# Intersection-Soft Filters inR0-Algebras

Authors: Young Bae Jun; Sun Shin Ahn; Kyoung Ja Lee
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

The notions of (strong) intersection-soft filters inR0-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.


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## Body

## 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.R
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 R0-algebras is discussed in [7].In this paper, we apply the notion of intersection-soft sets to the filter theory inR0-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 onR0-Algebras

Definition 1 (see [5]).
LetL 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.
LetL 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 xn 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]).
LetL be an R0-algebra. Then the following properties hold:
(1)(∀x,y∈L)(x≤y⟺x⟶y=1),(2)(∀x,y∈L)(x≤y⟶x),(3)(∀x∈L)(¬x=x⟶0),(4)(∀x,y∈L)((x⟶y)∨(y⟶x)=1),(5)(∀x,y∈L)×(x≤y⇒y⟶z≤x⟶z,z⟶x≤z⟶y),(6)(∀x,y∈L)(((x⟶y)⟶y)⟶y=x⟶y),(7)(∀x,y∈L)×(x∨y=((x⟶y)⟶y)∧((y⟶x)⟶x)),(8)(∀x∈L)(x⊙¬x=0,x⊕¬x=1),(9)(∀x,y∈L)(x⊙y≤x∧y,x⊙(x⟶y)≤x∧y),(10)(∀x,y,z∈L)((x⊙y)⟶z=x⟶(y⟶z)),(11)(∀x,y∈L)(x≤y⟶(x⊙y)),(12)(∀x,y,z∈L)(x⊙y≤z⟺x≤y⟶z),(13)(∀x,y,z∈L)(x≤y⇒x⊙z≤y⊙z),(14)(∀x,y,z∈L)(x⟶y≤(y⟶z)⟶(x⟶z)),(15)(∀x,y,z∈L)((x⟶y)⊙(y⟶z)≤x⟶z).Definition 3 (see [7]).
A nonempty subsetF 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]).
LetF 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, letU 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 ofE over U (a soft set of E for short) is any function fA:E→𝒫(U),suchthatfA(x)=∅ifx∉A,forA∈𝒫(E), or, equivalently, any set
(16)ℱA:={(x,fA(x))∣x∈E,fA(x)∈𝒫(U),fA(x)=∅ifx∉A},
for A∈𝒫(E).Definition 6 (see [9]).
LetℱA and ℱB be soft sets of E. We say that ℱA is a soft subset of ℱB, denoted by ℱA⊆~ℱB, if fA(x)⊆fB(x) for all x∈E.


## 2.1. Basic Results onR0-Algebras

Definition 1 (see [5]).
LetL 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.
LetL 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 xn 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]).
LetL be an R0-algebra. Then the following properties hold:
(1)(∀x,y∈L)(x≤y⟺x⟶y=1),(2)(∀x,y∈L)(x≤y⟶x),(3)(∀x∈L)(¬x=x⟶0),(4)(∀x,y∈L)((x⟶y)∨(y⟶x)=1),(5)(∀x,y∈L)×(x≤y⇒y⟶z≤x⟶z,z⟶x≤z⟶y),(6)(∀x,y∈L)(((x⟶y)⟶y)⟶y=x⟶y),(7)(∀x,y∈L)×(x∨y=((x⟶y)⟶y)∧((y⟶x)⟶x)),(8)(∀x∈L)(x⊙¬x=0,x⊕¬x=1),(9)(∀x,y∈L)(x⊙y≤x∧y,x⊙(x⟶y)≤x∧y),(10)(∀x,y,z∈L)((x⊙y)⟶z=x⟶(y⟶z)),(11)(∀x,y∈L)(x≤y⟶(x⊙y)),(12)(∀x,y,z∈L)(x⊙y≤z⟺x≤y⟶z),(13)(∀x,y,z∈L)(x≤y⇒x⊙z≤y⊙z),(14)(∀x,y,z∈L)(x⟶y≤(y⟶z)⟶(x⟶z)),(15)(∀x,y,z∈L)((x⟶y)⊙(y⟶z)≤x⟶z).Definition 3 (see [7]).
