An Investigation on Algebraic Structure of Soft Sets and Soft Filters ...

Hindawi Publishing Corporation ISRN Algebra Volume 2014, Article ID 635783, 8 pages http://dx.doi.org/10.1155/2014/635783

Research Article An Investigation on Algebraic Structure of Soft Sets and Soft Filters over Residuated Lattices S. Rasouli1 and B. Davvaz2 1 2

Department of Mathematics, Persian Gulf University, Bushehr 75169, Iran Department of Mathematics, Yazd University, Yazd, Iran

Correspondence should be addressed to B. Davvaz; [email protected] Received 23 September 2013; Accepted 29 October 2013; Published 13 March 2014 Academic Editors: H. S. Kim, T. Nakatsu, and S. Yang 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, Soft𝑃 (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 [1], rough set theory [2, 3], vague set theory [4], and the interval mathematics [5] were established by researchers to modelling uncertainties appearing in the above fields. In 1992, Molodtsov [6] 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. [7], discussed the application of soft set theory to a decision making problem. Chen et al. [8] 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. [9] and Ali et al. [10] defined and studied several operations on soft sets. The algebraic structure of the soft sets has been studied by some authors. Aktas¸ and C ¸ a˘gman

[11] 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 [12] and some new operations are introduced in [13]. Jun et al. [14] introduced and investigated the notion of soft 𝑑-algebras. Zhan and Jun [15] studied soft BL-algebras on fuzzy sets. Also, Feng et al. [16] combined soft sets theory, fuzzy sets, and rough sets. Feng et al. [17] studied deeply the relation between soft set theory and rough set theory. Recently, Yamak et al. in [18] introduce and study the notion of soft hyperstructure. Residuated lattices were introduced in 1924 by Krull in [19] who discussed decomposition into isolated component ideals. After him, they were investigated by Ward and Dilworth in 1930s, 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 H´ajeks basic logic [20]. MV-algebras, G¨odel algebras, and product algebras are particular cases of BL-algebras. Also, there is an interesting connection between MV-algebras and BCKalgebras. 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

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lattices and some basic notions relevant to soft set theory will be used in the next sections, and we prove that Soft𝑃 (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: (𝑅1 ) (𝐴, ∨, ∧, 0, 1) is a bounded lattice, (𝑅2 ) (𝐴, ⊙, 1) is a commutative monoid, (𝑅3 ) ⊙ 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 𝑥0 = 1 and an 𝑥𝑛 = 𝑥𝑛−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 𝑥𝑛 = 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 [21], 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 BCKalgebra 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 BCKalgebra that satisfies the identity 𝑥 ∗ (𝑥 ∗ 𝑦) = 𝑦 ∗ (𝑦 ∗ 𝑥). By [22], MV-algebras are known to be term-wise equivalent to bounded commutative BCK-algebras. (a) 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. (b) 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: } { 𝐼 ⊙ 𝐽 = { ∑ 𝑥𝑦 : 𝑋, 𝑌 are finite subsets of 𝐼, 𝐽} . (1) } {𝑥∈𝑋,𝑦∈𝑌 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 [23]. (ii) Define on the real unit interval [0, 1] the binary operations “⊙” and “ → ” by 1 {0, if 𝑥 + 𝑦 ≤ , 𝑥⊙𝑦={ 2 𝑥 ∧ 𝑦, otherwise, {

(2) 1, if 𝑥 ≤ 𝑦, { 𝑥 󳨀→ 𝑦 = { 1 max { − 𝑥, 𝑦} , otherwise. 2 { Then, ([0, 1], max, min, ⊙, → , 0, 1) is a bounded residuated lattice. (iii) Let R denote the set of real numbers and Q denote the set of rationals. Then, the unit interval [0, 1] of R endowed with the following operations 𝑥⊙R 𝑦 := max (0, 𝑥 + 𝑦 − 1) , 𝑥󳨀→R 𝑦 := min (1, 1 − 𝑥 + 𝑦)

