# Implicative Int-Soft Filters ofR0-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 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.


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## Body

## 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.R
0-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]).
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 and 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 [9] 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 the following:(i)
1∈F,(ii)
(∀x∈F)(∀y∈L)(x→y∈F⇒y∈F).Definition 4 (see [10]).
A subsetF 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, letU be an initial universe set and let E be a set of parameters. We say that the pair (U,E) is asoft 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 ofE over U (a soft set of E for short) is
(16)anyfunctionfA:E→𝒫(U),suchthatfA(x)=∅ifx∉A,forA∈𝒫(E),
or, equivalently, any set
(17)ℱA:={(x,fA(x))∣x∈E,fA(x)∈𝒫(U),fA(x)=∅ifx∉A(x,fA(x))}
for A∈𝒫(E).


## 3. Implicative Int-Soft Filters

In what follows, denote byS(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 setℱL∈S(U,L) is called an int-soft filter of L if it satisfies
(18)(∀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.Lemma 7 (see [8]).
LetℱL∈S(U,L). Then ℱL is an int-soft filter of L if and only if the following assertions are valid:
(19)(∀x∈L)(fL(x)⊆fL(1)),(20)(∀x,y∈L)(fL(x⟶y)∩fL(x)⊆fL(y)).Definition 8.
LetℱL∈S(U,L). Then ℱL is called animplicative int-soft filter of L if and only if it satisfies (19) and
(21)(∀x,y,z∈L)(fL(x⟶z)⊇fL(x⟶(y⟶z))∩fL(x⟶y)).Example 9.
LetL={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 ℱL∈S(U,L) be given as follows:
(22)ℱL={(0,γ1),(a,γ1),(b,γ1),(c,γ2),(d,γ2),(1,γ2)},
where γ1 and γ2 are subsets of U with γ1⊊γ2. Then ℱL 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.
LetℱL be an implicative int-soft filter of L. If we take x=1 in (21), then
(23)fL(z)=fL(1⟶z)⊇fL(1⟶(y⟶z))∩fL(1⟶y)=fL(y⟶z)∩fL(y)
for all y,z∈L. Therefore, ℱL is an int-soft filter of L.The following example shows that the converse of Theorem10 is not true in general.Example 11.
LetL={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 ℱL∈S(U,L) be given as follows:
(24)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 (see [8]). But it is not an implicative int-soft filter of L since
(25)fL(b⟶a)=γ1⊋γ2=fL(b⟶(b⟶a))∩fL(b⟶b).We provide conditions for an int-soft filter to be an implicative int-soft filter.Theorem 12.
An int-soft filterℱL of L is implicative if and only if it satisfies
(26)fL(x⟶z)⊇fL(x⟶(¬z⟶y))∩fL(y⟶z)
for all x,y,z∈L.Proof.
LetℱL be an int-soft filter of L that satisfies condition (26). Using (R1), (R4), and (26), we have
(27)fL(x⟶z)=fL(¬z⟶¬x)⊇fL(¬z⟶(¬¬x⟶¬y))∩fL(¬y⟶¬x)=fL(x⟶(y⟶z))∩fL(x⟶y)
for all x,y,z∈L. Thus, ℱL is an implicative int-soft filter of L.
Conversely, suppose thatℱL is an implicative int-soft filter of L. Then
(28)fL(x⟶z)=fL(¬z⟶¬x)⊇fL(¬z⟶(¬y⟶¬x))∩fL(¬z⟶¬y)=fL(x⟶(¬z⟶y))∩fL(y⟶z)
for all x,y,z∈L by (R1), (R4), and (21).Lemma 13 (see [8]).
Every int-soft filterℱL is order preserving; that is,
(29)(∀x,y∈L)(x≤y⟹fL(x)⊆fL(y)),(30)(∀x,y,z∈L)(fL(x⟶(¬z⟶y))∩fL(y⟶z)⊆fL(x⟶(¬z⟶z))),(31)(∀x,y∈L)(fL(x⊙y)=fL(x)∩fL(y)=fL(x∧y)).Proposition 14.
