# (

Authors: Xueling Ma; Hee Sik Kim
Journal: The Scientific World Journal
 (2014)

---

## Abstract

The purpose of this paper is to give a foundation for providing a new soft algebraic tool in considering many problems containing uncertainties. In order to provide these new soft algebraic structures, we discuss a new soft set-(M, N)-soft intersection set, which is a generalization of soft intersection sets. We introduce the concepts of (M, N)-SI filters of BL-algebras and establish some characterizations. Especially, (M, N)-soft congruences in BL-algebras are concerned.


---

## Body

## 1. Introduction

It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. In making inference levels, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. BL-algebra has been introduced by Hájek as the algebraic structures for his Basic Logic [1]. A well-known example of a BL-algebra is the interval [0,1] endowed with the structure induced by a continuous t-norm. In fact, the MV-algebras, Gödel algebras, and product algebras are the most known classes of BL-algebras. BL-algebras are further discussed by many researchers; see [2–12].We note that the complexities of modeling uncertain data in economics, engineering, environmental science, sociology, information sciences, and many other fields cannot be successfully dealt with by classical methods. Based on this reason, Molodtsov [13] proposed a completely new approach for modeling vagueness and uncertainty, which is called soft set theory. We note that soft set theory emphasizes a balanced coverage of both theory and practice. Nowadays, it has promoted a breath of the discipline of information sciences, intelligent systems, expert and decision support systems, knowledge systems and decision making, and so on. For example, see [14–24]. In particular, Çag˘man et al., Sezgin et al., and Jun et al. applied soft intersection theory to groups [25], near-rings [26], and BL-algebras [27], respectively.In this paper, we organize the recent paper as follows. In Section2, we recall some concepts and results of BL-algebras and soft sets. In Section 3, we investigate some characterizations of (M,N)-SI filters of BL-algebras. In particular, some important properties of (M,N)-soft congruences of BL-algebras are discussed in Section 4.


## 2. Preliminaries

Recall that an algebraL=(L,≤,∧,∨,⊙,→,0,1) is a BL-algebra [1] if it is a bounded lattice such that the following conditions are satisfied:(i)
(L,⊙,1) is a commutative monoid;(ii)
⊙ and → form an adjoin pair; that is, z≤x→y if and only if x⊙z≤y for all x,y,z∈L;(iii)
x∧y=x⊙(x→y);(iv)
(x→y)∨(y→x)=1.In what follows,L is a BL-algebra unless otherwise specified.In any BL-algebraL, the following statements are true (see [1, 5, 6]):(a1)
x≤y⇔x→y=1;(a2)
x→(y→z)=(x⊙y)→z=y→(x→z);(a3)
x⊙y≤x∧y;(a4)
x→y≤(z→x)→(z→y),x→y≤(y→z)→(x→z);(a5)
x→x′=x′′→x;(a6)
x∨x′=1⇒x∧x′=0;(a7)
(x→y)⊙(y→z)≤x→z;(a8)
x≤y⇒x→z≥y→z;(a9)
x≤y⇒z→x≤z→y,where x′=x→0.A nonempty subsetA of L is called afilter of L if it satisfies the following conditions:(I1)
1∈A,(I2)
∀x∈A, ∀y∈L, x→y∈A⇒y∈A.It is easy to check that a nonempty subsetA of L is a filter of L if and only if it satisfies(I3)
∀x,y∈L, x⊙y∈A,(I4)
∀x∈A, ∀y∈L, x≤y⇒y∈A (see [6]).From now on, we letL be a BL-algebra, U an initial universe, E a set of parameters, and P(U) the power set of U and A,B,C⊆E.Definition 1 (see [13, 16]).
A soft setfA over U is a set defined by fA:E→P(U) such that fA(x)=∅ if x∉A. Here fA is also called anapproximate function. A soft set over U can be represented by the set of ordered pairs fA={(x,fA(x))∣x∈E,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 [16]).
LetfA,fB∈S(U).(1)
fA is said to be asoft subset of fB and denoted by fA⊆~fB if fA(x)⊆fB(x), for all x∈E. fA and fB are said to besoft equal, denoted by fA=fB, if fA⊆~fB and fA⊇~fB.(2)
The union offA and fB, denoted by fA∪~fB, is defined as fA∪~fB=fA∪B, where fA∪B(x)=fA(x)∪fB(x), for all x∈E.(3)
The intersection offA and fB, denoted by fA∩~fB, is defined as fA∩~fB=fA∩B, where fA∩B(x)=fA(x)∩fB(x), for all x∈E.Definition 3 (see [27]).
A soft setfL over U is called an SI-filter of L over U if it satisfies(S1)fL(x)⊆fL(1) for any x∈L,(S2)fL(x→y)∩fL(x)⊆fL(y) for all x,y∈L.


