DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 950897 10.1155/2013/950897 950897 Research Article Intersection-Soft Filters in R0-Algebras Jun Young Bae 1 Ahn Sun Shin 2 Lee Kyoung Ja 3 Zhou Yong 1 Department of Mathematics Education (and RINS) Gyeongsang National University Chinju 660-701 Republic of Korea gnu.ac.kr 2 Department of Mathematics Education Dongguk University Seoul 100-715 Republic of Korea dongguk.edu 3 Department of Mathematics Education Hannam University Daejeon 306-791 Republic of Korea hnu.kr 2013 28 3 2013 2013 26 01 2013 01 03 2013 06 03 2013 2013 Copyright © 2013 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 (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 . Maji et al.  and Molodtsov  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  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.  described the application of soft set theory to a decision making problem. Maji et al.  also studied several operations on the theory of soft sets. Chen et al.  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  in order to get an algebraic proof of the completeness theorem of a formal deductive system . The filter theory in R0-algebras is discussed in .

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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Algebras Definition 1 (see [<xref ref-type="bibr" rid="B9">5</xref>]).

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) xy=¬y¬x,

(R2) 1x=x,

(R3) (yz)((xy)(xz))=yz,

(R4) x(yz)=y(xz),

(R5) x(yz)=(xy)(xz),

(R6) (xy)((xy)(¬xy))=1.

Let L be an R0-algebra. For any x,yL, we define xy=¬(x¬y) and xy=¬xy. It is proven that and are commutative, associative, and xy=¬(¬x¬y), and (L,,,,,0,1) is a residuated lattice. In the following, let xn denote xxx where x appears n times for n.

We refer the reader to the book  for further information regarding R0-algebras.

Lemma 2 (see [<xref ref-type="bibr" rid="B8">7</xref>]).

Let L be an R0-algebra. Then the following properties hold: (1)(x,yL)(xyxy=1),(2)(x,yL)(xyx),(3)(xL)(¬x=x0),(4)(x,yL)((xy)(yx)=1),(5)(x,yL)×(xyyzxz,zxzy),(6)(x,yL)(((xy)y)y=xy),(7)(x,yL)×(xy=((xy)y)((yx)x)),(8)(xL)(x¬x=0,x¬x=1),(9)(x,yL)(xyxy,x(xy)xy),(10)(x,y,zL)((xy)z=x(yz)),(11)(x,yL)(xy(xy)),(12)(x,y,zL)(xyzxyz),(13)(x,y,zL)(xyxzyz),(14)(x,y,zL)(xy(yz)(xz)),(15)(x,y,zL)((xy)(yz)xz).

Definition 3 (see [<xref ref-type="bibr" rid="B8">7</xref>]).

A nonempty subset F of L is called a filter of L if it satisfies

1F,

(xF)(yL)(xyF    yF).

Lemma 4 (see [<xref ref-type="bibr" rid="B8">7</xref>]).

Let F be a nonempty subset of L. Then F is a filter of L if and only if it satisfies

(xF)(yL)(xy    yF),

(x,yF)(xyF).

2.2. Basic Results on Soft Set Theory

Soft set theory was introduced by Molodtsov  and Çağman and Enginoğlu .

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 [<xref ref-type="bibr" rid="B7">1</xref>, <xref ref-type="bibr" rid="B1">9</xref>]).

A soft set of E over U (a soft set of E for short) is any function fA:E𝒫(U),suchthatfA(x)=ifxA,forA𝒫(E), or, equivalently, any set (16)A:={(x,fA(x))xE,fA(x)𝒫(U),fA(x)=ifxA}, for A𝒫(E).

Definition 6 (see [<xref ref-type="bibr" rid="B1">9</xref>]).

Let A and B be soft sets of E. We say that A is a soft subset of B, denoted by A~B, if fA(x)fB(x) for all xE.

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 LS(U,L) is called an int-soft filter of L if it satisfies (17)(γ𝒫(U))(LγLγisafilterofL), where Lγ={xLγfL(x)} which is called the γ-inclusive set of L.

If L is an int-soft filter of L, every γ-inclusive set Lγ 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: (18)x¬x01acbbca100abc1011111ac1111bbb111caab1110abc1 Then (L,,,¬,) is an R0-algebra (see ) where xy=min{x,y} and xy=max{x,y}. Let LS(U,L) be given as follows: (19)L={(0,γ1),(a,γ1),(b,γ1),(c,γ2),(1,γ2)}, where γ1 and γ2 are subsets of U with γ1γ2. Then L is an int-soft filter of L.