A nonempty subsetF 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]).
LetF 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, letU 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 ofE over U (a soft set of E for short) is any function fA:E→𝒫(U),suchthatfA(x)=∅ifx∉A,forA∈𝒫(E), or, equivalently, any set
(16)ℱA:={(x,fA(x))∣x∈E,fA(x)∈𝒫(U),fA(x)=∅ifx∉A},
for A∈𝒫(E).Definition 6 (see [9]).
LetℱA and ℱB be soft sets of E. We say that ℱA is a soft subset of ℱB, denoted by ℱA⊆~ℱB, if fA(x)⊆fB(x) for all x∈E.


## 3. Intersection-Soft Filters

In what follows, we denote byS(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 setℱL∈S(U,L) is called an int-soft filter of L if it satisfies
(17)(∀γ∈𝒫(U))(ℱLγ≠∅⇒ℱLγisafilterofL),
where ℱLγ={x∈L∣γ⊆fL(x)} which is called the γ-inclusive set of ℱL.
IfℱL is an int-soft filter of L, every γ-inclusive set ℱLγ is called an inclusive filter of L.Example 8.
LetL={0,a,b,c,1} be a set with the order 0<a<b<c<1, and the following Cayley tables:
(18)x¬x01acbbca10⟶0abc1011111ac1111bbb111caab1110abc1
Then (L,∧,∨,¬,→) is an R0-algebra (see [10]) where x∧y=min{x,y} and x∨y=max{x,y}. Let ℱL∈S(U,L) be given as follows:
(19)ℱL={(0,γ1),(a,γ1),(b,γ1),(c,γ2),(1,γ2)},
where γ1 and γ2 are subsets of U with γ1⊊γ2. Then ℱL is an int-soft filter of L.We provide characterizations of an int-soft filter.Theorem 9.
LetℱL∈S(U,L). Then ℱL is an int-soft filter of L if and only if the following assertions are valid: (1)
(∀x∈L)(fL(x)⊆fL(1)),(2)
(∀x,y∈L)(fL(x→y)∩fL(x)⊆fL(y)).Proof.
Assume thatℱL is an int-soft filter of L. For any x∈L, let fL(x)=γ. Then x∈ℱLγ. Since ℱLγ is a filter of L, we have 1∈ℱLγ and so fL(x)=γ⊆fL(1). For any x,y∈L, let fL(x→y)∩fL(x)=γ. Then x→y∈ℱLγ and x∈ℱLγ. Since ℱLγ is a filter of L, it follows that y∈ℱLγ. Hence fL(x→y)∩fL(x)=γ⊆fL(y).
Conversely, suppose thatℱL satisfies two conditions (1) and (2). Let γ∈𝒫(U) such that ℱLγ≠∅. Then there exists a∈ℱLγ, and so γ⊆fL(a). It follows from (1) that γ⊆fL(a)⊆fL(1). Thus 1∈ℱLγ. Let x,y∈L such that x→y∈ℱLγ and x∈ℱLγ. Then γ⊆fL(x→y) and γ⊆fL(x). It follows from (2) that
(20)γ⊆fL(x⟶y)∩fL(x)⊆fL(y),
that is, y∈ℱLγ. Thus ℱLγ(≠∅) is a filter of L, and hence ℱL is an int-soft filter of L.Proposition 10.