Every implicative int-soft filterℱL of L satisfies the following assertions:
(32)(∀x,y∈L)(fL(x⟶y)=fL(x⟶(¬y⟶y))),(33)(∀x,y,z∈L)(fL(x⟶y)⊇fL(z⟶(x⟶(¬y⟶y)))∩fL(z))(34)(∀x,y∈L)(fL(x⟶y)=fL(x⟶(x⟶y))),(35)(∀x,y∈L)((z→(x→(x→y)))∩fL(z)fL(x⟶y)⊇fL(z⟶(x⟶(x⟶y)))∩fL(z)),(36)(∀x∈L)(∀n∈ℕ)(fL(nx)=fL(x)),(37)(∀x∈L)(fL(x∨¬x)⊇fL(1)).Proof.
If we putz=y in (26), then
(38)fL(x⟶y)⊇fL(x⟶(¬y⟶y))∩fL(y⟶y)=fL(x⟶(¬y⟶y))∩fL(1)=fL(x⟶(¬y⟶y))
for all x,y∈L. Since x→y≤¬y→(x→y) for all x,y∈L, it follows from (29) and (R4) that
(39)fL(x⟶y)⊆fL(¬y⟶(x⟶y))=fL(x⟶(¬y⟶y))
for all x,y∈L. Consequently, we get fL(x→y)=fL(x→(¬y→y)) for all x,y∈L. Equation (33) follows from (32) and (20).
If we putz=y and y=x in (21), then
(40)fL(x⟶y)⊇fL(x⟶(x⟶y))∩fL(x⟶x)=fL(x⟶(x⟶y))∩fL(1)=fL(x⟶(x⟶y))
for all x,y∈L. Since x→y≤x→(x→y) for all x,y∈L, (29) implies that
(41)fL(x⟶y)⊆fL(x⟶(x⟶y)).
Combining (40) and (41), we have fL(x→y)=fL(x→(x→y)) for all x,y∈L.
Using (20) and (34), we have
(42)fL(x⟶y)=fL(x⟶(x⟶y))⊇fL(z⟶(x⟶(x⟶y)))∩fL(z)
for all x,y,z∈L. The proof of (36) is by induction on n. For n=2, if we use (34), then
(43)fL(2x)=fL(x⊕x)=fL(¬x⟶x)=fL(¬x⟶(¬x⟶0))=fL(¬x⟶0)=fL(x)
for allx∈L. Suppose (36) holds for n=k; that is, fL(kx)=fL(x) for all x∈L. It follows from (34) that
(44)fL((k+1)x)=fL(x⊕kx)=fL(¬x⟶kx)=fL(¬x⟶(x⊕(k-1)x))=fL(¬x⟶(¬x⟶(k-1)x))=fL(¬x⟶(k-1)x)=fL(x⊕(k-1)x)=fL(kx)=fL(x)
for allx∈L. Therefore, (36) is valid. Note that
(45)1=((¬x⟶x)⟶x)⟶((¬x⟶x)⟶x)=¬x⟶(((¬x⟶x)⟶x)⟶¬(¬x⟶x))
and ¬x→((¬x→x)→x)=1 for all x∈L. It follows from (21) and (R1) that
(46)fL(1)=fL(¬x⟶(((¬x⟶x)⟶x)⟶¬(¬x⟶x)))∩fL(¬x⟶((¬x⟶x)⟶x))⊆fL(¬x⟶¬(¬x⟶x))=fL((¬x⟶x)⟶x)
for allx∈L. Similarly fL(1)⊆fL((x→¬x)→¬x) for all x∈L. It follows from (31) that
(47)fL(1)⊆fL((x⟶¬x)⟶¬x)∩fL((¬x⟶x)⟶x)=fL(((x⟶¬x)⟶¬x)∧((¬x⟶x)⟶x))=fL(x∨¬x)
for all x∈L.Theorem 15.
LetℱL be an int-soft filter of L. If ℱL satisfies condition (33), then ℱL is an implicative int-soft filter of L.Proof.