(3)

for all 𝑥, 𝑦 ∈ [0, 1] becomes an MV-algebra which is called the standard MV-algebra. Also, for each 𝑛 ∈ N, if we set 𝑆(𝑛) = {0, 1/𝑛, . . . , (𝑛 − 1)/𝑛, 1}, 𝑆(Q) = Q ∩ [0, 1], then (𝑆(𝑛), max, min, ⊙𝑛 , → 𝑛 , 0, 1) and (𝑆(Q), max, min, ⊙Q , → Q , 0, 1) are MV-algebras := (𝑥 ⊙R ( → R )𝑦) ∩ 𝑆(𝑛) and where 𝑥⊙𝑛 ( → 𝑛 )𝑦 𝑥⊙Q ( → Q )𝑦 := (𝑥⊙R ( → R )𝑦) ∩ 𝑆(Q).

ISRN Algebra

3

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: (fi1) 𝑥, 𝑦 ∈ 𝐹 implies 𝑥 ⊙ 𝑦 ∈ 𝐹, (fi2) 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 [24] 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 [24]). Let 𝐹 and 𝐺 be filters of BL-algebra 𝐴 and 𝑥, 𝑦 ∈ 𝐴. Then, (1) (𝐹, 𝑥) is a filter of 𝐴;

Definition 5 (see [6]). A pair (ϝ, ℘) is called a soft set over 𝑈, when ℘ ⊆ 𝑃, and ϝ : ℘ → P(𝑈) is a set-valued mapping. In [25], for a soft set (ϝ, ℘), the set Supp(ϝ, ℘) = {𝜖 ∈ ℘ | 𝐹(𝜖) ≠ 0} is called the support of the soft set (ϝ, ℘). The soft set (ϝ, ℘) is called nonnull if Supp(ϝ, ℘) ≠ 0, and it is called a relative null soft set (with respect to the parameter set ℘), denoted by, 0℘ , if Supp(ϝ, ℘) = 0. 00 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 ∈ ℘. 𝑈𝑃 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 [6]. These examples were also discussed in [9, 11]. Now, we give an example of a soft set. Example 6. Let 𝐷 = {0, 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 1}. Define on 𝐷 the following operations:

(2) 𝐹 ⊆ (𝐹, 𝑥);

→

0

x1 x2 x3 x4

1

⊙

0

x1 x2 x3

x4

1

(3) 𝑥 ≤ 𝑦 implies (𝐹, 𝑥) ⊆ (𝐹, 𝑦);