LetℱL∈S(U,L) be an int-soft filter of L. Then the following properties are valid. (1)
ℱL is order preserving, that is,
(21)(∀x,y∈L)(x≤y⇒fL(x)⊆fL(y)).(2)
(∀x,y∈L)(fL(x→y)=fL(1)⇒fL(x)⊆fL(y)).(3)
(∀x,y∈L)(fL(x⊙y)=fL(x)∩fL(y)=fL(x∧y)).(4)
(∀x∈L)(∀n∈ℕ)(fL(xn)=fL(x)).(5)
(∀x∈L)(fL(0)=fL(x)∩fL(¬x)).(6)
(∀x,y∈L)(fL(x→y)∩fL(y→z)⊆fL(x→z)).(7)
(∀x,y,z∈L)(x⊙y≤z⇒fL(x)∩fL(y)⊆fL(z)).(8)
(∀x,y∈L)(fL(x)∩fL(x→y)=fL(y)∩fL(y→x)=fL(x)∩fL(y)).(9)
(∀x,y∈L)(fL(x⊙(x→y))=fL(y⊙(y→x))=fL(x)∩fL(y)).(10)
(∀x,y,z∈L)(fL(x→(¬z→y))∩fL(y→z)⊆fL(x→(¬z→z))).(11)
(∀x,y,z∈L)(fL(x→(y→z))∩fL(x→y)⊆fL(x→(x→z))).Proof.
(
1
) Let x,y∈L such that x≤y. Then x→y=1, and so
(22)fL(x)=fL(x)∩fL(1)=fL(x)∩fL(x⟶y)⊆fL(y)
by (1) and (2) of Theorem 9.
(
2
) Let x,y∈L such that fL(x→y)=fL(1). Then
(23)fL(x)=fL(x)∩fL(1)=fL(x)∩fL(x⟶y)⊆fL(y)
by (1) and (2) of Theorem 9.
(
3
) Since x⊙y≤x∧y for all x,y∈L, it follows from (1) that fL(x⊙y)⊆fL(x)∩fL(y). Using (11) and (1), we have fL(x)⊆fL(y→(x⊙y)). It follows from Theorem 9(2) that fL(x)∩fL(y)⊆fL(y→(x⊙y))∩fL(y)⊆fL(x⊙y). Therefore fL(x⊙y)=fL(x)∩fL(y). Since y≤x→y and x⊙(x→y)≤x∧y for all x,y∈L, we have fL(y)⊆fL(x→y) and fL(x)∩fL(y)⊆fL(x)∩fL(x→y)=fL(x⊙(x→y))⊆fL(x∧y)⊆fL(x)∩fL(y) by (1). Hence fL(x∧y)=fL(x)∩fL(y) for all x,y∈L.
(
4
) It follows from (3).
(
5
) Note that x⊙¬x= for all x∈L. Using (3), we have
(24)fL(0)=fL(x⊙¬x)=fL(x)∩fL(¬x)
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
(25)fL(x)∩fL(y)⊆fL(x)∩fL(x⟶y).
Since x⊙(x→y)≤x∧y for all x,y∈L, we have
(26)fL(x)∩fL(x⟶y)=fL(x⊙(x⟶y))⊆fL(x∧y)=fL(x)∩fL(y)
by (3) and (1). Hence fL(x)∩fL(y)=fL(x)∩fL(x→y). Similarly, fL(y)  ∩  fL(y→x)  =  fL(x)∩fL(y) for all x,y∈L.
(
9
) Using (3), we have
(27)fL(x⊙(x⟶y))=fL(x)∩fL(x⟶y),fL(y⊙(y⟶x))=fL(y)∩fL(y⟶x),
for all x,y∈L. It follows from (8) that fL(x⊙(x→y))=fL(y⊙(y→x))=fL(x)∩fL(y).
(
10
) Note that
(28)(x⟶(¬z⟶y))⊙(y⟶z)=((x⊙¬z)⟶y)⊙(y⟶z)≤(x⊙¬z)⟶z=x⟶(¬z⟶z)
for all x,y,z∈L. Using (1) and (3), we have
(29)fL(x⟶(¬z⟶y))∩fL(y⟶z)=fL((x⟶(¬z⟶y))⊙(y⟶z))⊆fL(x⟶(¬z⟶z))
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
(30)fL(x⟶(y⟶z))∩fL(x⟶y)⊆fL(x⟶(x⟶z))
for all x,y,z∈L.Theorem 11.
LetℱL∈S(U,L). Then ℱL is an int-soft filter of L if and only if the following assertions are valid: (1)
ℱL is order preserving,(2)
(∀x,y∈L)(fL(x⊙y)=fL(x)∩fL(y)).Proof.