LetℱL 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
(48)fL(x⟶z)⊇fL(1⟶(x⟶(¬z⟶z)))∩fL(1)=fL(x⟶(¬z⟶z))⊇fL(x⟶(¬z⟶y))∩fL(y⟶z)
for all x,y,z∈L. It follows from Theorem 12 that ℱL is an implicative int-soft filter of L.Corollary 16.
LetℱL be an int-soft filter of L. If ℱL satisfies condition (32), then ℱL is an implicative int-soft filter of L.Theorem 17.
If an int-soft filterℱL of L satisfies condition (36), then it is implicative.Proof.
Sincex→y≤x→(¬y→y) for all x,y∈L, it follows from (29) that
(49)fL(x⟶y)⊆fL(x⟶(¬y⟶y))
for all x,y∈L. Now condition (36) implies that
(50)fL(x⟶y)=fL(2(x⟶y))=fL(¬(x⟶y)⟶(x⟶y))=fL(x⟶(¬(x⟶y)⟶y))⊇fL(x⟶(¬y⟶y))
for allx,y∈L. Combining (49) and (50) induces fL(x→y)=fL(x→(¬y→y)) for all x,y∈L. Therefore, ℱL is an implicative int-soft filter of L by Corollary 16.Theorem 18.
If an int-soft filterℱL of L satisfies condition (37), then it is implicative.Proof.
Using (19), (20), (37), and (R4), we have
(51)fL(x⟶y)⊇fL((y∨¬y)⟶(x⟶y))∩fL(y∨¬y)⊇fL((y∨¬y)⟶(x⟶y))∩fL(1)=fL((y∨¬y)⟶(x⟶y))=fL(y⟶(x⟶y))∩fL(¬y⟶(x⟶y))=fL(x⟶(y⟶y))∩fL(¬y⟶(x⟶y))=fL(x⟶1)∩fL(¬y⟶(x⟶y))=fL(¬y⟶(x⟶y))for all x,y∈L. Since x→y≤¬y→(x→y) for all x,y∈L, it follows from (29) that fL(x→y)⊆fL(¬y→(x→y)) for all x,y∈L. Therefore,
(52)fL(x⟶y)=fL(¬y⟶(x⟶y))
for allx,y∈L, and so ℱL is an implicative int-soft filter of L by Corollary 16.Theorem 19.
LetℱL∈S(U,L) satisfy condition (19) and
(53)(∀x,y,z∈L)(fL(x)⊇fL(z⟶((x⟶y)⟶x))∩fL(z)((x→y)→x)),(54)(∀x,y,z∈L)(fL(x⟶z)⊇fL(x⟶y)∩fL(y⟶z)).
Then ℱL is an implicative int-soft filter of L.Proof.
Using (R4) and (6), we have
(55)((x⟶z)⟶z)⟶(x⟶z)=x⟶(((x⟶z)⟶z)⟶z)=x⟶(x⟶z)
for all x,z∈L. It follows from (R2), (R4), (19), (53), and (54) that
(56)fL(x⟶z)⊇fL(1⟶(((x⟶z)⟶z)⟶(x⟶z)))∩fL(1)=fL(((x⟶z)⟶z)⟶(x⟶z))=fL(x⟶(x⟶z))⊇fL(y⟶(x⟶z))∩fL(x⟶y)=fL(x⟶(y⟶z))∩fL(x⟶y)
for allx,y,z∈L. Thus, ℱL is an implicative int-soft filter of L.Corollary 20.
Every int-soft filter satisfying condition (53) is an implicative int-soft filter.Proof.
LetℱL be an int-soft filter of L that satisfies condition (53). Since ℱL satisfies two conditions (19) and (54) (see [8]), we know that ℱL is an implicative int-soft filter of L.Theorem 21.
A soft setℱL of L is an implicative int-soft filter of L if and only if the nonempty γ-inclusive set ℱLγ is an implicative filter of L for all γ∈𝒫(U).The implicative filtersℱLγ in Theorem 21 are called inclusive implicative filters of L.Proof.