0 x1 x2 x3 x4 1

1 0 0 0 0 0

1 1 1 1 1 1 1 1 x3 1 x1 1 x1 x1 1 1 x1 x2 x3 1 x1 x2 x3 x4

1 1 1 1 1 1

0 x1 x2 x3 x4 1

0 0 0 0 0 0

0 x1 x1 x1 x1 x1

0 x1 x1 x3 x3 x3

0 x1 x2 x3 x4 x4

0 x1 x2 x3 x4 1

∨

0

x1 x2 x3 x4

1

∧

0

x1 x2 x3

x4

1

0 x1 x2 x3 x4 1

0 x1 x2 x3 x4 1

x1 x1 x2 x3 x4 1

1 1 1 1 1 1

0 x1 x2 x3 x4 1

0 0 0 0 0 0

0 x1 x1 x1 x1 x1

0 x1 x2 x3 x4 x4

0 x1 x2 x3 x4 1

(4) 𝐹 ⊆ 𝐺 implies (𝐹, 𝑥) ⊆ (𝐺, 𝑥); (5) (𝐹, 𝑥) = 𝐴 if and only if 𝑥 ∈ 𝐹; (6) (𝐹, 𝑥) ∩ (𝐹, 𝑦) = (𝐹, 𝑥 ∧ 𝑦) = (𝐹, 𝑥 ⊙ 𝑦); (7) (𝐹, 𝑥) ∪ (𝐺, 𝑥) = (𝐹 ∪ 𝐺, 𝑥) and (𝐹, 𝑥) ∩ (𝐺, 𝑥) = (𝐹 ∩ 𝐺, 𝑥); (8) ((𝐹, 𝑥), 𝑦) = ((𝐹, 𝑦), 𝑥) = (𝐹, 𝑥 ∨ 𝑦). (9) (𝑥, 𝑥) = 𝐴; (10) (𝑥, 𝑦) = (𝑥, 𝑥 ∧ 𝑦) = (𝑥, 𝑥 ⊙ 𝑦); (11) (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 0⊥ = {1}⊥ = 𝐴. For any subset 𝑋 of 𝐴, (𝑋⊥ )⊥ is denoted by 𝑋⊥⊥ . Proposition 4 (see [24]). Let 𝑋 and 𝑌 be two nonempty subsets of BL-algebra 𝐴, and let {𝑋𝑖 }𝑖∈𝐼 be a nonempty family subset of 𝐴 and 𝐹 ∈ 𝐹(𝐴). Then, (1) 𝑋⊥ is a filter of 𝐴; (2) If 𝑋 ⊆ 𝑌, then 𝑌⊥ ⊆ 𝑋⊥ and 𝑋⊥⊥ ⊆ 𝑌⊥⊥ ; (3) 𝑋 ⊆ 𝑋⊥⊥ , 𝑋⊥ = 𝑋⊥⊥⊥ , 𝑋⊥ = ⟨𝑋⟩⊥ ; ⊥

⊥

(4) ⟨𝑋⟩ ∩ 𝑋 = {1}, 𝐹 ∩ 𝐹 = {1}; (5) 𝐹⊥ is a prime filter if and only if 𝐹 is a chain and 𝐹 ≠ 1; (6) ⋂𝑖∈𝐼 𝑋𝑖 ⊥ = (⋃𝑖∈𝐼 𝑋𝑖 )⊥ . 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 𝑃𝑈 (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).

x2 x2 x2 x4 x4 1

x3 x3 x4 x3 x4 1

x4 x4 x4 x4 x4 1

0 x1 x2 x1 x2 x2 0 x1 x2 x1 x2 x2

0 x1 x1 x3 x3 x3

One can see that D = (𝐷; ∨, ∧, ⊙, → , 0, 1) is a divisible residuated lattice. Furthermore, 0 ≤ 𝑥1 ≤ 𝑥2 , 𝑥3 ≤ 𝑥4 ≤ 1, but 𝑥2 , 𝑥3 are incomparable; thus D is not a chain. Also, (𝑥2 → 𝑥3 )∨(𝑥3 → 𝑥2 ) = 𝑥1 so D is not an MTL-algebra. Moreover, 𝐹(D) = {𝐹1 = 𝐷, 𝐹2 = {𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 1}, 𝐹3 = {𝑥2 , 𝑥4 , 1}, 𝐹4 = {𝑥3 , 𝑥4 , 1}, 𝐹5 = {𝑥4 , 1}, and 𝐹6 = {1}}. Now, let ℘ = 𝐹(D) and ϝ : ℘ → P(𝐷) which is defined by ϝ(𝐹𝑖 ) = 𝑥𝜎(𝑖) /𝐹𝑖 , where 𝜎 is a permutation on {1, 2, 3, 4, 5, 6}. Then, (ϝ, ℘) is a soft set over D. Maji et al. [9], Feng et al. [25], and Ali et al. [10] introduced and investigated several binary operations. Definition 7 (see [6]). Let (ϝ1 , ℘1 ) and (ϝ2 , ℘2 ) be two soft sets over a common universe 𝑈. (i) (ϝ1 , ℘1 ) is said to be a soft subset of (ϝ2 , ℘2 ) and is ̃ (ϝ2 , ℘2 ) if ℘1 ⊆ ℘2 and ϝ1 (𝜖) ⊆ denoted by (ϝ1 , ℘1 )⊆ ϝ2 (𝜖) for all 𝜖 ∈ ℘1 . (ii) (ϝ1 , ℘1 ) and (ϝ2 , ℘2 ) are said to be soft equal and is ̃ (ϝ2 , ℘2 ) and denoted by (ϝ1 , ℘1 ) = (ϝ2 , ℘2 ) if (ϝ1 , ℘1 )⊆ ̃ (ϝ2 , ℘2 )⊆(ϝ1 , ℘1 ). Example 8. Consider divisible residuated lattice D in Example 6. Let ℘1 = {𝑥2 , 𝑥3 } and ℘2 = {𝑥1 , 𝑥2 , 𝑥3 }. Now, we define ϝ1 : ℘1 → P(𝐷) by ϝ1 (𝜖) = (𝐹5 , 𝜖) and ϝ2 : ℘2 →

