TSWJ The Scientific World Journal 1537-744X 2356-6140 Hindawi Publishing Corporation 10.1155/2014/517039 517039 Research Article A New Extended Soft Intersection Set to M,N-SI Implicative Fitters of BL-Algebras http://orcid.org/0000-0003-2510-9515 Zhan Jianming 1 http://orcid.org/0000-0003-1324-7080 Liu Qi 1 Kim Hee Sik 2 Wang Guoyin 1 Department of Mathematics Hubei Minzu University Enshi Hubei 445000 China muc.edu.cn 2 Department of Mathematics Hanyang University Seoul 133-791 Republic of Korea hanyang.ac.kr 2014 2072014 2014 24 03 2014 09 07 2014 21 7 2014 2014 Copyright © 2014 Jianming Zhan 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.

Molodtsov’s soft set theory provides a general mathematical framework for dealing with uncertainty. The concepts of (M,N)-SI implicative (Boolean) filters of BL-algebras are introduced. Some good examples are explored. The relationships between (M,N)-SI filters and (M,N)-SI implicative filters are discussed. Some properties of (M,N)-SI implicative (Boolean) filters are investigated. In particular, we show that (M,N)-SI implicative filters and (M,N)-SI Boolean filters are equivalent.

1. Introduction

We know that dealing with uncertainties is a major problem in many areas such as economics, engineering, medical sciences, and information science. These kinds of problems cannot be dealt with by classical methods because some classical methods have inherent difficulties. To overcome them, Molodtsov  introduced the concept of a soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Since then, especially soft set operations have undergone tremendous studies; for examples, see . At the same time, soft set theory has been applied to algebraic structures, such as . We also note that soft set theory emphasizes balanced coverage of both theory and practice. Nowadays, it has promoted a breath of the discipline of information sciences, decision support systems, knowledge systems, decision-making, and so on; see .

B L -algebras, which have been introduced by Hájek  as algebraic structures of basic logic, arise naturally in the analysis of the proof theory of propositional fuzzy logic. Turunen  proposed the concepts of implicative filters and Boolean filters in BL-algebras. Liu et al. [16, 17] applied fuzzy set theory to BL-algebras. After that, some researchers have further investigated some properties of BL-algebras. Further, Ma et al. investigated some kinds of generalized fuzzy filters BL-algebras and obtained some important results; see [18, 19]. Zhang et al. [20, 21] described the relations between pseudo-BL, pseudo-effect algebras, and BCC-algebras, respectively. The other related results can be found in [22, 23].

Recently, Çağman et al. put forward soft intersection theory; see [24, 25]. Jun and Lee  applied this theory to BL-algebras. Ma and Kim  introduced a new concept: (M,N)-soft intersection set. They introduced the concept of (M,N)-soft intersection filters of BL-algebras and investigated some related properties.

In this paper, we introduce the concept of (M,N)-soft intersection implicative filters of BL-algebras. Some related properties are investigated. In particular, we show that (M,N)-SI implicative filters and (M,N)-SI Boolean filters are equivalent.

2. Preliminaries

Recall that an algebra L=(L,,,,,,0,1) is a BL-algebra  if it is a bounded lattice such that the following conditions are satisfied:

(L,,1) is a commutative monoid,

and form an adjoin pair, that is, zxy if and only if xzy for all x,y,zL,

xy=x(xy),

(xy)(yx)=1.

In what follows, L is a BL-algebra unless otherwise is specified.

In any BL-algebra L, the following statements are true (see [14, 15]):

xyxy=1,

x(yz)=(xy)z=y(xz),

xyxy,

xy(zx)(zy),xy(yz)(xz),

xx=x′′x,

xx=1xx=0,

(xy)(yz)xz,

xyxzyz,

xyzxzy,

xy=((xy)y)((yx)x),

where x=x0.

A nonempty subset A of L is called a filter of L if it satisfies the following conditions: (I1) 1A, (I2) for all xA, for all yL,xyAyA.

It is easy to check that a nonempty subset A of L is a filter of L if and only if it satisfies (I3) for all x,yL, xyA, (I4) for all xA, for all yL, xyyA (see ).

Now, we call a nonempty subset A of L an implicative filter if it satisfies (I1) and (I5) x(zy)A, yzAxzA.