The necessity follows from(1) and (3) of Proposition 10.
Conversely, suppose thatℱL satisfies two conditions (1) and (2). Let x,y∈L. Since x≤1, we have fL(x)⊆fL(1) by (1). Note that x⊙(x→y)≤y. It follows from (2) and (1) that
(31)fL(x)∩fL(x⟶y)=fL(x⊙(x⟶y))⊆fL(y).
Therefore ℱL is an int-soft filter of L.Proposition 12.
Letb∈L such that ¬b=b. If ℱL∈S(U,L) is an int-soft filter of L, then fL(b)=fL(x) for all x∈{a∈L∣0≤a≤b}.Proof.
Suppose that there existsy∈{a∈L∣0≤a≤b} such that fL(b)≠fL(y). Then fL(y)⊊fL(b), and so b∈ℱLγ and y∉ℱLγ where γ=fL(b). Since ℱLγ is a filter of L, we have 0=b⊙b∈ℱLγ. This shows that ℱLγ=L, and it is a contradiction. Hence fL(b)=fL(x) for all x∈{a∈L∣0≤a≤b}.Theorem 13.
LetℱL∈S(U,L). Then ℱL is an int-soft filter of L if and only if the following assertion is valid:
(32)(∀x,y,z∈L)×(fL(y)∩fL((x⟶y)⟶z)⊆fL(x⟶z)).Proof.
Assume thatℱL 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
(33)fL(y)∩fL((x⟶y)⟶z)⊆fL(x⟶y)∩fL((x⟶y)⟶z)⊆fL(z)⊆fL(x⟶z).
Conversely, suppose thatℱL satisfies the inclusion (32). Let x,y∈L. Then
(34)fL(x)=fL(x)∩fL((0⟶x)⟶x)⊆fL(0⟶x)=fL(1),fL(x)∩fL(x⟶y)=fL(x)∩fL((1⟶x)⟶y)⊆fL(1⟶y)=fL(y)
by (32). Therefore ℱL is an int-soft filter of L by Theorem 9.Theorem 14.
LetℱL∈S(U,L). Then ℱL is an int-soft filter of L if and only if the following assertion is valid:
(35)(∀x,y,z∈L)(x≤y⟶z⇒fL(x)∩fL(y)⊆fL(z)).Proof.
Suppose thatℱL is an int-soft filter of L. Let x,y,z∈L such that x≤y→z. Then fL(x)⊆fL(y→z) by Proposition 10(1), and so
(36)fL(x)∩fL(y)⊆fL(y⟶z)∩fL(y)⊆fL(z)
by Theorem 9 (2).
Conversely, assume thatℱL satisfies the condition (35). Let x,y∈L. Since x≤1=x→1, we have fL(x)⊆fL(1) by (35). Note that x→y≤x→y. It follows from (35) that fL(x→y)∩fL(x)⊆fL(y). Therefore ℱL is an int-soft filter of L by Theorem 9.Proposition 15.
Every int-soft filterℱL of L satisfies. 							(37)(∀x,y,z∈L)×(fL((x⟶y)⟶z)⊆fL(x⟶(y⟶z))).Proof.
Letx,y,z∈L. Since 1=y→(x→y)≤((x→y)→z)→(y→z), we have
(38)fL(1)⊆fL(((x⟶y)⟶z)⟶(y⟶z))
by Proposition 10  (1). It follows from (2) and Theorem 9 that
(39)fL((x⟶y)⟶z)=fL((x⟶y)⟶z)∩fL(1)⊆fL((x⟶y)⟶z)∩fL×(((x⟶y)⟶z)⟶(y⟶z))⊆fL(y⟶z)⊆fL(x⟶(y⟶z)).
This completes the proof.The following example shows that the converse of Proposition15 may not be true in general.Example 16.
