓^∗_𝐿(𝑥⟶𝑧)⊇𝛿=𝑓^∗_𝐿𝑥⟶𝑦⟶𝑧∩𝑓^∗_𝐿𝑥⟶𝑦. (63)
Therefore, ℱ^∗_𝐿 is an implicative int-soft filter of 𝐿.

Theorem 24. For any implicative filter 𝐹 of 𝐿, there exists an implicative int-soft filter of 𝐿 such that its inclusive implicative filter is 𝐹.

Proof. Let ℱ_𝐿 be a soft set of 𝐿 over 𝑈 in which 𝑓_𝐿 is given by 𝑓_𝐿∶𝐿⟶𝒫(𝑈),𝑥↦𝛾,if𝑥∈𝐹,∅,otherwise, (64) where 𝛾 is a nonempty subset of 𝑈. Since ∈𝐹, we have 𝑓_𝐿(1)=𝛾⊇𝑓_𝐿(𝑥) for all 𝑥∈𝐿. For every 𝑥,𝑦,𝑧∈𝐿, if 𝑥→(𝑦→𝑧)∈𝐹 and 𝑥→𝑦∈𝐹, then 𝑥→𝑧∈𝐹. Hence, 𝑓_𝐿(𝑥⟶𝑧)=𝛾=𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦. (65)
If 𝑥→(𝑦→𝑧)∉𝐹 or 𝑥→𝑦∉𝐹, then 𝑓_𝐿(𝑥→(𝑦→𝑧))=∅ or 𝑓_𝐿(𝑥→𝑦)=∅. Thus, 𝑓_𝐿(𝑥⟶𝑧)⊇∅=𝑓_𝐿𝑥⟶𝑦⟶𝑧∩𝑓_𝐿𝑥⟶𝑦. (66)
Therefore, ℱ_𝐿 is an implicative int-soft filter of 𝐿. Obviously, 𝐹=ℱ^𝛾_𝐿.

4. Conclusion

In [8], Jun et al. have applied the notion of intersection-soft sets to the filter theory in 𝑅₀-algebras. They have introduced the concept of strong int-soft filters in 𝑅₀-algebras and investigated related properties. They have established characterizations of a strong int-soft filter and provided a condition for an int-soft filter to be strong. They also have constructed an extension property of a strong int-soft filter.

In this paper, we have introduced a new notion which is called an implicative int-soft filter and investigated related properties. We have discussed a relation between an int-soft filter and an implicative int-soft filter. We have provided conditions for an int-soft filter to be an implicative int-soft filter. We have considered characterizations of an implicative int-soft filter and constructed a new implicative int-soft filter from an old one. We also have established the extension property of an implicative 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 algebraic structures with applications in soft set theory, and (3) to study the notions of the 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.