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P(𝐷) by ϝ2 (𝜖) = (𝐹4 , 𝜖) for each 𝜖 ∈ ℘𝑖 , where 𝑖 ∈ {1, 2}, 𝐹4 = {𝑥3 , 𝑥4 , 1}, and 𝐹5 = {𝑥4 , 1}. By Proposition 3, we obtain ̃ (ϝ2 , ℘2 ). that (ϝ1 , ℘1 )⊆

and ℶ1 ((𝑥2 , 𝑥4 )) = 𝐹4 × 𝐹5 . Therefore, we have (ℶ1 , ℘) = ̃ (ϝ2 , ℘2 ), (ℶ2 , ℘) = (ϝ1 , ℘1 )∨ ̃ (ϝ2 , ℘2 ), and (ℶ3 , ℘) = (ϝ1 , ℘1 )∧ (ϝ1 , ℘1 ) × (ϝ2 , ℘2 ).

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

Definition 13 (see [10]). Let (ϝ1 , ℘1 ) and (ϝ2 , ℘2 ) be two soft sets over a common universe 𝑈 such that ℘1 ∩ ℘2 ≠ 0.

Definition 9 (see [10]). Let {(ϝ𝑖 , ℘𝑖 ) : 𝑖 ∈ Λ} be a family of soft sets over a common universe 𝑈.

(i) The restricted difference of (ϝ1 , ℘1 ) and (ϝ2 , ℘2 ) is defined as the soft set (ϝ, ℘) = (ϝ1 , ℘1 )_(ϝ2 , ℘2 ), where ℘ = ℘1 ∩ ℘2 and ϝ(𝜖) = ϝ1 (𝜖) − ϝ2 (𝜖), for all 𝜖 ∈ ℘. If ℘1 ∩ ℘2 = 0, we define (ϝ1 , ℘1 )_(ϝ2 , ℘2 ) = 00