A nonempty subset A of L is said to be a Boolean filter of L if it satisfies xxA, for all xA. (see ).

From now on, we let L be an BL-algebra, U an initial universe, E a set of parameters, P(U) the power set of U, and A,B,CE. We let MNU.

Definition 1 (see [<xref ref-type="bibr" rid="B19">1</xref>]).

A soft set fA over U is a set defined by fA:EP(U) such that fA(x)= if xA. Here fA is also called an approximate function. A soft set over U can be represented by the set of ordered pairs fA={(x,fA(x))xE,fA(x)P(U)}. It is clear to see that a soft set is a parameterized family of subsets of U. Note that the set of all soft sets over U will be denoted by S(U).

Definition 2 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

Let fA,fBS(U).

fA is said to be a soft subset of fB and denoted by fA~fB if fA(x)fB(x), for all xE. fA and fB are said to be soft equally, denoted by fA=fB, if fA~fB and fA~fB.

The union of fA and fB, denoted by fA~fB, is defined as fA~fB=fAB, where fAB(x)=fA(x)fB(x), for all xE.

The intersection of fA and fB, denoted by fA~fB, is defined as fA~fB=fAB, where fAB(x)=fA(x)fB(x), for all xE.

Definition 3 (see [<xref ref-type="bibr" rid="B12">26</xref>]).

( 1 ) A soft set fL over U is called an SI- filter of L over U if it satisfies

fL(x)fL(1) for any xL,

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

( 2 ) A soft set fL over U is called an SI-implicative filter of L over U if it satisfies (S1) and

fL(x(zy))fl(yz)fL(xz), for all x,y,zL.

In , Ma and Kim introduced the concept of (M,N)-SI filters in BL-algebras.

Definition 4 (see [<xref ref-type="bibr" rid="B15">27</xref>]).

A soft set fS over U is called an (M,N)-soft intersection filter (briefly, (M,N)-SI filter) of L over U if it satisfies

fL(x)NfL(1)M for all xL,

fL(xy)fL(x)NfL(y)M for all x,yL.

Define an ordered relation “~(M,N)” on S(U) as follows. For any fL,gLS(U), MNU, we define fL~(M,N)gLfLN~(M,N)gLM.

And we define a relation “=(M,N)” as follows: fL=(M,N)gLfL~(M,N)gL and gL~(M,N)fL.

Definition 5 (see [<xref ref-type="bibr" rid="B15">27</xref>]).

A soft set fS over U is called an (M,N)-soft intersection filter (briefly, (M,N)-SI filter) of L over U if it satisfies

fL(x)~(M,N)fL(1) for all xL,

fL(xy)fL(x)~(M,N)fL(y) for all x,yL.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M184"><mml:mo mathvariant="bold">(</mml:mo><mml:mi>M</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>N</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M185"><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:math></inline-formula> Implicative (Boolean) Filters

In this section, we investigate some characterizations of (M,N)-SI implicative filters of BL-algebras. Finally, we prove that a soft set in BL-algebras is an (M,N)-SI implicative filter if and only if it is an (M,N)-SI Boolean filter.

Definition 6.

A soft set fL over U is called an (M,N)-soft intersection implicative filter (briefly, (M,N)-SI implicative filter) of L over U if it satisfies (SI1) and (SI3) fL(x(zy))fL(yz)N~(M,N)fL(yz)M for all x,y,zL.

Remark 7.

If fL is an (M,N)-SI implicative filter of L over U, then fL is an (,U)-SI implicative filter of L. Hence every SI-implicative filter of L is an (M,N)-SI implicative filter of L, but the converse need not be true in general. See the following example.

Example 8.

Assume that U=D2={x,yx2=y2=e,xy=yx}={e,x,y,yx}, dihedral group, is the universe set.

Let L={0,a,b,1}, where 0<a<b<1. Then we define xy=min{x,y},xy=max{x,y} and and as follows: (1)0ab100000a0aaab0aab10ab10ab101111a0111b0b1110ab1 Then (L,,,,,1) is a BL-algebra.

Let M={e,y} and N={e,x,y}.

Define a soft set fL over U by fL(1)={e,x}, fL(a)=fL(b)={e,x,y}, and fL(0)={e,y}. Then one can easily check that fL is an (M,N)-SI implicative filter of L over U, but it is not an SI implicative filter of L over U since fL(1)={e,x}fL(a).