We provide characterizations of an int-soft filter.

Theorem 9.

Let LS(U,L). Then L is an int-soft filter of L if and only if the following assertions are valid:

(xL)(fL(x)fL(1)),

(x,yL)(fL(xy)fL(x)fL(y)).

Proof.

Assume that L is an int-soft filter of L. For any xL, let fL(x)=γ. Then xLγ. Since Lγ is a filter of L, we have 1Lγ and so fL(x)=γfL(1). For any x,yL, let fL(xy)fL(x)=γ. Then xyLγ and xLγ. Since Lγ is a filter of L, it follows that yLγ. Hence fL(xy)fL(x)=γfL(y).

Conversely, suppose that L satisfies two conditions (1) and (2). Let γ𝒫(U) such that Lγ. Then there exists aLγ, and so γfL(a). It follows from (1) that γfL(a)fL(1). Thus 1Lγ. Let x,yL such that xyLγ and xLγ. Then γfL(xy) and γfL(x). It follows from (2) that (20)γfL(xy)fL(x)fL(y), that is, yLγ. Thus Lγ() is a filter of L, and hence L is an int-soft filter of L.

Proposition 10.

Let LS(U,L) be an int-soft filter of L. Then the following properties are valid.

L is order preserving, that is, (21)(x,yL)(xyfL(x)fL(y)).

(x,yL)(fL(xy)=fL(1)fL(x)fL(y)).

(x,yL)(fL(xy)=fL(x)fL(y)=fL(xy)).

(xL)(n)(fL(xn)=fL(x)).

(xL)(fL(0)=fL(x)fL(¬x)).

(x,yL)(fL(xy)fL(yz)fL(xz)).

(x,y,zL)(xyzfL(x)fL(y)fL(z)).

(x,yL)(fL(x)fL(xy)=fL(y)fL(yx)=fL(x)fL(y)).

(x,yL)(fL(x(xy))=fL(y(yx))=fL(x)fL(y)).

(x,y,zL)(fL(x(¬zy))fL(yz)fL(x(¬zz))).

(x,y,zL)(fL(x(yz))fL(xy)fL(x(xz))).

Proof.

( 1 ) Let x,yL such that xy. Then xy=1, and so (22)fL(x)=fL(x)fL(1)=fL(x)fL(xy)fL(y) by (1) and (2) of Theorem 9.

( 2 ) Let x,yL such that fL(xy)=fL  (1). Then (23)fL(x)=fL(x)fL(1)=fL(x)fL(xy)fL(y) by (1) and (2) of Theorem 9.

( 3 ) Since xyxy for all x,yL, it follows from (1) that fL(xy)fL(x)fL(y). Using (11) and (1), we have fL(x)fL(y(xy)). It follows from Theorem 9(2) that fL(x)fL(y)fL(y(xy))fL(y)fL(xy). Therefore fL(xy)=fL(x)fL(y). Since yxy and x(xy)xy for all x,yL, we have fL(y)fL(xy) and fL(x)fL(y)fL(x)fL(xy)=fL(x(xy))fL(xy)fL(x)fL(y) by (1). Hence fL(xy)=fL(x)fL(y) for all x,yL.

( 4 ) It follows from (3).

( 5 ) Note that x¬x= for all xL. Using (3), we have (24)fL(0)=fL(x¬x)=fL(x)fL(¬x) for all x,yL.

( 6 ) Combining (15), (1), and (3), we have the desired result.

( 7 ) It follows from (1) and (3).

( 8 ) Since yxy for all x,yL, it follows from (1) that (25)fL(x)fL(y)fL(x)fL(xy). Since x(xy)xy for all x,yL, we have (26)fL(x)fL(xy)=fL(x(xy))fL(xy)=fL(x)fL(y) by (3) and (1). Hence fL(x)fL(y)=fL(x)fL(xy). Similarly, fL(y)    fL(yx)  =  fL(x)fL(y) for all x,yL.

( 9 ) Using (3), we have (27)fL(x(xy))=fL(x)fL(xy),fL(y(yx))=fL(y)fL(yx), for all x,yL. It follows from (8) that fL(x(xy))=fL(y(yx))=fL(x)fL(y).

( 10 ) Note that (28)(x(¬zy))(yz)=((x¬z)y)(yz)(x¬z)z=x(¬zz) for all x,y,zL. Using (1) and (3), we have (29)fL(x(¬zy))fL(yz)=fL((x(¬zy))(yz))fL(x(¬zz)) for all x,y,zL.

