ISRN.ALGEBRA ISRN Algebra 2090-6293 Hindawi Publishing Corporation 635783 10.1155/2014/635783 635783 Research Article An Investigation on Algebraic Structure of Soft Sets and Soft Filters over Residuated Lattices Rasouli S. 1 Davvaz B. 2 Kim H. S. Nakatsu T. Yang S. 1 Department of Mathematics Persian Gulf University Bushehr 75169 Iran pgu.ac.ir 2 Department of Mathematics Yazd University Yazd Iran yazduni.ac.ir 2014 1332014 2014 23 09 2013 29 10 2013 13 3 2014 2014 Copyright © 2014 S. Rasouli and B. Davvaz. 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.

We introduce the notion of soft filters in residuated lattices and investigate their basic properties. We investigate relations between soft residuated lattices and soft filter residuated lattices. The restricted and extended intersection (union), and -intersection, cartesian product, and restricted and extended difference of the family of soft filters residuated lattices are established. Also, we consider the set of all soft sets over a universe set U and the set of parameters P with respect to U , S o f t P ( U ), and we study its structure.

1. Introduction

In economics, engineering, environmental science, medical science, and social science, there are complicated problems which to solve them methods in classical mathematics may not be successfully used because of various uncertainties arising in these problems. Alternatively, mathematical theories such as probability theory, fuzzy set theory , rough set theory [2, 3], vague set theory , and the interval mathematics  were established by researchers to modelling uncertainties appearing in the above fields. In 1992, Molodtsov  introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields. At present, works on soft set theory are progressing rapidly. Some authors, for example, Maji et al. , discussed the application of soft set theory to a decision making problem. 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. In theoretical aspects, Maji et al.  and Ali et al.  defined and studied several operations on soft sets. The algebraic structure of the soft sets has been studied by some authors. Aktaş and Çağman  studied the basic concepts of soft set theory and compared soft sets to the related concepts of fuzzy sets and rough sets. Soft set relations are defined and studied in  and some new operations are introduced in . Jun et al.  introduced and investigated the notion of soft d -algebras. Zhan and Jun  studied soft BL-algebras on fuzzy sets. Also, Feng et al.  combined soft sets theory, fuzzy sets, and rough sets. Feng et al.  studied deeply the relation between soft set theory and rough set theory. Recently, Yamak et al. in  introduce and study the notion of soft hyperstructure.

Residuated lattices were introduced in 1924 by Krull in  who discussed decomposition into isolated component ideals. After him, they were investigated by Ward and Dilworth in 1930 s , as the main tool in the abstract study of ideal theory in rings. Residuated lattices are the algebraic counterpart of logics without contraction rule.

An important class of residuated lattices is BL-algebras. BL-algebras constitute the algebraic structures for Hájeks basic logic . MV-algebras, Gödel algebras, and product algebras are particular cases of BL-algebras. Also, there is an interesting connection between MV-algebras and BCK-algebras.

In this paper, we study the concept of soft residuated lattices. The paper is organized in four sections. In Section 2, we gather the definitions and basic properties of residuated lattices and some basic notions relevant to soft set theory will be used in the next sections, and we prove that Sof t P ( U ) is a bounded commutative BCK-algebra with respect to suitable operations. In Section 3, we introduce the notion of soft filters in residuated lattices and study their properties. Section 4 is a conclusion.

2. A Brief Excursion into Residuated Lattices and Soft Sets 2.1. Residuated Lattices

In the following, we recall some basic definitions and properties of residuated lattices and give some examples in this concept.

Definition 1.

A residuated lattice is an algebra 𝔄 = ( A ; , , , , 0,1 ) of type ( 2 , 2 , 2 , 2 , 0 , 0 ) satisfying the following:

( R 1 ) ( A , , , 0 , 1 ) is a bounded lattice,

( R 2 ) ( A , , 1 ) is a commutative monoid,

( R 3 ) and form an adjoint pair; that is, x y z if and only if y x z , for all x , y , z B .

A is called a divisible residuated lattice if it satisfies the following:

(div) x y = x ( x y ) .

A is called an MTL-algebra if it satisfies the following:

(prel) ( x y ) ( y x ) = 1 .

A is called a BL-algebra if it satisfies the div and prel conditions.

A residuated lattice A is nontrivial if and only if 0 1 . We denote the set of natural numbers by ω and define x 0 = 1 and an x n = x n - 1 x for n ω { 0 } . In a bounded residuated lattice the order of x A , x 1 , in symbols ord ( x ) is the smallest n ω such that an x n = 0 ; if no such n exists, then ord ( x ) = . A BL-algebra is called locally finite if all nonunit elements in it has finite order. Also, in a bounded residuated lattice we define a negation,    * , by x * : = x 0 , for all x A . For any bounded residuated lattice A we denote ( x * ) * by x * * . A bounded residuated lattice verifying DN (double negation), that is, x * * = x , condition is also called a “Girard monoid”. An algebra ( A , , * , 0 ) is an MV-algebra if ( A , , 0 ) is a commutative monoid, ( x * ) * = x . x 0 * = 0 * and ( x y * ) * x = ( x * y ) * y , for all x , y A . It is well known that a BL-algebra 𝔄 is an MV-algebra if and only if 𝔄 satisfies the DN . Also, according to , a residuated lattice 𝔄 is an MV-algebra if and only if 𝔄 satisfies the additional condition ( x y ) y = ( y x ) x . Let ( A , , * , 0 ) be an MV-algebra. We define x y = ( x * y * ) * and 1 = 0 * . One can see that ( A , , * , 1 ) is an MV-algebra, too.

Also the structure ( A , * , 0 ) of type ( 2,0 ) is called a BCK-algebra if the following axioms are satisfied for all x , y , z A .

(BCK1) ( ( z * y ) * ( z * x ) ) * ( x * y ) = 0 .

(BCK2) x * 0 = x .

(BCK3) 0 * x = 0 .

(BCK4) x * y = 0 and y * x = 0 imply x = y .