By means of “~(M,N),” we can obtain the following equivalent concept.

Definition 9.

A soft set fL over U is called an (M,N)-SI implicative filter of L over U if it satisfies (SI1) and (SI3) fL(x(zy))fL(yz)~(M,N)fL(yz) for all x,y,zL.

From the above definitions, we have the following.

Proposition 10.

Every (M,N)-SI implicative filter of L over U is an (M,N)-SI filter, but the converse may not be true as shown in the following example.

Example 11.

Define xy=min{x,y} and (2)xy={1,ifxy,y,ifx>y.

Then L=([0,1],,,,,0,1) is a BL-algebra.

Let U=L, M={0.5,0.75}, and N={0.5,0.75,1}.

Define a soft set fL over U by (3)fL(x)={{0,0.5},ifx[0,12],{0.5,1},ifx[12,1].

Then one can easily check that fL is an (M,N)-SI filter of L over U, but it is not an (M,N)-SI implicative filter of L over U. Since fL(2/3((1/3)1/4))fL(1/41/3)N=fL(1)fL(1)N={0.5,1}{0.5,0.75,1}={0.5,1} and fL(2/31/4)M=fL(1/4)M={0,0.5}{0.5,0.75}={0,0.5,0.75}, this implies that fL(2/3((1/3)1/4))fL(1/41/3)NfL(xz)M.

Lemma 12 (see [<xref ref-type="bibr" rid="B15">27</xref>]).

If a soft set fL over U is an (M,N)-SI filter of L, then for any x,y,zL we have

xyfL(x)~(M,N)fL(y),

fL(xy)=fL(1)fL(x)~(M,N)fL(y),

fL(xy)=(M,N)fL(x)fL(y)=(M,N)fL(xy),

fL(0)=(M,N)fL(x)fL(x),

fL(xy)fL(yz)~(M,N)fL(xz),

fL(x)fL(y)~(M,N)fL(yzyz),

fL(xy)~(M,N)fL((yz)(xz)),

fL(xy)~(M,N)fL((zx)(zy)).

Theorem 13.

Let fL be an (M,N)-SI filter of L over U, then the following are equivalent:

fL is an (M,N)-SI implicative filter of L,

fL(xz)~(M,N)fL(x(zz)), for all x,y,zL,

fL(xz)=(M,N)fL(x(zz)), for all x,y,zL,

fL(xz)~(M,N)fL(y(x(zz)))fL(y), for all x,y,zL.

Proof.

(1) (2) Assume that fL is an (M,N)-SI filter of L over U. Putting y=z in (SI3), then (4)fL(xz)M=(fL(xz)M)M(fL(x(zz))fL(zz)N)M=(fL(x(zz))fL(1)N)MfL(x(zz))(fL(1)M)NfL(x(zz))N; that is, fL(xz)~(M,N)fL(x(zz)). Thus, (2) holds.

(2) (3) By (a1) and (a2), xzz(xz)=x(zz); then it follows from Lemma 12 (1) that fL(xz)~(M,N)fL(x(zz)). Thus, (3) holds.

(3) (4) Assume that (4) holds. By Lemma 12 (5), we have fL(xzy)fL(yz)~(M,N)fL(xzz). By (a2), fL(x(zy))fL(yz)~(M,N)fL(x(zz)).

(4) (1) Putting y=1 in (4), we have (5)fL(xz)~(M,N)fL(x(zz)). Hence (6)fL(zz)~(M,N)fL(x(zy))fL(yz).

Thus, (SI3) holds. This shows that fL is an (M,N)-SI implicative filter of L over U.

Now, we introduce the concept of (M,N)-SI Boolean filters of BL-algebras.

Definition 14.

Let fL be an (M,N)-SI filter of L over U, then fL is called an (M,N)-SI Boolean filter of L over U if it satisfies

fL(xx)=(M,N)fL(1) for all xL.

Theorem 15.

A soft set fL over U is an (M,N)-SI implicative filter of L if and only if it is an (M,N)-SI Boolean filter.

Proof.

Assume that fL over U is an (M,N)-SI Boolean filter of L over U. Then (7)fL(xz)~(M,N)fL((zz)(xz))fL(zz)=(M,N)fL((zz)(xz))fL(1)~(M,N)fL((zz)(xz)). By (a10) and (a1), we have (8)(zz)(xz)=(z(xz))(z(xz))=z(xz)=x(zz). Hence fL(xz)~(M,N)fL(x(zz)). It follows from Theorem 13 that fL is an (M,N)-SI implicative filter of L over U.