( 11 ) Note that (x(yz))(xy)=(y(xz))(xy)x(xz) for all x,y,zL. It follows from (1) and (3) that (30)fL(x(yz))fL(xy)fL(x(xz)) for all x,y,zL.

Theorem 11.

Let LS(U,L). Then L is an int-soft filter of L if and only if the following assertions are valid:

L is order preserving,

(x,yL)(fL(xy)=fL(x)fL(y)).

Proof.

The necessity follows from (1) and (3) of Proposition 10.

Conversely, suppose that L satisfies two conditions (1) and (2). Let x,yL. Since x1, we have fL(x)fL(1) by (1). Note that x(xy)y. It follows from (2) and (1) that (31)fL(x)fL(xy)=fL(x(xy))fL(y). Therefore L is an int-soft filter of L.

Proposition 12.

Let bL such that ¬b=b. If LS(U,L) is an int-soft filter of L, then fL(b)=fL(x) for all x{aL0ab}.

Proof.

Suppose that there exists y{aL0ab} such that fL(b)fL(y). Then fL(y)fL(b), and so bLγ and yLγ where γ=fL(b). Since Lγ is a filter of L, we have 0=bbLγ. This shows that Lγ=L, and it is a contradiction. Hence fL(b)=fL(x) for all x{aL0ab}.

Theorem 13.

Let LS(U,L). Then L is an int-soft filter of L if and only if the following assertion is valid: (32)(x,y,zL)×(fL(y)fL((xy)z)fL(xz)).

Proof.

Assume that L is an int-soft filter of L, and let x,y,zL. Since yxy, it follows from Proposition 10(1) and Theorem 9(2) that (33)fL(y)fL((xy)z)fL(xy)fL((xy)z)fL(z)fL(xz).

Conversely, suppose that L satisfies the inclusion (32). Let x,yL. Then (34)fL(x)=fL(x)fL((0x)x)fL(0x)=fL(1),fL(x)fL(xy)=fL(x)fL((1x)y)fL(1y)=fL(y) by (32). Therefore L is an int-soft filter of L by Theorem 9.

Theorem 14.

Let LS(U,L). Then L is an int-soft filter of L if and only if the following assertion is valid: (35)(x,y,zL)(xyzfL(x)fL(y)fL(z)).

Proof.

Suppose that L is an int-soft filter of L. Let x,y,zL such that xyz. Then fL(x)fL(yz) by Proposition 10(1), and so (36)fL(x)fL(y)fL(yz)fL(y)fL(z) by Theorem 9 (2).

Conversely, assume that L satisfies the condition (35). Let x,yL. Since x1=x1, we have fL(x)fL(1) by (35). Note that xyxy. It follows from (35) that fL(xy)fL(x)fL(y). Therefore L is an int-soft filter of L by Theorem 9.

Proposition 15.

Every int-soft filter L of L satisfies. (37)(x,y,zL)×(fL((xy)z)fL(x(yz))).

Proof.

Let x,y,zL. Since 1=y(xy)((xy)z)(yz), we have (38)fL(1)fL(((xy)z)(yz)) by Proposition 10  (1). It follows from (2) and Theorem 9 that (39)fL((xy)z)=fL((xy)z)fL(1)fL((xy)z)fL×(((xy)z)(yz))fL(yz)fL(x(yz)). 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: (40)x¬x01adbccbda100abcd10111111ad11111bcc1111cbbb111daabc1110abcd1 Then (L,,,¬,) is an R0-algebra (see ) where xy=min{x,y} and xy=max{x,y}. Let LS(U,L) be given as follows: (41)L={(0,γ1),(a,γ2),(b,γ2),(c,γ2),(d,γ1),(1,γ1)}, where γ1 and γ2 are subsets of U with γ1γ2. Then L satisfies the condition (37), but L is not an int-soft filter of L since fL(a)fL(1).

Proposition 17.

For an int-soft filter L of L, the following are equivalent: (42)(x,yL)(fL(y(yx))fL(yx)),(43)(x,y,zL)(fL(z(yx))fL((zy)(zx))).

Proof.

Assume that (42) is valid, and let x,y,zL. Using (R4), (5), and (14), we have (44)z(yx)z(z((zy)x)). It follows from Proposition 10 (1), (42), and (R4) that (45)fL(z(yx))fL(z(z((zy)x)))fL(z((zy)x))=fL((zy)(zx)).