LetL={0,a,b,c,d,1} be a set with the order 0<a<b<c<d<1, and the following Cayley tables:
(40)x¬x01adbccbda10⟶0abcd10111111ad11111bcc1111cbbb111daabc1110abcd1
Then (L,∧,∨,¬,→) is an R0-algebra (see [10]) where x∧y=min{x,y} and x∨y=max{x,y}. Let ℱL∈S(U,L) be given as follows:
(41)ℱL={(0,γ1),(a,γ2),(b,γ2),(c,γ2),(d,γ1),(1,γ1)},
where γ1 and γ2 are subsets of U with γ1⊊γ2. Then ℱL satisfies the condition (37), but ℱL is not an int-soft filter of L since fL(a)⊈fL(1).Proposition 17.
For an int-soft filterℱL of L, the following are equivalent:
(42)(∀x,y∈L)(fL(y⟶(y⟶x))⊆fL(y⟶x)),(43)(∀x,y,z∈L)(fL(z⟶(y⟶x))⊆fL((z⟶y)⟶(z⟶x))).Proof.
Assume that (42) is valid, and let x,y,z∈L. Using (R4), (5), and (14), we have
(44)z⟶(y⟶x)≤z⟶(z⟶((z⟶y)⟶x)).
It follows from Proposition 10 (1), (42), and (R4) that
(45)fL(z⟶(y⟶x))⊆fL(z⟶(z⟶((z⟶y)⟶x)))⊆fL(z⟶((z⟶y)⟶x))=fL((z⟶y)⟶(z⟶x)).
Conversely, suppose that (43) holds. If we use z instead of y in (43), then
(46)fL(z⟶(z⟶x))⊆fL((z⟶z)⟶(z⟶x))=fL(1⟶(z⟶x))=fL(z⟶x),
which proves (42).We make a new int-soft filter from old one.Theorem 18.
LetℱL∈S(U,L). For a subset γ of U, define a soft set ℱL* of L by
(47)fL*:L⟶𝒫(U),x↦{fL(x)ifx∈ℱLγ,∅otherwise.
If ℱL is an int-soft filter of L, then so is ℱL*.Proof.
Assume thatℱL is an int-soft filter of L. Then ℱLγ(≠∅) is a filter of L for all γ∈𝒫(U). Hence 1∈ℱLγ, and so fL*(1)=fL(1)⊇fL(x)⊇fL*(x) for all x∈L. Let x,y∈L. If x∈ℱLγ and x→y∈ℱLγ, then y∈ℱLγ. Hence
(48)fL*(x)∩fL*(x⟶y)=fL(x)∩fL(x⟶y)⊆fL(y)=fL*(y).
If x∉ℱLγ or x→y∉ℱLγ, then fL*(x)=∅ or fL*(x→y)=∅. Thus
(49)fL*(x)∩fL*(x⟶y)=∅⊆fL*(y).
Therefore ℱL* is an int-soft filter of L.Theorem 19.
Any filter ofL can be realized as an inclusive filter of some int-soft filter of L.Proof.
LetF be a filter of L. For a nonempty subset γ of U, let ℱL be a soft set of L defined by
(50)fL:L⟶𝒫(U),x↦{γifx∈F,∅otherwise.
Obviously fL(x)⊆fL(1) for all x∈L. For any x,y∈L, if x∈F and x→y∈F, then y∈F. Hence fL(x)∩fL(x→y)=γ=fL(y). If x∉F or x→y∉F, then fL(x)=∅ or fL(x→y)=∅. Thus fL(x)∩fL(x→y)=∅⊆fL(y). Therefore ℱL is an int-soft filter of L and clearly ℱLγ=F. This completes the proof.Definition 20.
An int-soft filterℱL of L is said to be strong if the following assertion is valid:
(51)(∀x,y∈L)(fL(y⟶x)⊆fL(((x⟶y)⟶y)⟶x)).Example 21.
The int-soft filterℱL in Example 8 is strong.Theorem 22.
LetℱL∈S(U,L). Then ℱL is a strong int-soft filter of L if and only if the following assertions are valid: (1)
(∀x∈L)(fL(x)⊆fL(1)),(2)
(∀x,y,z∈L)(fL(z→(y→x))∩fL(z)⊆fL(((x→y)→y)→x)).Proof.
