# A New Extended Soft Intersection Set toM,N-SI Implicative Fitters of BL-Algebras

Authors: Jianming Zhan; Qi Liu; Hee Sik Kim
Journal: The Scientific World Journal
 (2014)

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## Abstract

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.


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## Body

## 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 [1] 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 [2–5]. At the same time, soft set theory has been applied to algebraic structures, such as [6–8]. 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 [9–13].B
L-algebras, which have been introduced by Hájek [14] as algebraic structures of basic logic, arise naturally in the analysis of the proof theory of propositional fuzzy logic. Turunen [15] 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 [26] applied this theory to BL-algebras. Ma and Kim [27] 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 algebraL=(L,≤,∧,∨,⊙,→,0,1) is a BL-algebra [14] 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 is specified.In anyBL-algebra L, the following statements are true (see [14, 15]):(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,(a10)
x∨y=((x→y)→y)∧((y→x)→x),where x′=x→0.A nonempty subsetA of L is called afilter of L if it satisfies the following conditions: (I1) 1∈A, (I2) for all x∈A, for all 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) for all x,y∈L, x⊙y∈A, (I4) for all x∈A, for all y∈L, x≤y⇒y∈A (see [15]).Now, we call a nonempty subsetA of L an implicative filter if it satisfies (I1) and (I5) x→(z′→y)∈A, y→z∈A⇒x→z∈A.A nonempty subsetA of L is said to be a Boolean filter of L if it satisfies x∨x′∈A, for all x∈A. (see [15–18]).From now on, we letL be an BL-algebra, U an initial universe, E a set of parameters, P(U) the power set of U, and A,B,C⊆E. We let ∅⊆M⊂N⊆U.Definition 1 (see [1]).
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 an approximate 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 [9]).
LetfA,fB∈S(U).(1)
fA is said to be a soft subset of fB and denoted by fA⊆~fB if fA(x)⊆fB(x), for all x∈E. fA and fB are said to be soft equally, 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 [26]).
(
1
) A soft set fL 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.
(
2
) A soft set fL over U is called an SI-implicative filter of L over U if it satisfies (S1) and(S3)
fL(x→(z′→y))∩fl(y→z)⊆fL(x→z), for all x,y,z∈L.In [27], Ma and Kim introduced the concept of (M,N)-SI filters in BL-algebras.Definition 4 (see [27]).
A soft setfS over U is called an (M,N)-soft intersection filter (briefly, (M,N)-SI filter) 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.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⊆~(M,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.Definition 5 (see [27]).
A soft setfS over U is called an (M,N)-soft intersection filter (briefly, (M,N)-SI filter) 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.