## 3.(M,N)-SI Filters

In this section, we introduce the concept of(M,N)-SI filters in BL-algebras and investigate some characterizations. From now on, we let ∅⊆M⊂N⊆U.Definition 4.
A soft setfL over U is called an (M,N)-soft intersection filter (briefly, (M,N)-SIfilter) of L over U if it satisfies(SI1)
fL(x)∩N⊆fL(1)∪M for all x∈L,(SI2)
fL(x→y)∩fL(x)∩N⊆fL(y)∪M for all x,y∈L.Remark 5.
IffL is an (M,N)-SI filter of L over U, then fL is an (∅,U)-SI filter of L over U. Hence every SI-filter of L is an (M,N)-SI filter of L, but the converse need not be true in general. See the following example.Example 6.
Assume thatU=S3, the symmetric 3-group is the universal set, and let L={0,a,b,1}, where 0<a<b<1. We define x∧y:=min⁡{x,y},x∨y:=max⁡{x,y} and ⊙ and → as follows:
(1)⊙0ab100000a00aab0abb10ab1⟶0ab101111aa111b0a1110ab1
It is clear that (L,∧,∨,⊙,→,1) is a BL-algebra. Let M={(13),(123)} and N={(1),(12),(13),(123)}. Define a soft set fL over U by fL(1)={(1),(12),(123)},fL(b)={(1),(12),(13),(123)} and fL(a)=fL(0)={(1),(12)}. Then we can easily check that fL is an (M,N)-SI filter of L over U, but it is not SI-filter of L over U since fL(b)⊈fL(1).The following proposition is obvious.Proposition 7.
If a soft setfL over U is an (M,N)-SI filter of L over U, then
(2)(fS(1)∩N)∪M⊇(fS(x)∩N)∪M∀x∈S.Define an ordered relation “⊆~(M,N)” on S(U) as follows: for any fL,gL∈S(U),∅⊆M⊂N⊆U, we define fL⊆~(M,N)gL⇔fL∩N⊆~gL∪M. And we define a relation “=(M,N)” as follows: fL=(M,N)gL⇔fL⊆~(M,N)gL and gL⊆~(M,N)fL. Using this notion we state Definition 4 as follows.Definition 8.
A soft setfL over U is called an (M,N)-soft intersection filter(briefly, (M,N)-SIfilter) of L over U if it satisfies(SI1′)
fL(x)⊆~(M,N)fL(1) for all x∈L,(SI2′)
fL(x→y)∩fL(x)⊆~(M,N)fL(y) for all x,y∈L.Proposition 9.
IffL is an (M,N)-SI filter of L over U, then fL*={x∈L∣(fL(x)∩N)∪M=(fL(1)∩N)∪M} is a filter of L.Proof.
Assume thatfL is an (M,N)-SI filter of L over U. Then it is clear that 1∈fL*. For any x,x→y∈fL*, (fL(x)∩N)∪M=(fL(x→y)∩N)∪M=(fL(1)∩N)∪M. By Proposition 7, we have (fL(y)∩N)∪M⊆(fL(1)∩N)∪M. Since fL is an (M,N)-SI filter of L over U, we have