Assume thatℱL is an implicative int-soft filter of L. Let γ∈𝒫(U) be such that ℱLγ≠∅. Then ℱL is an int-soft filter of L (see Theorem 10), and so ℱLγ is a filter of L. Let x,y,z∈L be such that x→(y→z)∈ℱLγ and x→y∈ℱLγ. Then fL(x→(y→z))⊇γ and fL(x→y)⊇γ. It follows from (21) that
(57)fL(x⟶z)⊇fL(x⟶(y⟶z))∩fL(x⟶y)⊇γ,
and so that x→z∈ℱLγ. Therefore, ℱLγ is an implicative filter of L for all γ∈𝒫(U).
Conversely, suppose that the nonemptyγ-inclusive set ℱLγ is an implicative filter of L for all γ∈𝒫(U). Then ℱLγ is a filter of L, and so ℱL is an int-soft filter of L. For every x,y,z∈L, let γ=fL(x→(y→z))∩fL(x→y). Then x→(y→z)∈ℱLγ and x→y∈ℱLγ, which imply that x→z∈ℱLγ. Hence,
(58)fL(x⟶z)⊇γ=fL(x⟶(y⟶z))∩fL(x⟶y).
Thus, ℱL is an implicative int-soft filter of L.Theorem 22 (extension property).
LetℱL and 𝒢L be two int-soft filters of L such that fL(1)=gL(1) and fL(x)⊆gL(x) for all x∈L. If ℱL is implicative, then so is 𝒢L.Proof.
Assume thatℱL is implicative. Then fL(x∨¬x)⊇fL(1) by (37). It follows from the hypothesis that
(59)gL(x∨¬x)⊇fL(x∨¬x)⊇fL(1)=gL(1)
for all x∈L. Therefore, 𝒢L 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 anyℱL∈S(U,L), let ℱL* be a soft set of L over U defined by
(60)fL*:L⟶𝒫(U),x↦{fL(x),ifx∈ℱLγ,δ,otherwise,
where γ and δ are subsets of U with δ⊊fL(x). If ℱL is an implicative int-soft filter of L, then so is ℱL*.Proof.
Assume thatℱL is an implicative int-soft filter of L. Then ℱLγ is an implicative filter of L for all γ⊆U with ℱLγ≠∅. Hence, 1∈ℱLγ, and so
(61)fL*(1)=fL(1)⊇fL(x)⊇fL*(x)
for all x∈L. Let x,y,z∈L. If x→(y→z)∈ℱLγ and x→y∈ℱLγ, then x→z∈ℱLγ. Hence,
(62)fL*(x⟶z)=fL(x⟶z)⊇fL(x⟶(y⟶z))∩fL(x⟶y)=fL*(x⟶(y⟶z))∩fL*(x⟶y).
Ifx→(y→z)∉ℱLγ or x→y∉ℱLγ, then fL*(x→(y→z))=δ or fL*(x→y)=δ. Thus,
(63)fL*(x⟶z)⊇δ=fL*(x⟶(y⟶z))∩fL*(x⟶y).
Therefore,ℱL* is an implicative int-soft filter of L.Theorem 24.
For any implicative filterF of L, there exists an implicative int-soft filter of L such that its inclusive implicative filter is F.Proof.
LetℱL be a soft set of L over U in which fL is given by
(64)fL:L⟶𝒫(U),x↦{γ,ifx∈F,∅,otherwise,
where γ is a nonempty subset of U. Since ∈F, we have fL(1)=γ⊇fL(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,
(65)fL(x⟶z)=γ=fL(x⟶(y⟶z))∩fL(x⟶y).
Ifx→(y→z)∉F or x→y∉F, then fL(x→(y→z))=∅ or fL(x→y)=∅. Thus,
(66)fL(x⟶z)⊇∅=fL(x⟶(y⟶z))∩fL(x⟶y).
Therefore,ℱL is an implicative int-soft filter of L. Obviously, F=ℱLγ.


## 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.



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