By a bounded BCK-algebra we mean an algebra ( A , * , 0,1 ) , where ( A , * , 0 ) is a BCK-algebra and x * 1 = 0 , for each x A . A commutative BCK-algebra is a BCK-algebra that satisfies the identity x * ( x * y ) = y * ( y * x ) . By , MV-algebras are known to be term-wise equivalent to bounded commutative BCK-algebras.

Let ( A , , * , 1 ) be an MV-algebra. We define x * y = x y * and 0 = 1 * . Then, ( A , * , 0,1 ) is a bounded commutative BCK-algebra in which x y = y * ( 1 * x ) = x * ( 1 * y ) and x * = 1 * x .

Let ( A , * , 1 ) be a bounded commutative BCK-algebra. We define x y = x * ( 1 * y ) and x * = 1 * x . Then, ( A , , * , 1 ) is an MV-algebra in which x * y = x y * .

In the following, we give some examples of residuated lattice.

Example 2.

(i) Assume that R is a commutative ring with unit and let I ( R ) be the collection of all ideals of R . This set, ordered by inclusion, is a lattice. The meet of two ideals is their intersection and their join is the ideal generated by the union. We define multiplication of two ideals I , J in the usual way: (1) I J = { x X , y Y x y : X , Y    are    finite    subsets    of    I , J } . Then, I ( R ) forms a residuated lattice with unit of the ring R itself and divisions given by I J = { k R : I k J } . It was in this setting that residuated lattices were first defined by Ward and Dilworth .

(ii) Define on the real unit interval [ 0,1 ] the binary operations “ ” and “ ” by (2) x y = { 0 , if    x + y 1 2 , x y , otherwise , x y = { 1 , if    x y , max { 1 2 - x , y } , otherwise . Then, ( [ 0,1 ] , max , min , , , 0,1 ) is a bounded residuated lattice.

(iii) Let denote the set of real numbers and denote the set of rationals. Then, the unit interval [ 0,1 ] of endowed with the following operations (3) x    y : = max ( 0 , x + y - 1 ) , x    y : = min ( 1,1 - x + y ) for all x , y [ 0,1 ] becomes an MV-algebra which is called the standard MV-algebra. Also, for each n , if we set S ( n ) = { 0 , 1 / n , , ( n - 1 ) / n , 1 } , S ( ) = [ 0,1 ] , then ( S ( n ) , max , min , n , n , 0,1 ) and ( S ( ) , max , min , , , 0 ,    1 ) are MV-algebras where x n ( n ) y : = ( x    ( ) y ) S ( n ) and x ( ) y : = ( x ( ) y ) S ( ) .

Let A be a residuated lattice and F be a nonempty subset of A . F is called a filter of A if it satisfies the following conditions for all x , y A :

x , y F implies x y F ,

x y and x F imply y F .

Trivial examples of filters are { 1 } and A . We will denote by F ( A ) the set of filters.

Leustean in  introduced the notion of coannihilator of BL-algebras. Let F be a filter of A and x A . The coannihilator of x relative to F is the set ( F , x ) : = { y A x y F } . For any x , y A , we will denote by ( x , y ) the coannihilator ( x , y ) in which x is the generated principle ideal of x .

Proposition 3 (see [<xref ref-type="bibr" rid="B20">24</xref>]).

Let F and G be filters of BL-algebra A and x , y A . Then,

( F , x ) is a filter of A ;

F ( F , x ) ;

x y implies ( F , x ) ( F , y ) ;

F G implies ( F , x ) ( G , x ) ;

( F , x ) = A if and only if x F ;

( F , x ) ( F , y ) = ( F , x y ) = ( F , x y ) ;

( F , x ) ( G , x ) = ( F G , x ) and ( F , x ) ( G , x ) = ( F G , x ) ;

( ( F , x ) , y ) = ( ( F , y ) , x ) = ( F , x y ) .

( x , x ) = A ;

( x , y ) = ( x , x y ) = ( x , x y ) ;

( x , y ) = ( x y , y ) .

For any nonempty subset X of A , the coannihilator of X is the set X : = { y A x y = 1    for    any    x X } . It is easy to see that A = { 1 } and = { 1 } = A . For any subset X of A , ( X ) is denoted by X .

Proposition 4 (see [<xref ref-type="bibr" rid="B20">24</xref>]).

Let X and Y be two nonempty subsets of BL-algebra A , and let { X i } i I be a nonempty family subset of A and F F ( A ) . Then,

X is a filter of A ;

If X Y , then Y X and X Y ;

X X , X = X , X = X ;

X X = { 1 } , F F = { 1 } ;

F is a prime filter if and only if F is a chain and F 1 ;

i I X i = ( i I X i ) .

2.2. Soft Sets

In this subsection, we recall some basic notions relevant to soft set. Let U be an initial universe set and let P U (simply denoted by P ) be the set of parameters with respect to U . Usually, parameters are attributes, characteristics, or properties of the objects in U . The family of all subsets of U is denoted by 𝒫 ( U ) .

Definition 5 (see [<xref ref-type="bibr" rid="B23">6</xref>]).

A pair ( Ϝ , ) is called a soft set over U , when P , and Ϝ : 𝒫 ( U ) is a set-valued mapping.

In , for a soft set ( Ϝ , ) , the set Supp ( Ϝ , ) = { ϵ F ( ϵ ) } is called the support of the soft set ( Ϝ , ) . The soft set ( Ϝ , ) is called nonnull if Supp ( Ϝ , ) , and it is called a relative null soft set (with respect to the parameter set ), denoted by, , if Supp ( Ϝ , ) = . is called the empty soft set over U . The soft set ( Ϝ , ) is called relative whole soft set (with respect to the parameter set ), denoted by U , if Ϝ ( ϵ ) = U , for all, ϵ . U P is called the whole soft set. In the following, for a soft set ( Ϝ , ) by Dom ( Ϝ ) we mean the parameterized set .

For illustration, Molodtsov considered several examples in . These examples were also discussed in [9, 11]. Now, we give an example of a soft set.

Example 6.