(1) The restricted intersection (union) of {(ϝ𝑖 , ℘𝑖 ) : 𝑖 ∈ Λ} ̃ 𝑖∈Λ (𝐹𝑖 , ℘𝑖 ) ((ϝ, ℘) = is defined as the soft set (ϝ, ℘) = ⊓ ̃ 𝑖∈Λ (𝐹𝑖 , ℘𝑖 )), where ℘ = ∩𝑖∈Λ ℘𝑖 and ϝ(𝜖) = ∩𝑖∈Λ ϝ𝑖 (𝜖) ⊔ (ϝ(𝜖) = ∪𝑖∈Λ ϝ𝑖 (𝜖)), for all 𝜖 ∈ ℘. If ℘ = ∩𝑖∈Λ ℘𝑖 = 0, we ̃ 𝑖∈Λ (𝐹𝑖 , ℘𝑖 ) = 00 . ̃ 𝑖∈Λ (𝐹𝑖 , ℘𝑖 ) = 00 and ⊔ define ⊓ (2) The extended intersection (union) of {(ϝ𝑖 , ℘𝑖 ) : 𝑖 ∈ Λ} ̃ 𝑖∈Λ (𝐹𝑖 , ℘𝑖 ) ((ϝ, ℘) = is defined as the soft set (ϝ, ℘) = ∩ ̃ ∪𝑖∈Λ (𝐹𝑖 , ℘𝑖 )), where ℘ = ∪𝑖∈Λ ℘𝑖 and ϝ(𝜖) = ∩𝑖∈Λ(𝜖) ϝ𝑖 (𝜖) (ϝ(𝜖) = ∪𝑖∈Λ(𝜖) ϝ𝑖 (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 = {𝑥1 , 𝑥2 , 1}, ℘2 = {𝑥1 , 𝑥3 , 𝑥4 }, and ℘3 = {0, 𝑥1 , 𝑥2 , 𝑥3 , 1}. Now, we define ϝ𝑖 : ℘𝑖 → P(𝐷) with ϝ𝑖 (𝜖) = (𝐹𝑖+1 , 𝜖) for each 𝜖 ∈ ℘𝑖 , where 𝑖 ∈ {1, 2, 3}. Now, we assume that ℘ = {𝑥1 } and ℘󸀠 = 𝐷. Also, we suppose that ϝ : ℘ → P(𝐷) in which ϝ(𝜖) = (𝐹5 , 𝜖) and ϝ󸀠 : ℘󸀠 → P(𝐷) in which ϝ󸀠 (𝜖) = (𝐹2 , 𝜖). By Proposition 3, we ̃ 𝑖∈Λ (ϝ𝑖 , ℘𝑖 ), ̃ 𝑖∈Λ (ϝ𝑖 , ℘𝑖 ), (ϝ󸀠 , ℘) = ⊔ can obtain that (ϝ, ℘) = ⊓ 󸀠 󸀠 ̃ 𝑖∈Λ (ϝ𝑖 , ℘𝑖 ), where Λ = {1, 2, 3}. Also, if we and (ϝ , ℘ ) = ∪ ̃ 𝑖∈Λ (ϝ𝑖 , ℘𝑖 ), we get ϝ(0) = 𝐹4 , ϝ(𝑥1 ) = (𝐹5 , 𝑥1 ), let ϝ = ∪ ϝ(𝑥2 ) = (𝐹3 , 𝑥2 ), ϝ(𝑥3 ) = (𝐹5 , 𝑥3 ), ϝ(𝑥4 ) = (𝐹3 , 𝑥4 ), and ϝ(1) = 𝐷. Definition 11 (see [25]). Let {(ϝ𝑖 , ℘𝑖 ) : 𝑖 ∈ Λ} be a family of soft sets over a common universe 𝑈. (1) The ∧-intersection of {(ϝ𝑖 , ℘𝑖 ) : 𝑖 ∈ Λ} is defined as the ̃ 𝑖∈Λ (ϝ𝑖 , ℘𝑖 ), where ℘ = ∏𝑖∈Λ ℘𝑖 and soft set (ϝ, ℘) = ∧ ϝ(𝜖) = ∩𝑖∈Λ ϝ𝑖 (𝜖𝑖 ), for all 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ ℘. (2) The ∨-intersection of {(ϝ𝑖 , ℘𝑖 ) : 𝑖 ∈ Λ} is defined as the ̃ 𝑖∈Λ (ϝ𝑖 , ℘𝑖 ), where ℘ = ∏𝑖∈Λ ℘𝑖 and soft set (ϝ, ℘) = ∨ ϝ(𝜖) = ∪𝑖∈Λ ϝ𝑖 (𝜖𝑖 ), for all 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ ℘. (3) Let (ϝ𝑖 , ℘𝑖 ) be a soft set over universe 𝑈𝑖 , where 𝑖 ∈ Λ. The cartesian product of {(ϝ𝑖 , ℘𝑖 ) : 𝑖 ∈ Λ} is defined as ̃ (ϝ , ℘ ), where ℘ = ∏ ℘ the soft set (ϝ, ℘) = ∏ 𝑖 𝑖∈Λ 𝑖 𝑖∈Λ 𝑖 and ϝ(𝜖) = ∏𝑖∈Λ ϝ𝑖 (𝜖𝑖 ), for all (𝜖𝑖 )𝑖∈Λ ∈ ℘. Example 12. Consider divisible residuated lattice D in Example 6. Let ℘1 = {𝑥2 } and ℘2 = {𝑥3 , 𝑥4 }. Now, we define ϝ1 : ℘1 → P(𝐷) with ϝ1 (𝑥2 ) = (𝐹5 , 𝑥2 ). Also, we define ϝ2 : ℘2 → P(𝐷) with ϝ1 (𝑥3 ) = (𝐹5 , 𝑥3 ) and ϝ1 (𝑥4 ) = (𝐹5 , 𝑥4 ) = 𝐷. Now, we assume that ℘ = ℘1 × ℘2 , ℶ1 : ℘ → P(𝐷) in which ℶ1 ((𝑥2 , 𝑥3 )) = (𝐹5 , 𝑥2 ∧ 𝑥3 ) = (𝐹5 , 𝑥1 ) = 𝐹5 and ℶ1 ((𝑥2 , 𝑥4 )) = (𝐹5 , 𝑥2 ∧ 𝑥4 ) = (𝐹5 , 𝑥2 ) = 𝐹4 , ℶ2 : ℘ → P(𝐷) in which ℶ2 ((𝑥2 , 𝑥3 )) = {𝑥2 , 𝑥3 , 𝑥4 , 1} and ℶ2 ((𝑥2 , 𝑥4 )) = 𝐹4 , and ℶ3 : ℘ → P(𝐷) in which ℶ3 ((𝑥2 , 𝑥3 )) = 𝐹4 × 𝐹3