(3)(fL(y)∩N)∪M=((fL(y)∪M)∩N)∪M⊇(fL(x)∩fL(x⟶y)∩N)∪M=((fL(y)∩N)∪M)∩((fL(x⟶y)∩N)∪M)=(fL(1)∩N)∪M.
Hence, (fL(y)∩N)∪M=(fL(1)∩N)∪M, which implies y∈fL*. This shows that fL* is a filter of L.Proposition 10.
If a soft setfL over U is an (M,N)-SI filter of L, then for any x,y,z∈L,(1)
x≤y⇒fL(x)⊆~(M,N)fL(y),(2)
fL(x→y)=fL(1)⇒fL(x)⊆~(M,N)fL(y),(3)
fL(x⊙y)=(M,N)fL(x)∩fL(y)=(M,N)fL(x∧y),(4)
fL(0)=(M,N)fL(x)∩fL(x′),(5)
fL(x→y)∩fL(y→z)⊆~(M,N)fL(x→z),(6)
fL(x)∩fL(y)⊆~(M,N)fL(x⊙z→y⊙z),(7)
fL(x→y)⊆~(M,N)fL((y→z)→(x→z)),(8)
fL(x→y)⊆~(M,N)fL((z→x)→(z→y)).Proof.
(1) Letx,y∈L be such that x≤y. Then x→y=1, and hence
(4)(fL(x)∩N)=(fL(x)∩N)∩(fL(1)∪M)=(fL(y)∩N)∩(fL(x⟶y)∪M)⊆(fL(x)∩fL(x⟶y)∩N)∪M⊆fL(y)∪M,
which implies fL(x)⊆~(M,N)fL(y).
(2) Letx,y∈L be such that fL(x→y)=fL(1). Then,
(5)fL(x)∩N=(fL(x)∩N)∩(fL(1)∪M)=(fL(x)∩N)∩(fL(x⟶y)∪M)⊆(fL(x)∩fL(x⟶y)∩N)∪M⊆fL(y)∪M;
that is, fL(x)⊆~(M,N)fL(y).
(3) By(a3), we have x⊙y≤x∧y for all x,y∈L. By (1), fL(x⊙y)⊆~(M,N)fL(x)∩fL(y). Since x≤y→x⊙y, we obtain fL(x)⊆~(M,N)fL(y→(x⊙y)). It follows from (SI2) that fL(x)∩fL(y)⊆~(M,N)fL(y→(x⊙y))∩fL(y)⊆fL(x⊙y). Hence, fL(x⊙y)=(M,N)fL(x)∩fL(y).
Sincey≤x→y and x⊙(x→y)≤x∧y, we have fL(y)⊆~(M,N)fL(x→y) and fL(x⊙(x→y))⊆~(M,N)fL(x∧y). Hence we have
f
L
(
x
)
∩
f
L
(
y
)
⊆
~
(
M
,
N
)
f
L
(
x
)
∩
f
L
(
x
→
y
)
=
(
M
,
N
)
f
L
(
x
⊙
(
x
→
y
)
)
⊆
~
(
M
,
N
)
f
L
(
x
∧
y
)
⊆
~
(
M
,
N
)
f
L
(
x
)
∩
f
L
(
y
), which implies fL(x)∩fL(y)=(M,N)fL(x∧y). Thus fL(x⊙y)=(M,N)fL(x)∩fL(y)=(M,N)fL(x∧y).
(4) It is a consequence of (3), sincex⊙x′=0.
(5) By(a4).
(6) By(a7).
(7) By(a8).
(8) By(a9).By Definition4 and Proposition 10, we can deduce the following result.Proposition 11.
A soft setfL over U is an (M,N)-SI filter of L over U if and only if it satisfies
(6)(SI3)x⟶(y⟶z)=1⟹fL(x)∩fL(y)⊆~(M,N)fL(z).Proposition 12.
A soft setfL over U is an (M,N)-SI filter of L over U if and only if it satisfies(SI4)∀x,y∈L,x≤y⇒fL(x)⊆~(M,N)fL(y),(SI5)∀x,y∈L,fL(x⊙y)=(M,N)fL(x)∩fL(y).Proof.
(⇒) By Proposition 10(1) and (3).
(⇐) Let x,y∈L. Since x≤1, by (SI3), we have fL(x)⊆~(M,N)fL(1). Hence (SI1′) holds. Since x⊙(x→y)≤y, by (SI3) and (SI4), we have fL(x)∩fL(x→y)=(M,N)fL(x⊙(x→y))⊆~(M,N)fL(y); that is, (SI2′) holds. Therefore, fL is an (M,N)-SI filter of L over U.