Let D = { 0 , x 1 , x 2 , x 3 , x 4 , 1 } . Define on D the following operations:

One can see that 𝔇 = ( D ; , , , , 0,1 ) is a divisible residuated lattice. Furthermore, 0 x 1 x 2 , x 3 x 4 1 , but x 2 , x 3 are incomparable; thus 𝔇 is not a chain. Also, ( x 2 x 3 ) ( x 3 x 2 ) = x 1 so 𝔇 is not an MTL-algebra. Moreover, F ( 𝔇 ) = { F 1 = D , F 2 = { x 1 , x 2 , x 3 , x 4 , 1 } , F 3 = { x 2 , x 4 , 1 } , F 4 = { x 3 , x 4 , 1 } , F 5 = { x 4 , 1 } , and    F 6 = { 1 } } . Now, let = F ( 𝔇 ) and Ϝ : 𝒫 ( D ) which is defined by Ϝ ( F i ) = x σ ( i ) / F i , where σ is a permutation on { 1,2 , 3,4 , 5,6 } . Then, ( Ϝ , ) is a soft set over 𝔇 .

Maji et al. , Feng et al. , and Ali et al.  introduced and investigated several binary operations.

Definition 7 (see [<xref ref-type="bibr" rid="B23">6</xref>]).

Let ( Ϝ 1 , 1 ) and ( Ϝ 2 , 2 ) be two soft sets over a common universe U .

( Ϝ 1 , 1 ) is said to be a soft subset of ( Ϝ 2 , 2 ) and is denoted by ( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) if 1 2 and Ϝ 1 ( ϵ ) Ϝ 2 ( ϵ ) for all ϵ 1 .

( Ϝ 1 , 1 ) and ( Ϝ 2 , 2 ) are said to be soft equal and is denoted by ( Ϝ 1 , 1 ) = ( Ϝ 2 , 2 ) if ( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) and ( Ϝ 2 , 2 ) ~ ( Ϝ 1 , 1 ) .

Example 8.

Consider divisible residuated lattice 𝔇 in Example 6. Let 1 = { x 2 , x 3 } and 2 = { x 1 , x 2 , x 3 } . Now, we define Ϝ 1 : 1 𝒫 ( D ) by Ϝ 1 ( ϵ ) = ( F 5 , ϵ ) and Ϝ 2 : 2 𝒫 ( D ) by Ϝ 2 ( ϵ ) = ( F 4 , ϵ ) for each ϵ i , where i { 1,2 } , F 4 = { x 3 , x 4 , 1 } , and F 5 = { x 4 , 1 } . By Proposition 3, we obtain that ( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) .

In the following, let { ( Ϝ i , i ) : i Λ } be a nonempty family of soft sets over a common universe U .

Definition 9 (see [<xref ref-type="bibr" rid="B12">10</xref>]).

Let { ( Ϝ i , i ) : i Λ } be a family of soft sets over a common universe U .

The restricted intersection (union) of { ( Ϝ i , i ) : i Λ } is defined as the soft set ( Ϝ , ) = ~ i Λ ( F i , i ) ( ( Ϝ , ) = ~ i Λ ( F i , i ) ), where = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ ) ( Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ ) ),  for all ϵ . If = i Λ i = , we define ~ i Λ ( F i , i ) = and ~ i Λ ( F i , i ) = .

The extended intersection (union) of { ( Ϝ i , i ) : i Λ } is defined as the soft set ( Ϝ , ) = ~ i Λ ( F i , i ) ( ( Ϝ , ) = ~ i Λ ( F i , i ) ), where = i Λ i and Ϝ ( ϵ ) = i Λ ( ϵ ) Ϝ i ( ϵ ) ( Ϝ ( ϵ ) = i Λ ( ϵ ) Ϝ i ( ϵ ) ), where Λ ( ϵ ) = { i Λ ϵ i } .

In order to make the above definition more clear, we present the following example.

Example 10.

Consider divisible residuated lattice 𝔇 in Example 6. Let 1 = { x 1 , x 2 , 1 } , 2 = { x 1 , x 3 , x 4 } , and 3 = { 0 , x 1 , x 2 , x 3 , 1 } . Now, we define Ϝ i : i 𝒫 ( D ) with Ϝ i ( ϵ ) = ( F i + 1 , ϵ ) for each ϵ i , where i { 1,2 , 3 } . Now, we assume that = { x 1 } and = D . Also, we suppose that Ϝ : 𝒫 ( D ) in which Ϝ ( ϵ ) = ( F 5 , ϵ ) and Ϝ : 𝒫 ( D ) in which Ϝ ( ϵ ) = ( F 2 , ϵ ) . By Proposition 3, we can obtain that ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , and ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , where Λ = { 1,2 , 3 } . Also, if we let Ϝ = ~ i Λ ( Ϝ i , i ) , we get Ϝ ( 0 ) = F 4 , Ϝ ( x 1 ) = ( F 5 , x 1 ) , Ϝ ( x 2 ) = ( F 3 , x 2 ) , Ϝ ( x 3 ) = ( F 5 , x 3 ) , Ϝ ( x 4 ) = ( F 3 , x 4 ) , and Ϝ ( 1 ) = D .

Definition 11 (see [<xref ref-type="bibr" rid="B4">25</xref>]).

Let { ( Ϝ i , i ) : i Λ } be a family of soft sets over a common universe U .

The -intersection of { ( Ϝ i , i ) : i Λ } is defined as the soft set ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , where = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ i ) ,  for all ϵ = ( ϵ i ) i Λ .

The -intersection of { ( Ϝ i , i ) : i Λ } is defined as the soft set ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , where = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ i ) , for all ϵ = ( ϵ i ) i Λ .

Let ( Ϝ i , i ) be a soft set over universe U i , where i Λ . The cartesian product of { ( Ϝ i , i ) : i Λ } is defined as the soft set ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , where = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ i ) , for all ( ϵ i ) i Λ .

Example 12.