(ii) The extended difference of (ϝ1 , ℘1 ) and (ϝ2 , ℘2 ) is defined as the soft set (ϝ, ℘) = (ϝ1 , ℘1 )S(ϝ2 , ℘2 ), where ℘ = ℘1 ∪ ℘2 , and we have ϝ (𝜖) , if 𝜖 ∈ ℘1 − ℘2 , { { 1 ϝ (𝜖) = {ϝ2𝑐 (𝜖) , if 𝜖 ∈ ℘2 − ℘1 , { − ϝ , if 𝜖 ∈ ℘1 ∩ ℘2 . ϝ (𝜖) (𝜖) 2 { 1

(4)

Example 14. Consider Example 10. Let ℘ = {𝑥1 } and ℘󸀠 = 𝐹2 . We suppose that ϝ : ℘ → P(𝐷) in which ϝ(𝑥1 ) = {0, 𝑥1 , 𝑥3 , 1} and ϝ󸀠 : ℘󸀠 → P(𝐷) in which ϝ󸀠 (𝑥1 ) = ϝ󸀠 (𝑥3 ) = {0, 𝑥1 , 𝑥3 , 1} and ϝ󸀠 (𝑥2 ) = ϝ󸀠 (𝑥4 ) = ϝ󸀠 (1) = 𝐷. Then, we can obtain that (ϝ, ℘) = (ϝ1 , ℘1 )_(ϝ2 , ℘2 ) and (ϝ󸀠 , ℘) = (ϝ1 , ℘1 )S(ϝ2 , ℘2 ). Definition 15 (see [10]). The complement of a soft set (ϝ, ℘) over 𝑈 is denoted by (ϝ, ℘)𝑐 and is defined by (ϝ, ℘)𝑐 = (ϝ𝑐 , ℘) where ϝ𝑐 : ℘ → P(𝑈) is a mapping given by ϝ𝑐 (𝜖) = 𝑈−ϝ(𝜖), for all 𝜖 ∈ ℘. Clearly, (ϝ, ℘)𝑐 = 𝑈𝑃 _(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 Soft𝑃 (𝑈). Let 𝑈 be a universal set and let 𝑃 be the set of parameters with respect to 𝑈. One can see that (Soft𝑃 (𝑈), 𝛼, 𝛽, 𝑈𝑃 , 00 ), ̃, ∪ ̃ } and 𝛽 ∈ {⊓ ̃, ∩ ̃ }, is a distributive where 𝛼 ∈ {⊔ bounded complete lattice if Soft𝑃 (𝑈) is closed under 𝛼 and 𝛽. Furthermore, the partial relation defined by lattice ̃ . Since ̃, ⊓ ̃ , 𝑈𝑃 , 00 ) coincides with ⊆ operations (Soft𝑃 (𝑈), ∪ ̃ ̃ (Soft𝑃 (𝑈), ∪, ⊓, 𝑈𝑃 , 00 ) is a distributive bounded complete lattice, we can define a new operation → as follows: ̃ (ϝ2 , ℘2 )} . ̃ {(ϝ, ℘) | (ϝ1 , ℘1 ) ⊓ ̃ (ϝ, ℘) ⊆ (ϝ1 , ℘) 󳨀→ (ϝ2 , ℘2 ) = ∪ (5) Also, we let (ϝ, ℘)∗ = (ϝ, ℘) → 00 . Clearly, we have ̃ (ϝ, ℘) = 00 . Hence, we obtain the following corollary. (ϝ, ℘)∗ ⊓ Corollary 16. Let 𝑈 be an universal set and let 𝑃 be the set of parameters with respect to 𝑈. Then, 𝑆𝑃 (𝑈) = ̃, ⊓ ̃, ⊓ ̃ , → , 𝑈𝑃 , 00 ) is a bounded residuated lattice. (𝑆𝑜𝑓𝑡𝑃 (𝑈), ∪ Example 17. Consider divisible residuated lattice D in Example 6. Let ℘1 = {𝑥1 , 𝑥2 }, ℘2 = {𝑥3 , 𝑥4 }, ℘3 = {𝑥3 }, ℘4 = {0, 𝑥1 , 𝑥3 , 𝑥4 }, and ℘5 = {0, 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 }. Now, we define ϝ𝑖 : ℘𝑖 → P(𝐷) with ϝ𝑖 (𝜖) = (𝐹𝑖+1 , 𝜖), for 𝑖 ∈ {1, 2, 4, 5} and 𝜖 ∈ ℘𝑖 . Also, let ϝ3 : ℘3 → P(𝐷) with ϝ3 (𝜖) = 𝜖⊥ , for