Suppose thatℱL is a strong int-soft filter of L. Obviously, (1) is valid. For every x,y,z∈L, we have
(52)fL(z⟶(y⟶x))∩fL(z)⊆fL(y⟶x)⊆fL(((x⟶y)⟶y)⟶x)
by Theorem 9 (2) and (51).
Conversely, assume thatℱL satisfies two conditions (1) and (2). If we take y=1 in (2), then fL(z)∩fL(z→x)⊆fL(x) for all x,z∈L. Hence ℱL is an int-soft filter of L. Now if we put z=1 in (2), then
(53)fL(y⟶x)=fL(1⟶(y⟶x))=fL(1⟶(y⟶x))∩fL(1)⊆fL(((x⟶y)⟶y)⟶x)
by (R2) and (1). Therefore ℱL is a strong int-soft filter of L.Example 23.
LetL=[0,1]. For any a,b∈L, we define
(54)¬a=1-a,a∧b=min{a,b},a∨b=max{a,b}a⟶b={1a≤b¬a∨botherwise.
Then (L,∧,∨,¬,→) is an R0-algebra (see [5]). Let ℱL∈S(U,L) be given as follows:
(55)ℱL={(1,γ2),(x,γ1)∣x∈L∖{1}},
where γ1 and γ2 are subsets of U with γ1⊊γ2. Then ℱL is an int-soft filter of L. But
(56)fL(1⟶(0.3⟶0.8))∩fL(1)=γ2⊈γ1=fL(((0.8⟶0.3)⟶0.3)⟶0.8),
and so ℱL 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.
LetL be an R0-algebra satisfying the following inequality:
(57)(∀x,y∈L)((x⟶y)⟶y≤(y⟶x)⟶x).
Then every int-soft filter of L is strong.Proof.
LetℱL be an int-soft filter of L. Using (5), (6), and (57), we have
(58)y⟶x=((y⟶x)⟶x)⟶x≤((x⟶y)⟶y)⟶x
for all x,y∈L. It follows from Proposition 10 (1) that
(59)fL(y⟶x)⊆fL(((x⟶y)⟶y)⟶x)
for all x,y∈L. Therefore ℱL is a strong int-soft filter of L.We consider an extension property of a strong int-soft filter.Theorem 25.
LetℱL and 𝒢L be two int-soft filters of L such that ℱL⊆~𝒢L and fL(1)=gL(1). If ℱL is strong, then so is 𝒢L.Proof.
Assume thatℱL is a strong int-soft filter of L. For any x,y∈L, let a=y→x. Since ℱL is a strong int-soft filter of L, we have
(60)gL(1)=fL(1)=fL(y⟶(a⟶x))⊆fL((((a⟶x)⟶y)⟶y)⟶(a⟶x))⊆gL((((a⟶x)⟶y)⟶y)⟶(a⟶x)),
by (51) and assumption, and so
(61)gL(1)=gL((((a⟶x)⟶y)⟶y)⟶(a⟶x))=gL(a⟶((((a⟶x)⟶y)⟶y)⟶x)).
Since 𝒢L is an int-soft filter of L, it follows that
(62)gL(a)=gL(a)∩gL(1)=gL(a)∩gL×(a⟶((((a⟶x)⟶y)⟶y)⟶x))⊆gL((((a⟶x)⟶y)⟶y)⟶x).
Using (R4) and (14), we have
(63)1=x⟶(a⟶x)≤((a⟶x)⟶y)⟶(x⟶y)≤((x⟶y)⟶y)⟶(((a⟶x)⟶y)⟶y)≤((((a⟶x)⟶y)⟶y)⟶x)⟶(((x⟶y)⟶y)⟶x).
It follows from (62) and Theorem 14 that
(64)gL(y⟶x)=gL(a)⊆gL((((a⟶x)⟶y)⟶y)⟶x)=gL(1)∩gL((((a⟶x)⟶y)⟶y)⟶x)⊆gL(((x⟶y)⟶y)⟶x).
Therefore 𝒢L 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 inR0-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.



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