Consider divisible residuated lattice 𝔇 in Example 6. Let 1 = { x 2 } and 2 = { x 3 , x 4 } . Now, we define Ϝ 1 : 1 𝒫 ( D ) with Ϝ 1 ( x 2 ) = ( F 5 , x 2 ) . Also, we define Ϝ 2 : 2 𝒫 ( D ) with Ϝ 1 ( x 3 ) = ( F 5 , x 3 ) and Ϝ 1 ( x 4 ) = ( F 5 , x 4 ) = D . Now, we assume that = 1 × 2 , 1 : 𝒫 ( D ) in which 1 ( ( x 2 , x 3 ) ) = ( F 5 , x 2 x 3 ) = ( F 5 , x 1 ) = F 5 and 1 ( ( x 2 , x 4 ) ) = ( F 5 , x 2 x 4 ) = ( F 5 , x 2 ) = F 4 , 2 : 𝒫 ( D ) in which 2 ( ( x 2 , x 3 ) ) = { x 2 , x 3 , x 4 , 1 } and 2 ( ( x 2 , x 4 ) ) = F 4 , and 3 : 𝒫 ( D ) in which 3 ( ( x 2 , x 3 ) ) = F 4 × F 3 and 1 ( ( x 2 , x 4 ) ) = F 4 × F 5 . Therefore, we have ( 1 , ) = ( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) , ( 2 , ) = ( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) , and ( 3 , ) = ( Ϝ 1 , 1 ) × ( Ϝ 2 , 2 ) .

Definition 13 (see [<xref ref-type="bibr" rid="B12">10</xref>]).

Let ( Ϝ 1 , 1 ) and ( Ϝ 2 , 2 ) be two soft sets over a common universe U such that 1 2 .

The restricted difference of ( Ϝ 1 , 1 ) and ( Ϝ 2 , 2 ) is defined as the soft set ( Ϝ , ) = ( Ϝ 1 , 1 ) ( Ϝ 2 , 2 ) , where = 1 2 and Ϝ ( ϵ ) = Ϝ 1 ( ϵ ) - Ϝ 2 ( ϵ ) , for all ϵ . If 1 2 = , we define ( Ϝ 1 , 1 ) ( Ϝ 2 , 2 ) =

The extended difference of ( Ϝ 1 , 1 ) and ( Ϝ 2 , 2 ) is defined as the soft set ( Ϝ , ) = ( Ϝ 1 , 1 ) ( Ϝ 2 , 2 ) , where = 1 2 , and we have (4) Ϝ ( ϵ ) = { Ϝ 1 ( ϵ ) , if    ϵ 1 - 2 , Ϝ 2 c ( ϵ ) , if    ϵ 2 - 1 , Ϝ 1 ( ϵ ) - Ϝ 2 ( ϵ ) , if    ϵ 1 2 .

Example 14.

Consider Example 10. Let = { x 1 } and = F 2 . We suppose that Ϝ : 𝒫 ( D ) in which Ϝ ( x 1 ) = { 0 , x 1 , x 3 , 1 } and Ϝ : 𝒫 ( D ) in which Ϝ ( x 1 ) = Ϝ ( x 3 ) = { 0 , x 1 , x 3 , 1 } and Ϝ ( x 2 ) = Ϝ ( x 4 ) = Ϝ ( 1 ) = D . Then, we can obtain that ( Ϝ , ) = ( Ϝ 1 , 1 ) ( Ϝ 2 , 2 ) and ( Ϝ , ) = ( Ϝ 1 , 1 ) ( Ϝ 2 , 2 ) .

Definition 15 (see [<xref ref-type="bibr" rid="B12">10</xref>]).

The complement of a soft set ( Ϝ , ) over U is denoted by ( Ϝ , ) c and is defined by ( Ϝ , ) c = ( Ϝ c , ) where Ϝ c : 𝒫 ( U ) is a mapping given by Ϝ c ( ϵ ) = U - Ϝ ( ϵ ) , for all ϵ . Clearly, ( Ϝ , ) c = U P ( Ϝ , ) and ( ( Ϝ , ) c ) c = ( Ϝ , ) .

In the following, the set of all soft sets ( Ϝ , ) over U , in which P and Ϝ : 𝒫 ( U ) is a map, is denoted by Sof t P ( U ) .

Let U be a universal set and let P be the set of parameters with respect to U . One can see that ( Sof t P ( U ) , α , β , U P , ) , where α { ~ , ~ } and β { ~ , ~ } , is a distributive bounded complete lattice if Sof t P ( U ) is closed under α and β . Furthermore, the partial relation defined by lattice operations ( Sof t P ( U ) , ~ , ~ , U P , ) coincides with ~ . Since ( Sof t P ( U ) , ~ , ~ , U P , ) is a distributive bounded complete lattice, we can define a new operation as follows: (5) ( Ϝ 1 , ) ( Ϝ 2 , 2 ) = ~ { ( Ϝ , ) ( Ϝ 1 , 1 ) ~ ( Ϝ , ) ~ ( Ϝ 2 , 2 ) } . Also, we let ( Ϝ , ) * = ( Ϝ , ) . Clearly, we have ( Ϝ , ) * ~ ( Ϝ , ) = . Hence, we obtain the following corollary.

Corollary 16.

Let U be an universal set and let P be the set of parameters with respect to U . Then, S P ( U ) = ( S o f t P ( U ) , ~ , ~ , ~ , , U P , ) is a bounded residuated lattice.

Example 17.

Consider divisible residuated lattice 𝔇 in Example 6. Let 1 = { x 1 , x 2 } , 2 = { x 3 , x 4 } , 3 = { x 3 } , 4 = { 0 , x 1 , x 3 , x 4 } , and 5 = { 0 , x 1 , x 2 , x 3 , x 4 } . Now, we define Ϝ i : i 𝒫 ( D ) with Ϝ i ( ϵ ) = ( F i + 1 , ϵ ) , for i { 1,2 , 4,5 } and ϵ i . Also, let Ϝ 3 : 3 𝒫 ( D ) with Ϝ 3 ( ϵ ) = ϵ , for ϵ 3 . If we let = { x 3 , x 4 } and we consider the soft set Ϝ : 𝒫 ( D ) with Ϝ ( x 3 ) = F 5 and Ϝ ( x 4 ) = D , then we have ( Ϝ 1 , 1 ) * = ( Ϝ , ) . Also, it is clear that ( Ϝ 1 , 1 ) ~ ( Ϝ 1 , 1 ) * U P .