ISRN Algebra

5

𝜖 ∈ ℘3 . If we let ℘ = {𝑥3 , 𝑥4 } and we consider the soft set ϝ : ℘ → P(𝐷) with ϝ(𝑥3 ) = 𝐹5 and ϝ(𝑥4 ) = 𝐷, then we have ̃ (ϝ1 , ℘1 )∗ ≠ 𝑈𝑃 . (ϝ1 , ℘1 )∗ = (ϝ, ℘). Also, it is clear that (ϝ1 , ℘1 )∪ In the next proposition, we show that 𝑆𝑃 (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 (ϝ1 , ℘1 ), (ϝ2 , ℘2 ) ∈ 𝑆𝑜𝑓𝑡𝑃 (𝑈), we have (i) 00 → (ϝ, ℘) = 𝑈𝑃 , ̃ (ϝ2 , ℘2 ) = (ϝ1 , ℘1 )⊓ ̃ ((ϝ1 , ℘1 ) → (ϝ2 , ℘2 )). (ii) (ϝ1 , ℘1 )⊓ 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 (ϝ1 ∩ ϝ2 )(𝜖) = ϝ1 (𝜖) ∩ (∪𝜖∈Dom(ϝ) {ϝ(𝜖) | (ϝ1 ∩ ϝ)(𝜖) ⊆ ϝ2 (𝜖)}). Hence, the result holds. Example 19. Consider Example 17. Clearly, we have (ϝ1 , ℘1 ) → (ϝ2 , ℘2 ) = (ϝ2 , ℘2 ) and (ϝ2 , ℘2 ) → (ϝ1 , ℘1 ) = (ϝ1 , ℘1 ). Hence, 𝑆𝑃 (𝑈) is not an MTL-algebra. Also, (ϝ3 , ℘3 )∗ = (ϝ1 , ℘1 ) and it implies that 𝑆𝑃 (𝑈) is not a Girard monoid. Therefore, 𝑆𝑃 (𝑈) is not an MV-algebra. ̃ ,𝑐 , 0𝑃 ) Recently, Ali et al. in [13] show that (Soft𝑃 (𝑈), ