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.

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 Section 2, 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 algebra L=(L,≤,∧,∨,⊙,→,0,1) is a BL-algebra [1] 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, z≤x→y if and only if x⊙z≤y for all x,y,z∈L;

x∧y=x⊙(x→y);

(x→y)∨(y→x)=1.

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

In any BL-algebra L, the following statements are true (see [1, 5, 6]):

x≤y⇔x→y=1;

x→(y→z)=(x⊙y)→z=y→(x→z);

x⊙y≤x∧y;

x→y≤(z→x)→(z→y),x→y≤(y→z)→(x→z);

x→x′=x′′→x;

x∨x′=1⇒x∧x′=0;

(x→y)⊙(y→z)≤x→z;

x≤y⇒x→z≥y→z;

x≤y⇒z→x≤z→y,

where x′=x→0.

A nonempty subset A of L is called a filter of L if it satisfies the following conditions:

1∈A,

∀x∈A, ∀y∈L, x→y∈A⇒y∈A.

It is easy to check that a nonempty subset A of L is a filter of L if and only if it satisfies

∀x,y∈L, x⊙y∈A,

∀x∈A, ∀y∈L, x≤y⇒y∈A (see [6]).

From now on, we let L 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 [<xref ref-type="bibr" rid="B14">13</xref>, <xref ref-type="bibr" rid="B4">16</xref>]).

A soft set fA 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 [<xref ref-type="bibr" rid="B4">16</xref>]).

Let fA,fB∈S(U).

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 equal, 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=fA∪B, where fA∪B(x)=fA(x)∪fB(x), for all x∈E.

The intersection of fA 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 [<xref ref-type="bibr" rid="B10">27</xref>]).

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

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

fL(x)∩N⊆fL(1)∪M for all x∈L,

fL(x→y)∩fL(x)∩N⊆fL(y)∪M for all x,y∈L.

Remark 5.

If fL 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 that U=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 set fL 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 set fL 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 x∈L,

fL(x→y)∩fL(x)⊆~(M,N)fL(y) for all x,y∈L.

Proposition 9.

If fL 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 that fL 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 set fL over U is an (M,N)-SI filter of L, then for any x,y,z∈L,

x≤y⇒fL(x)⊆~(M,N)fL(y),

fL(x→y)=fL(1)⇒fL(x)⊆~(M,N)fL(y),

fL(x⊙y)=(M,N)fL(x)∩fL(y)=(M,N)fL(x∧y),

fL(0)=(M,N)fL(x)∩fL(x′),

fL(x→y)∩fL(y→z)⊆~(M,N)fL(x→z),

fL(x)∩fL(y)⊆~(M,N)fL(x⊙z→y⊙z),

fL(x→y)⊆~(M,N)fL((y→z)→(x→z)),

fL(x→y)⊆~(M,N)fL((z→x)→(z→y)).

Proof.

(1) Let x,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) Let x,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).

Since y≤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

fL(x)∩fL(y)⊆~(M,N)fL(x)∩fL(x→y)=(M,N)fL(x⊙(x→y))⊆~(M,N)fL(x∧y)⊆~(M,N)fL(x)∩fL(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), since x⊙x′=0.

(5) By (a4).

(6) By (a7).

(7) By (a8).

(8) By (a9).

By Definition 4 and Proposition 10, we can deduce the following result.

Proposition 11.

A soft set fL 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 set fL 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.

In this section, we investigate (M,N)-soft congruences, (M,N)-soft congruences classes, and quotient soft BL-algebras.

Definition 13.

A soft relation θ 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.

Let θ 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 a quotient soft set by θ.

Lemma 15.

If θ 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.

If θ is an (M,N)-congruence in L over U×U, then θ1 is an (M,N)-SI filter of L over U.

Proof.

For any x∈L, we have
(7)θ1(1)=θ(1,1)⊇~(M,N)θ(1,x)=θ1(x).
This proves that (SI1′) holds.

For any x,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.

Let fL 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.

(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.

(C4) 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.

(C5) 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.

Let fL 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.

If fL 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.

If fL 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 Proposition 10, 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 denote ff(1) by ff(1):={x∈L∣f(x)=(M,N)f(1)}.

Corollary 19.

If f 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).

Let f 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.

If f 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 on L/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.

If fL is an (M,N)-SI filter of L over U, then L/f≅L/ff(1).

Proof.

Define φ: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∈Kerφ⟺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.

Conflict of Interests

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

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