In the next proposition, we show that S P ( U ) is a divisible residuated lattice.

Proposition 18.

Let U be an universal set and let P be the set of parameters with respect to U . Then, for each soft sets ( Ϝ 1 , 1 ) , ( Ϝ 2 , 2 ) S o f t P ( U ) , we have

( Ϝ , ) = U P ,

( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) = ( Ϝ 1 , 1 ) ~ ( ( Ϝ 1 , 1 ) ( Ϝ 2 , 2 ) ) .

Proof.

(i) It is obvious.

(ii) First we show that 1 2 = 1 ( { 1 2 } ) . Obviously, We have 1 ( { 1 2 } ) 1 2 . On the other hand, 2 { 1 2 } , so 1 2 1 ( { 1 2 } ) and it shows the equality. Similarly, we can show that, for each ϵ 1 2 , we get ( Ϝ 1 Ϝ 2 ) ( ϵ ) = Ϝ 1 ( ϵ ) ( ϵ Dom ( Ϝ ) { Ϝ ( ϵ ) ( Ϝ 1 Ϝ ) ( ϵ ) Ϝ 2 ( ϵ ) } ) . Hence, the result holds.

Example 19.

Consider Example 17. Clearly, we have ( Ϝ 1 , 1 ) ( Ϝ 2 , 2 ) = ( Ϝ 2 , 2 ) and ( Ϝ 2 , 2 ) ( Ϝ 1 , 1 ) = ( Ϝ 1 , 1 ) . Hence, S P ( U ) is not an MTL-algebra. Also, ( Ϝ 3 , 3 ) * = ( Ϝ 1 , 1 ) and it implies that S P ( U ) is not a Girard monoid. Therefore, S P ( U ) is not an MV-algebra.

Recently, Ali et al. in  show that ( Sof t P ( U ) , ~ , c , P ) is an MV-algebra. Also, they show directly that ( Sof t P ( U ) , ~ , c , U P ) is an MV-algebra and ( Sof t P ( U ) , , P , U P ) is a bounded BCK-algebra whose every element is an involution.

By , if ( Ϝ 1 , 1 ) and ( Ϝ 2 , 2 ) are two soft sets over the same universe U , then (6) ( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) = ( ( Ϝ 1 , 1 ) c ~ ( Ϝ 2 , 2 ) c ) c . So ( Sof t P ( U ) , ~ , c , U P ) is an MV-algebra.

Lemma 20.

Let ( Ϝ 1 , 1 ) and ( Ϝ 2 , 2 ) be two soft sets over the same universe U . Then,

( Ϝ 1 , 1 ) ( Ϝ 2 , 2 ) = ( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) c .

Proof.

Let ( Ϝ , ) = ( Ϝ 1 , 1 ) ~ ( Ϝ 2 , 2 ) c . Then, = 1 2 , and for each ϵ we have Ϝ ( ϵ ) = Ϝ 1 ( ϵ ) Ϝ 2 ( ϵ ) c = Ϝ 1 ( ϵ ) Ϝ 2 ( ϵ ) . The result holds.

Corollary 21.

Let U be an universal set and let P be the set of parameters with respect to U . Then, ( S o f t P ( U ) , , P ) is a bounded commutative BCK-algebra.

Proof.

By Lemma 20, it is straightforward.

3. Soft Residuated Lattices and Soft Filters

In what follows, let R and be a residuated lattice and a nonempty set, respectively, and will refer to a arbitrary relation between an element of and an element of R ; that is is a subset of × R unless otherwise specified. A set valued function Ϝ : 𝒫 ( R ) can be defined as Ϝ ( ϵ ) : = { r R ( ϵ , r ) } , for all ϵ . Obviously, the pair ( Ϝ , ) is a soft set over R which is derived from the relation .

Definition 22.

Let ( Ϝ , ) be a nonnull soft set over residuated lattice R . Then, ( Ϝ , ) is called a soft residuated lattice over R , if Ϝ ( ϵ ) is a subalgebra of R for each ϵ Supp ( Ϝ , ) .

Example 23.

Let R 5 = { 0 , x 1 , x 2 , x 3 , 1 } . Define on R 5 the following operations:

Then, R 5 is a distributive divisible residuated lattice which it is not an MTL-algebra since ( x 1 x 2 ) ( x 2 x 1 ) = x 3 1 . Now, let ( Ϝ , ) be a soft set over residuated lattice R 5 , where = R 5 and Ϝ : 𝒫 ( R 5 ) is a set valued function defined by (7) Ϝ ( x ) = { y R 5 y x } { 0 } for all x . Then, Ϝ ( 0 ) = R 5 , Ϝ ( x 1 ) = { 0 , x 1 , x 3 , 1 } , Ϝ ( x 2 ) = { 0 , x 2 , 1 } , Ϝ ( x 3 ) = { 0 , x 3 , 1 } and Ϝ ( 1 ) = { 0,1 } . So Ϝ ( ϵ ) is a subalgebra of R 5 , for each ϵ R 5 . Therefore, ( Ϝ , ) is a soft residuated lattice over R 5 .

Example 24.

Consider the standard MV-algebra [ 0,1 ] . Let = . We define Ϝ : 𝒫 ( [ 0,1 ] ) , in which Ϝ ( ϵ ) = 𝕊 ϵ . Obviously, the pair ( Ϝ , ) is a soft MV-algebra over [ 0,1 ] .

Definition 25.

Let ( Ϝ , ) be a soft residuated lattice over residuated lattice R . A nonnull soft set ( Ϝ , ) is called a soft filter of ( Ϝ , ) if it satisfies the following conditions:

(SF1) ( Ϝ , ) ~ ( Ϝ , ) ;

(SF2) Ϝ ( ϵ ) is a filter of Ϝ ( ϵ ) , for each ϵ Supp ( Ϝ , ) .