## 4.(M,N)-Soft Congruences

In this section, we investigate(M,N)-soft congruences, (M,N)-soft congruences classes, and quotient soft BL-algebras.Definition 13.
A soft relationθ from fL×fL to P(U×U) is called an (M,N)-congruence in L over U×U if it satisfies(C1)θ(1,1)=(M,N)θ(x,x),∀x∈L,(C2)θ(x,y)=(M,N)θ(y,x),∀x∈L,(C3)θ(x,y)∩θ(y,z)⊆~(M,N)θ(x,z),∀x,y,z∈L,(C4)θ(x,y)⊆~(M,N)θ(x⊙z,y⊙z),∀x,y,z∈L,(C5)θ(x,y)⊆~(M,N)θ(x→z,y→z)∩θ(z→x,z→y),∀x,y,z∈L.Definition 14.
Letθ be an (M,N)-congruence in BL-algebra L over U×U and x∈L. Define θx in L as θx(y)=θ(x,y),∀y∈L. The set θx is called an (M,N)-congruence class of x by θ in L. The set L/θ={θx∣x∈L} is called aquotient soft set by θ.Lemma 15.
Ifθ is an (M,N)-congruence in L over U×U, then θ(x,y)⊆~(M,N)θ(1,1),∀x,y∈L.Proof.
By(C1) and (C3), we have θ(1,1)=θ(x,x)⊇~(M,N)θ(x,y)∩θ(y,x)=θ(x,y).Lemma 16.
Ifθ is an (M,N)-congruence in L over U×U, then θ1 is an (M,N)-SI filter of L over U.Proof.
For anyx∈L, we have
(7)θ1(1)=θ(1,1)⊇~(M,N)θ(1,x)=θ1(x).
This proves that (SI1′) holds.
For anyx,y∈L, by (C3) and (C5), we obtain
(8)θ(1,y)⊇~(M,N)θ(1,x⟶y)∩θ(x⟶y,y),θ(x⟶y,y)=θ(x⟶y,1⟶y)⊇~(M,N)θ(x,1).
It follows that
(9)θ(1,y)⊇~(M,N)θ(1,x⟶y)∩θ(x,1)wwwii=θ(1,x)∩θ(1,x⟶y);
that is, θ1(y)⊇~(M,N)θ1(x)∩θ1(x→y). This proves that (SI2′) holds. Thus, θ1 is an (M,N)-SI filter of L over U.Lemma 17.
LetfL be an (M,N)-SI filter of L over U. Then θ(x,y)=fL(x→y)∩fL(y→x) is an (M,N)-soft congruence in L.Proof.
For anyx,y,z∈L, we have the following.(C1) Consider
(10)θf(1,1)=fL(1⟶1)∩fL(1⟶1)=fL(1)=fL(x⟶x)∩fL(x⟶x)=θf(x,x).This proves that (C1) holds.(C2) It is clear that (C2) holds.(C3) By Proposition 10(5), we have
(11)θf(x,y)∩θf(y,z)=(fL(x⟶y)∩fL(y⟶x))∩(fL(y⟶z)∩fL(z⟶y))=(fL(x⟶y)∩fL(y⟶z))∩(fL(y⟶x)∩fL(z⟶y))⊆~(M,N)fL(x⟶z)∩fL(z⟶x)=θf(x,z).Thus (C3) holds.
(
C
4
) Since x→y≤(x⊙z)→(y⊙z) and y→x≤(y⊙z)→(x⊙z), we have
(12)fL(x⟶y)⊆~(M,N)fL((x⊙z)⟶(y⊙z)),fL(y⟶z)⊆~(M,N)fL((y⊙z)⟶(x⊙z)).
Thus, we have
(13)fL(x⟶y)∩fL(y→x)⊆~(M,N)fL((x⊙z)⟶(y⊙z))∩fL((y⊙z)⟶(x⊙z)),
which implies
(14)θf(x,y)⊆~(M,N)θf(x⊙z,y⊙z).
This implies that (C4) holds.
(
C
5
) Finally, we prove condition (C5):
(15)θf(x⟶z,y⟶z)∩θf(z⟶x,z⟶y)=fL((x⟶z)⟶(y⟶z))∩fL((y⟶z)⟶(x⟶z))∩fL((z⟶x)⟶(z⟶y))∩fL((z⟶y)⟶(z⟶x))⊇~(M,N)fL(y⟶x)∩fL(x⟶y)=θf(x,y).
Thus, (C5) holds. Therefore θf is an (M,N)-soft congruence in L.LetfL be an (M,N)-SI filter of L over U and x∈L. In the following, let fx denote the (M,N)-congruence class of x by θf in L and let L/f be the quotient soft set by θf.Lemma 18.
IffL is an (M,N)-SI filter of L over U, then fx=(M,N)fy if and only if fL(x→y)=(M,N)fL(y→x)=(M,N)fL(1) for all x,y∈L.Proof.