## 3.(M,N)-SI 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 setfL 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→(z′→y))∩fL(y→z)∩N⊆~(M,N)fL(y→z)∪M for all x,y,z∈L.Remark 7.
IffL 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 thatU=D2={〈x,y〉∣x2=y2=e,xy=yx}={e,x,y,yx}, dihedral group, is the universe set.
LetL={0,a,b,1}, where 0<a<b<1. Then we define x∧y=min⁡{x,y},x∨y=max⁡{x,y} and ⊙ and → as follows:
(1)⊙0ab100000a0aaab0aab10ab1⟶0ab101111a0111b0b1110ab1
Then (L,∧,∨,⊙,→,1) is a BL-algebra.
LetM={e,y} and N={e,x,y}.
Define a soft setfL 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 setfL over U is called an (M,N)-SI implicative filter of L over U if it satisfies (SI1′) and (SI3′) fL(x→(z′→y))∩fL(y→z)⊆~(M,N)fL(y→z) for all x,y,z∈L.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.
Definex⊙y=min⁡{x,y} and
(2)x⟶y={1,ifx≤y,y,ifx>y.
ThenL=([0,1],∧,∨,⊙,→,0,1) is a BL-algebra.
LetU=L, M={0.5,0.75}, and N={0.5,0.75,1}.
Define a soft setfL over U by
(3)fL(x)={{0,0.5},ifx∈[0,12],{0.5,1},ifx∈[12,1].
Then one can easily check thatfL 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(1/4→1/3)∩N=fL(1)∩fL(1)∩N={0.5,1}∩{0.5,0.75,1}={0.5,1} and fL(2/3→1/4)∪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(1/4→1/3)∩N⊈fL(x→z)∪M.Lemma 12 (see [27]).
If a soft setfL over U is an (M,N)-SI filter of L, then for any x,y,z∈L we have(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(y⊙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)).Theorem 13.
LetfL be an (M,N)-SI filter of L over U, then the following are equivalent:(1)
fL is an (M,N)-SI implicative filter of L,(2)
fL(x→z)⊇~(M,N)fL(x→(z′→z)), for all x,y,z∈L,(3)
fL(x→z)=(M,N)fL(x→(z′→z)), for all x,y,z∈L,(4)
fL(x→z)⊇~(M,N)fL(y→(x→(z′→z)))∩fL(y), for all x,y,z∈L.Proof.
(1)⇒ (2) Assume that fL is an (M,N)-SI filter of L over U. Putting y=z in (SI3), then
(4)fL(x⟶z)∪M=(fL(x⟶z)∪M)∩M⊇(fL(x⟶(z′⟶z))∩fL(z⟶z)∩N)∪M=(fL(x⟶(z′⟶z))∩fL(1)∩N)∪M⊇fL(x⟶(z′⟶z))∩(fL(1)∪M)∩N⊇fL(x⟶(z′⟶z))∩N;
that is, fL(x→z)⊇~(M,N)fL(x→(z′→z)). Thus, (2) holds.
(2)⇒ (3) By (a1) and (a2), x→z≤z′→(x→z)=x→(z′→z); then it follows from Lemma 12 (1) that fL(x→z)⊆~(M,N)fL(x→(z′→z)). Thus, (3) holds.
(3)⇒ (4) Assume that (4) holds. By Lemma 12 (5), we have fL(x⊙z′→y)∩fL(y→z)⊆~(M,N)fL(x⊙z′→z). By (a2), fL(x→(z′→y))∩fL(y→z)⊆~(M,N)fL(x→(z′→z)).
(4)⇒(1) Putting y=1 in (4), we have
(5)fL(x⟶z)⊇~(M,N)fL(x⟶(z′⟶z)).
Hence
(6)fL(z⟶z)⊇~(M,N)fL(x⟶(z′⟶y))∩fL(y⟶z).
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.
LetfL 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(SI4)
fL(x∨x′)=(M,N)fL(1) for all x∈L.Theorem 15.
A soft setfL 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 thatfL over U is an (M,N)-SI Boolean filter of L over U. Then
(7)fL(x⟶z)⊇~(M,N)fL((z∨z′)⟶(x⟶z))∩fL(z∨z′)=(M,N)fL((z∨z′)⟶(x⟶z))∩fL(1)⊇~(M,N)fL((z∨z′)⟶(x⟶z)).
By (a10) and (a1), we have
(8)(z∨z′)⟶(x→z)=(z⟶(x⟶z))∧(z′⟶(x⟶z))=z′⟶(x⟶z)=x⟶(z′⟶z).
Hence fL(x→z)⊇~(M,N)fL(x→(z′→z)). It follows from Theorem 13 that fL is an (M,N)-SI implicative filter of L over U.
Conversely, assume thatfL is an (M,N)-SI implicative filter of L over U. By Theorem 13, we have
(9)fL((x′⟶x)⟶x)=(M,N)fL((x′⟶x)⟶(x′→x))=fL(1),fL((x⟶x′)⟶x′)=(M,N)fL((x⟶x′)⟶(x′′⟶x′))=fL((x⟶x′)⟶(x⟶x))=fL(1).
By Lemma 12, we have
(10)fL(x∨x′)=(M,N)fL((x⟶x′)⟶x′)∩fL((x′⟶x)⟶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.
LetfL be an (M,N)-SI filter of L over U, then the following are equivalent:(1)
fL is an (M,N)-SI implicative (Boolean) filter,(2)
fL(x)=(M,N)fL(x′→x), for all x∈L,(3)
fL((x→y)→x)⊆(M,N)fL(x), for all x,y∈L,(4)
fL((x→y)→x)=(M,N)fL(x), for all x,y∈L,(5)
fL(x)⊇~(M,N)fL(z→((x→y)→x))∩fL(z), for all x,y,z∈L.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(1⟶x)=(M,N)fL(1→(x′⟶x))=fL(x′⟶x).
Thus, (2) holds.
(2)⇒ (3). By (a1), (a2), and (a8), we have x′≤x→y and so (x→y)→x≤x′→x. By Lemma 12,   fL((x→y)→x)⊆~(M,N)fL(x′→x). Combining (2), fL(x)=(M,N)fL(x′→x)⊇~(M,N)fL((x→y)→x). Thus, (3) holds.
(3)⇒ (4). Since x≤(x→y)→x, then by Lemma 12  fL(x)⊆~(M,N)fL((x→y)→x). Combining (3), fL(x)=(M,N)fL((x→y)→x).
(4)⇒ (5). By (SI2), fL((x→y)→x)⊇~(M,N)fL(z→((x→y)→x))∩fL(z). Combining (4), we have fL(x)⊇~(M,N)fL(z→((x→y)→x))∩fL(z). Thus, (5) holds.
(5)⇒ (1). By (a1), z≤x→z. By (a8), (x→z)′≤z′ and so z′→(x→z)≤(x→z)′→(x→z). Then by Lemma 12, fL(z′→(x→z))⊆~(M,N)fL((x→z)′→(x→z))=(M,N)fL(1→((x→z)′→(x→z)))∩fL(1). By (5), fL(x→z)⊇~(M,N)fL(z′→(x→z)) and so fL(x→z)⊇~(M,N)fL(x→(z′→z)). 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).
LetfL 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 x∈L. If fL is an (M,N)-SI implicative (Boolean) filter of L, then so is gL.Proof.
Assuming thatfL is an (M,N)-SI implicative (Boolean) filter of L over U, then fL(x∨x′)=(M,N)fL(1) for all x∈L. By hypothesis, gL(x∨x′)⊇~(M,N)fL(x∨x′)=(M,N)fL(1)=(M,N)gL(1). By (SI1′), we have gL(1)⊇~(M,N)gL(x∨x′). Thus, gL(x∨x′)=(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.



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