Example 26.

Let ( A , , , , , 1 ) be a residuated lattice. Suppose that = A . We set Ϝ : 𝒫 ( A ) given by Ϝ ( ϵ ) = ϵ . By Proposition 4, ( Ϝ , ) is a soft filter over the soft residuated lattice ( , P ) .

Example 27.

Consider soft residuated lattice ( Ϝ , ) over R 5 in Example 23. We define Ϝ : { x 1 , x 2 , x 3 , 1 } 𝒫 ( A ) in which we have Ϝ ( x ) = { y R 5 y x } . Therefore, one can see that soft set ( Ϝ , ) is a soft filter of ( Ϝ , ) .

Let ( A , , , , , 1 ) be a residuated lattice and let F be a filter of A and x , y F . Then, x    y x y x y , x y and x y F imply that F is a subalgebra of A . Thus, each soft filter of soft residuated lattice ( Ϝ , ) is a soft residuated lattice.

Theorem 28.

Let ( Ϝ , ) be a soft residuated lattice over A and let { ( Ϝ i , i ) i Λ } be a nonempty family of soft filter of ( Ϝ , ) . Then, the restricted intersection ~ i Λ ( Ϝ i , i ) and the extended intersection ~ i Λ ( ϵ ) ( Ϝ i , i ) are soft filters of ( Ϝ , ) if they are nonnull.

Proof.

Suppose that { ( Ϝ i , i ) i Λ } is a nonempty family of soft filters of ( Ϝ , ) . By Definition 9, let ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) where = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ ) , for all ϵ . Clearly, we have . Now, let ϵ Supp ( Ϝ , ) , so ϵ Supp ( Ϝ i , i ) for each i Λ . By hypothesis Ϝ i ( ϵ ) is a filter of Ϝ ( ϵ ) , for each i Λ and ϵ Supp ( Ϝ , ) . It follows that Ϝ ( ϵ ) is a filter of Ϝ ( ϵ ) . Hence, ( Ϝ , ) is a soft filter of ( Ϝ , ) . Similarly, we can show that the extended intersection ~ i Λ ( Ϝ i , i ) is a soft filter of ( Ϝ , ) .

Theorem 29.

Let ( Ϝ , ) be a soft residuated lattice over A and let { ( Ϝ i , i ) i Λ } be a nonempty family of soft filters of ( Ϝ , ) . Then, the restricted union ~ i Λ ( Ϝ i , i ) and the extended union ~ i Λ ( ϵ ) ( Ϝ i , i ) are soft filters of ( Ϝ , ) if Ϝ i ( ϵ ) Ϝ j ( ϵ ) or Ϝ j ( ϵ ) Ϝ i ( ϵ ) , for all i , j Λ ( ϵ ) , and they are nonnull.

Proof.

Suppose that { ( Ϝ i , i ) i Λ } is a nonempty family of soft filters of ( Ϝ , ) . By Definition 9, let ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) where    = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ ) , for all ϵ . Clearly, we have ( Ϝ , ) ~ ( Ϝ , ) . First, note that ( Ϝ , ) is nonnull since Supp ( Ϝ , ) = i Λ Supp ( Ϝ i , i ) . Let ϵ Supp ( Ϝ , ) . Then, Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ ) and so we have Ϝ j ( ϵ ) for some j Λ ( ϵ ) . Now, let x , y Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ ) . Therefore, x Ϝ i ( ϵ ) and y Ϝ j ( ϵ ) , for some i , j Λ ( ϵ ) . We can suppose that Ϝ i ( ϵ ) Ϝ j ( ϵ ) . So by hypothesis we obtain that x y Ϝ j ( ϵ ) Ϝ ( ϵ ) . On the other hand, let x y , x Ϝ ( ϵ ) , and y Ϝ ( ϵ ) . Then, x Ϝ i ( ϵ ) , for some i Λ ( ϵ ) , and it implies that y Ϝ i ( ϵ ) Ϝ ( ϵ ) . Thus, Ϝ ( ϵ ) is a filter of Ϝ ( ϵ ) , for each ϵ Supp ( Ϝ , ) . Hence, ( Ϝ , ) is a soft filter of ( Ϝ , ) .

Let ( Ϝ , ) be a soft residuated lattice over A and let { ( Ϝ i , i ) i Λ } be a nonempty family of soft filters of ( Ϝ , ) . Assume that i j = for all i , j Λ , i j . Therefore, Λ ( ϵ ) has only one element. Let Λ ( ϵ ) = { i ϵ } . So for each ϵ , Ϝ ( ϵ ) = Ϝ i ϵ ( ϵ ) , and it implies that, for each i , j Λ and ϵ , either Ϝ i ( ϵ ) = or Ϝ j ( ϵ ) = . Hence, by Theorem 29 we can conclude that the union ~ i Λ ( Ϝ i , I i ) is a soft filter of ( Ϝ , ) .

In the following, we give an example to see that Theorem 29 is not established in general.

Example 30.

Consider divisible residuated lattice 𝔇 in Example 6. Let 1 = { x 2 , x 3 } and 2 = { x 1 , x 2 , x 3 } . Now, we define Ϝ 1 : 1 𝒫 ( D ) by Ϝ 1 ( ϵ ) = F 3 and Ϝ 2 : 2 𝒫 ( D ) by Ϝ 2 ( ϵ ) = F 4 for each ϵ i , where i { 1,2 } . Obviously, the restricted union ~ i Λ ( Ϝ i , i ) and the extended union ~ i Λ ( ϵ ) ( Ϝ i , i ) are not soft filters of U , where = { x 2 , x 3 , x 4 , 1 } .

Theorem 31.

Let ( Ϝ , ) be a soft residuated lattice over A and let { ( Ϝ i , i ) i Λ } be a nonempty family of soft filters of ( Ϝ , ) . Then, the -intersection ~ i Λ ( Ϝ i , i ) is a soft filter of ~ i Λ ( Ϝ , ) if ~ i Λ ( Ϝ i , i ) is nonnull.