Conversely, suppose that (43) holds. If we use z instead of y in (43), then (46)fL(z(zx))fL((zz)(zx))=fL(1(zx))=fL(zx), which proves (42).

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

Theorem 18.

Let LS(U,L). For a subset γ of U, define a soft set L* of L by (47)fL*:L𝒫(U),x{fL(x)ifxLγ,otherwise. If L is an int-soft filter of L, then so is L*.

Proof.

Assume that L is an int-soft filter of L. Then Lγ() is a filter of L for all γ𝒫(U). Hence 1Lγ, and so fL*(1)=fL(1)fL(x)fL*(x) for all xL. Let x,yL. If xLγ and xyLγ, then yLγ. Hence (48)fL*(x)fL*(xy)=fL(x)fL(xy)fL(y)=fL*(y).   If xLγ or xyLγ, then fL*(x)= or fL*(xy)=. Thus (49)fL*(x)fL*(xy)=fL*(y). Therefore 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 L be a soft set of L defined by (50)fL:L𝒫(U),x{γifxF,otherwise. Obviously fL(x)fL(1) for all xL. For any x,yL, if xF and xyF, then yF. Hence fL(x)fL(xy)=γ=fL(y). If xF or xyF, then fL(x)= or fL(xy)=. Thus fL(x)fL(xy)=fL(y). Therefore L is an int-soft filter of L and clearly Lγ=F. This completes the proof.

Definition 20.

An int-soft filter L of L is said to be strong if the following assertion is valid: (51)(x,yL)(fL(yx)fL(((xy)y)x)).

Example 21.

The int-soft filter L in Example 8 is strong.

Theorem 22.

Let LS(U,L). Then L is a strong int-soft filter of L if and only if the following assertions are valid:

(xL)(fL(x)fL(1)),

(x,y,zL)(fL(z(yx))fL(z)fL(((xy)y)x)).

Proof.

Suppose that L is a strong int-soft filter of L. Obviously, (1) is valid. For every x,y,zL, we have (52)fL(z(yx))fL(z)fL(yx)fL(((xy)y)x) by Theorem 9 (2) and (51).

Conversely, assume that L satisfies two conditions (1) and (2). If we take y=1 in (2), then fL(z)fL(zx)fL(x) for all x,zL. Hence L is an int-soft filter of L. Now if we put z=1 in (2), then (53)fL(yx)=fL(1(yx))=fL(1(yx))fL(1)fL(((xy)y)x) by (R2) and (1). Therefore L is a strong int-soft filter of L.

Example 23.

Let L=[0,1]. For any a,bL, we define (54)¬a=1-a,ab=min{a,b},ab=max{a,b}ab={1ab¬abotherwise. Then (L,,,¬,) is an R0-algebra (see ). Let LS(U,L) be given as follows: (55)L={(1,γ2),(x,γ1)xL{1}}, where γ1 and γ2 are subsets of U with γ1γ2. Then L is an int-soft filter of L. But (56)fL(1(0.30.8))fL(1)=γ2γ1=fL(((0.80.3)0.3)0.8), and so 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: (57)(x,yL)((xy)y(yx)x). Then every int-soft filter of L is strong.

Proof.

Let L be an int-soft filter of L. Using (5), (6), and (57), we have (58)yx=((yx)x)x((xy)y)x for all x,yL. It follows from Proposition 10 (1) that (59)fL(yx)fL(((xy)y)x) for all x,yL. Therefore L is a strong int-soft filter of L.

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

Theorem 25.

Let L and 𝒢L be two int-soft filters of L such that L~  𝒢L and fL(1)=gL(1). If L is strong, then so is 𝒢L.

Proof.

Assume that L is a strong int-soft filter of L. For any x,yL, let a=yx. Since L is a strong int-soft filter of L, we have (60)gL(1)=fL(1)=fL(y(ax))fL((((ax)y)y)(ax))gL((((ax)y)y)(ax)), by (51) and assumption, and so (61)gL(1)=gL((((ax)y)y)(ax))=gL(a((((ax)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((((ax)y)y)x))gL((((ax)y)y)x). Using (R4) and (14), we have (63)1=x(ax)((ax)y)(xy)((xy)y)(((ax)y)y)((((ax)y)y)x)(((xy)y)x). It follows from (62) and Theorem 14 that (64)gL(yx)=gL(a)gL((((ax)y)y)x)=gL(1)gL((((ax)y)y)x)gL(((xy)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 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.

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