IffL is an (M,N)-SI filter of L over U, then fμ(ν)=θfμ(ν)=θf(μ,ν)=fL(μ→ν)∩fL(ν→μ); that is, fμ(ν)=fL(μ→ν)∩fL(ν→μ) for all x,y∈L. If fx=(M,N)fy, then fx(x)=(M,N)fy(x), and hence fL(x→x)=fL(1)=(M,N)fL(y→x)∩fL(x→y). Thus, fL(y→x)=(M,N)fL(x→y)=(M,N)fL(1).
Conversely, assume the given condition holds. By Proposition10, we have fL(x→z)⊇~(M,N)fL(x→y)∩fL(y→z) and fL(y→z)⊇~(M,N)fL(y→x)∩fL(x→z). If fL(y→x)=(M,N)fL(x→y)=(M,N)fL(1), then fL(x→z)⊇(M,N)fL(y→z) and fL(y→z)⊇(M,N)fL(x→z). Thus fL(x→z)=(M,N)fL(y→z). Similarly, we can prove that fL(z→x)=(M,N)fL(z→y). This implies that
(16)fx(z)=fL(x⟶z)∩fL(z⟶x)=(M,N)fL(y⟶z)∩fL(z⟶y)=fLy(z),
for all z∈L. Hence, fx=(M,N)fy.We denoteff(1) by ff(1):={x∈L∣f(x)=(M,N)f(1)}.Corollary 19.
Iff is an (M,N)-SI filter of L over U, then fx=(M,N)fy if and only if x~ff(1)y, where x~ff(1)y if and only if x→y∈ff(1) and y→x∈ff(1).
Letf be an (M,N)-SI filter of L over U. For any fx,fy∈L/f, we define
(17)fx∨fy=(M,N)fx∨y,fx∧fy=(M,N)fx∧y,fx⊙fy=(M,N)fx⊙y,fx⟶fy=(M,N)fx→y.Theorem 20.
Iff is an (M,N)-SI filter of L over U, then L/f=(L/f,∧,∨,′,→,f0,f1) is a BL-algebra.Proof.
We claim that the above operations onL/f are well defined. In fact, if fx=(M,N)fy and fa=(M,N)fb, by Corollary 19, we have x~f(f(1))y and a~ff(1)b, and so x∨a~ff(1)y∨b. Thus fx∨a=(M,N)fy∨b. Similarly, we prove fx∧a=(M,N)fy∧b,fx⊙a=(M,N)fy⊙b, and fx→a=(M,N)fy→b. Then it is easy to see that L/f is a BL-algebra. Especially, we prove the divisibility in L/f as follows. Define a lattice ordered relation “≼(M,N)” on L/f as follows:
(18)fx≼(M,N)fy⟺fx∨fy=(M,N)f1.
By Corollary 19, we have fL(x→y)=(M,N)fL(1). If fx,fy,fz∈L/f, then
(19)fx⊙fy≼(M,N)fz⟺fx⊙y≼(M,N)fz⟺fL((x⊙y)⟶z)=(M,N)fL(1)⟺fL(x→(y⟶z))=(M,N)fL(1)⟺fx≼(M,N)fy→z⟺fx≼(M,N)fy⟶fz.Theorem 21.
IffL is an (M,N)-SI filter of L over U, then L/f≅L/ff(1).Proof.
Defineφ:L→L/f by φ(x)=fx for all x∈L. For any x,y∈L, we have
(20)φ(x∨y)=fx∨y=(M,N)fx∨fy=φ(x)∨φ(y),φ(x∧y)=fx∧y=(M,N)fx∧fy=φ(x)∧φ(y),φ(x⊙y)=fx⊙y=(M,N)fx⊙fy=φ(x)⊙φ(y),φ(x⟶y)=fx→y=(M,N)fx⟶fy=φ(x)⟶φ(y).
Hence, φ is an epic. Moreover, we have
(21)x∈Ker⁡φ⟺φ(x)=f1⟺fx=(M,N)f1⟺x~ff(1)1⟺x∈ff(1),
which shows that L/f≅L/ff(1).


## 5. Conclusions

As a generalization of soft intersection filters of BL-algebras, we introduce the concept of(M,N)-SI (implicative) filters of BL-algebras. We investigate their characterizations. In particular, we describe (M,N)-soft congruences in BL-algebras.To extend this work, one can further investigate(M,N)-SI prime (semiprime) filters of BL-algebras. Maybe one can apply this idea to decision making, data analysis, and knowledge based systems.



---

*Source: 461060-2014-04-01.xml*