Proof.

By Definition 11, we can let ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , where = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ i ) , for all ϵ = ( ϵ i ) i Λ . Also, we let ( G , p ) = ~ i Λ ( Ϝ , ) , where p = i Λ and G ( ϵ ) = i Λ Ϝ ( ϵ i ) , for all ϵ = ( ϵ i ) i Λ p . Clearly, we have ( Ϝ , ) ~ ( G , p ) . Now, let ( Ϝ , ) be nonnull and ϵ = ( ϵ i ) i Λ Supp ( Ϝ , ) . We have Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ i ) , and since each Ϝ i ( ϵ i ) is a filter of Ϝ ( ϵ i ) , we obtain that Ϝ ( ϵ ) is a filter of G ( ϵ ) .

Theorem 32.

Let ( Ϝ , ) be a soft residuated lattice over A and let { ( Ϝ i , i ) i Λ } be a nonempty family of soft filters of ( Ϝ , ) . If Ϝ i ( ϵ ) Ϝ j ( ϵ ) or Ϝ j ( ϵ ) Ϝ i ( ϵ ) , for all i , j Λ and ϵ i Λ i , then the -intersection ~ i Λ ( Ϝ i , i ) is a soft filter of ~ i Λ ( Ϝ , ) if ( Ϝ i i ) is nonnull, for some i Λ .

Proof.

Suppose that ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , in which = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ i ) , for all ϵ = ( ϵ i ) i Λ . Also, we let ( G , P ) = ~ i Λ ( Ϝ , ) , where P = i Λ and G ( ϵ ) = i Λ Ϝ ( ϵ i ) , for all ϵ = ( ϵ i ) i Λ P . Clearly, we have ( Ϝ , ) ~ ( G , P ) . Now, let ( Ϝ j , j ) be nonnull, for some j Λ . Assume that ϵ j Supp ( Ϝ j , j ) . Obviously, ( Ϝ , ) is a nonnull soft set because, for each ϵ , where π j ( ϵ ) = ϵ j , we have ϵ Supp ( Ϝ , ) . Now, suppose that ϵ Supp ( Ϝ , ) . We have Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ i ) . Similar to the proof of Theorem 29, we can obtain that Ϝ ( ϵ ) is a filter of G ( ϵ ) . Hence, the -intersection ~ i Λ ( Ϝ i , i ) is a soft filter of ~ i Λ ( Ϝ , ) .

Theorem 33.

Let ( Ϝ , ) be a soft residuated lattice over A and let { ( Ϝ i , j ) i Λ } be a nonempty family nonnull soft filters of ( Ϝ , ) . Then, the cartesian product ~ i Λ ( Ϝ i , i ) is a soft filter of ~ i Λ ( Ϝ , ) .

Proof.

Suppose that ( Ϝ , ) = ~ i Λ ( Ϝ i , i ) , in which = i Λ i and Ϝ ( ϵ ) = i Λ Ϝ i ( ϵ i ) , for all ϵ = ( ϵ i ) i Λ . Also, let ( H , P ) = ~ i Λ ( Ϝ , ) , where P = i Λ and H ( ϵ ) = i Λ Ϝ ( ϵ i ) , for all ϵ = ( ϵ i ) i Λ P . Clearly, we have ( Ϝ , ) ~ ( H , P ) . By , for each ϵ P , H ( ϵ ) is a residuated lattice so ( H , P ) is a soft residuated lattice over i Λ A . Now, let ϵ = ( ϵ i ) i Λ i Λ Supp ( Ϝ i , i ) . Therefore, by  Ϝ ( ϵ ) is a filter of H ( ϵ ) . Hence, ~ i Λ ( Ϝ i , i ) is a soft filter of ~ i Λ ( Ϝ , ) on i Λ Supp ( Ϝ i , i ) .

In the following, we study the connection between soft sets and residuated lattice homomorphisms. Also, we consider a function between two soft residuated lattices and investigate its properties.

Proposition 34.

Let f : A A be a residuated lattice homomorphism. If ( Ϝ , ) is a soft filter over A , then ( f - 1 ( Ϝ ) , ) is a soft filter over A . Also, if f is onto and ( Ϝ , ) is a soft filter over A , then ( f ( Ϝ ) , ) is a soft filter over A .

Proof.

Since ( Ϝ , ) is a nonnull soft set by Definition 25 and ( Ϝ , ) is a soft filter over A , we observe that ( f - 1 ( Ϝ ) , ) is a nonnull soft set over A . We see that, for all ϵ Supp ( f - 1 ( Ϝ ) , ) , f - 1 ( Ϝ ) ( ϵ ) = f - 1 ( Ϝ ( ϵ ) ) . Since the nonempty set Ϝ ( ϵ ) is a filter of A and f is a homomorphism, so f - 1 ( Ϝ ( ϵ ) ) is a filter of A . Therefore, f - 1 ( Ϝ ( ϵ ) ) is a filter of A for all ϵ Supp ( f - 1 ( Ϝ ) , ) . Consequently, ( f - 1 ( Ϝ ) , ) is a soft filter over A .

Similarly, we can show that ( f ( Ϝ ) , ) is a soft filter over A where ( Ϝ , ) is a soft filter over A and f is an epimorphism.

A soft filter ( Ϝ , ) over a residuated lattice A is said to be trivial if Ϝ ( ϵ ) = { 1 } for every ϵ . A soft filter ( Ϝ , ) over A is said to be whole if Ϝ ( ϵ ) = A for each ϵ .

Theorem 35.

Let ( Ϝ , ) be a soft filter over A and let f : A A be a residuated lattice epimorphism.

If Ϝ ( ϵ ) = ker f for all ϵ , then ( f ( Ϝ ) , ) is the trivial soft filter over A .

If ( Ϝ , ) is whole, then ( f ( Ϝ ) , ) is the whole soft filter over A .

Proof.