Conversely, assume that fL is an (M,N)-SI implicative filter of L over U. By Theorem 13, we have (9)fL((xx)x)=(M,N)fL((xx)(xx))=fL(1),fL((xx)x)=(M,N)fL((xx)(x′′x))=fL((xx)(xx))=fL(1). By Lemma 12, we have (10)fL(xx)=(M,N)fL((xx)x)fL((xx)x)=(M,N)fL(1). Hence fL is an (M,N)-SI Boolean filter of L over U.

Remark 16.

Every (M,N)-SI implicative filter and (M,N)-SI Boolean filter in BL-algebras are equivalent.

Next, we give some characterizations of (M,N)-SI implicative (Boolean) filters in BL-algebras.

Theorem 17.

Let fL be an (M,N)-SI filter of L over U, then the following are equivalent:

fL is an (M,N)-SI implicative (Boolean) filter,

fL(x)=(M,N)fL(xx), for all xL,

fL((xy)x)(M,N)fL(x), for all x,yL,

fL((xy)x)=(M,N)fL(x), for all x,yL,

fL(x)~(M,N)fL(z((xy)x))fL(z), for all x,y,zL.

Proof.

(1) (2). Assume that fL is an (M,N)-SI implicative (Boolean) filter of L over U. By Theorem 13, we have (11)fL(x)=fL(1x)=(M,N)fL(1(xx))=fL(xx). Thus, (2) holds.

(2) (3). By (a1), (a2), and (a8), we have xxy and so (xy)xxx. By Lemma 12,   fL((xy)x)~(M,N)fL(xx). Combining (2), fL(x)=(M,N)fL(xx)~(M,N)fL((xy)x). Thus, (3) holds.

(3) (4). Since x(xy)x, then by Lemma 12fL(x)~(M,N)fL((xy)x). Combining (3), fL(x)=(M,N)fL((xy)x).

(4) (5). By (SI2), fL((xy)x)~(M,N)fL(z((xy)x))fL(z). Combining (4), we have fL(x)~(M,N)fL(z((xy)x))fL(z). Thus, (5) holds.

(5) (1). By (a1), zxz. By (a8), (xz)z and so z(xz)(xz)(xz). Then by Lemma 12, fL(z(xz))~(M,N)fL((xz)(xz))=(M,N)fL(1((xz)(xz)))fL(1). By (5), fL(xz)~(M,N)fL(z(xz)) and so fL(xz)~(M,N)fL(x(zz)). Therefore, it follows from Theorem 13 that fL is an (M,N)-SI implicative filter of L.

Finally, we investigate extension properties of (M,N)-SI implicative filters of BL-algebras.

Theorem 18 (extension property).

Let fL and gL be two (M,N)-SI filters of L over U such that fL(1)=(M,N)gL(1) and fL(x)~(M,N)gL(x) for all xL. If fL is an (M,N)-SI implicative (Boolean) filter of L, then so is gL.

Proof.

Assuming that fL is an (M,N)-SI implicative (Boolean) filter of L over U, then fL(xx)=(M,N)fL(1) for all xL. By hypothesis, gL(xx)~(M,N)fL(xx)=(M,N)fL(1)=(M,N)gL(1). By (SI1), we have gL(1)~(M,N)gL(xx). Thus, gL(xx)=(M,N)gL(1). Hence gL is an (M,N)-SI implicative (Boolean) filter of L.

4. Conclusions

In this paper, we introduce the concepts of (M,N)-SI implicative filters and (M,N)-SI Boolean filters of BL-algebras. Then we show that every (M,N)-SI Boolean filter is equivalent to (M,N)-SI implicative filters. In particular, some equivalent conditions for (M,N)-SI Boolean filters are obtained. We hope it can lay a foundation for providing a new soft algebraic tool in many uncertainties problems.

To extend this work, one can apply this theory to other fields, such as algebras, topology, and other mathematical branches. To promote this work, we can further investigate (M,N)-SI prime (semiprime) Boolean filters of BL-algebras. Maybe one can apply this idea to decision-making, data analysis, and knowledge based systems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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