(i) Suppose that Ϝ ( ϵ ) = ker f for all ϵ . Then, f ( Ϝ ( ϵ ) ) = 1 A for all ϵ . So, ( f ( Ϝ ) , ) is the trivial soft filter over A .

(ii) Assume that ( Ϝ , ) is whole. Then, Ϝ ( ϵ ) = A for each ϵ . Hence, f ( Ϝ ( ϵ ) ) = A for all ϵ . Hence, the result holds.

4. Conclusion

In this study, we have proposed the new concept of soft residuated lattice and have introduced their initial basic properties such as soft filters by using soft set theory. Also, the study of algebraic structures of soft sets with respect to new operations gives us a deep insight into their application. In fact, we establish a connection between the set of all soft set on a common universe and its lattice structure. It also provides new examples of these structures on the other hand. Residuated lattices, MV-algebras, and BCK-algebras of soft sets are indicated towards possible applications of soft sets in classical and nonclassical logic. To extend this work, one could study the properties of soft sets in other algebraic structures.

Conflict of Interests

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

Zadeh L. A. Fuzzy sets Information and Computation 1965 8 338 353 MR0219427 ZBL0139.24606 Pawlak Z. Rough sets International Journal of Computer and Information Sciences 1982 11 5 341 356 10.1007/BF01001956 MR703291 ZBL0525.04005 Pawlak Z. Skowron A. Rudiments of rough sets Information Sciences 2007 177 1 3 27 10.1016/j.ins.2006.06.003 MR2272732 ZBL1142.68549 Gau W. L. Buehrer D. J. Vague sets IEEE Transactions on Systems, Man and Cybernetics 1993 23 2 610 614 10.1109/21.229476 Gorzałczany M. B. A method of inference in approximate reasoning based on interval-valued fuzzy sets Fuzzy Sets and Systems 1987 21 1 1 17 10.1016/0165-0114(87)90148-5 MR868351 ZBL0635.68103 Molodtsov D. Soft set theory—first results Computers & Mathematics with Applications 1999 37 4-5 19 31 10.1016/S0898-1221(99)00056-5 MR1677178 ZBL0936.03049 Maji P. K. Roy A. R. Biswas R. An application of soft sets in a decision making problem Computers & Mathematics with Applications 2002 44 8-9 1077 1083 10.1016/S0898-1221(02)00216-X MR1937568 ZBL1044.90042 Chen D. Tsang E. C. C. Yeung D. S. Wang X. The parameterization reduction of soft sets and its applications Computers & Mathematics with Applications 2005 49 5-6 757 763 10.1016/j.camwa.2004.10.036 MR2135211 ZBL1074.03510 Maji P. K. Biswas R. Roy A. R. Soft set theory Computers & Mathematics with Applications 2003 45 4-5 555 562 10.1016/S0898-1221(03)00016-6 MR1968545 ZBL1032.03525 Ali M. I. Feng F. Liu X. Min W. K. Shabir M. On some new operations in soft set theory Computers & Mathematics with Applications 2009 57 9 1547 1553 10.1016/j.camwa.2008.11.009 MR2509967 ZBL1186.03068 Aktaş H. Çağman N. Soft sets and soft groups Information Sciences 2007 177 13 2726 2735 10.1016/j.ins.2006.12.008 MR2323240 ZBL1119.03050 Babitha K. V. Sunil J. J. Soft set relations and functions Computers & Mathematics with Applications 2010 60 7 1840 1849 10.1016/j.camwa.2010.07.014 MR2719702 ZBL1205.03060 Ali M. I. Shabir M. Naz M. Algebraic structures of soft sets associated with new operations Computers & Mathematics with Applications 2011 61 9 2647 2654 10.1016/j.camwa.2011.03.011 MR2795014 ZBL1221.03056 Jun Y. B. Lee K. J. Park C. H. Soft set theory applied to ideals in d -algebras Computers & Mathematics with Applications 2009 57 3 367 378 10.1016/j.camwa.2008.11.002 MR2488608 Zhan J. Jun Y. B. Soft B L -algebras based on fuzzy sets Computers & Mathematics with Applications 2010 59 6 2037 2046 10.1016/j.camwa.2009.12.008 MR2595978 Feng F. Li C. Davvaz B. Ali M. I. Soft sets combined with fuzzy sets and rough sets: a tentative approach Soft Computing 2010 14 9 899 911 10.1007/s00500-009-0465-6 ZBL1201.03046 Feng F. Liu X. Leoreanu-Fotea V. Jun Y. B. Soft sets and soft rough sets Information Sciences 2011 181 6 1125 1137 10.1016/j.ins.2010.11.004 MR2765314 ZBL1211.68436 Yamak S. Kazancı O. Davvaz B. Soft hyperstructure Computers & Mathematics with Applications 2011 62 2 797 803 10.1016/j.camwa.2011.06.009 MR2817916 ZBL1228.03035 Krull W. Axiomatische Begrundung der allgemeinen Idealtheo rie Sitzungsberichte der Physikalisch Medizinischen Societat der Erlangen 1924 56 47 63 Hájek P. Metamathematics of Fuzzy Logic 1998 4 Dodrecht, The Netherlands Kluwer Academic Trends in Logic 10.1007/978-94-011-5300-3 MR1900263 Turunen E. Mathematics behind Fuzzy Logic 1999 Heidelberg, Germany Physica Advances in Soft Computing MR1716958 Mundici D. MV-algebras are categorically equivalent to bounded commutative BCK-algebras Mathematica Japonica 1986 31 6 889 894 MR870978 ZBL0633.03066 Ward M. Dilworth R. P. Residuated lattices Transactions of the American Mathematical Society 1939 45 3 335 354 10.2307/1990008 MR1501995 ZBL0021.10801 Leustean L. Representations of many-valued algebras [Ph.D. thesis] 2003 Bucharest, Romania Faculty of Mathematics and Computer Science, University of Bucharest Feng F. Jun Y. B. Zhao X. Soft semirings Computers & Mathematics with Applications 2008 56 10 2621 2628 10.1016/j.camwa.2008.05.011 MR2460071 ZBL1165.16307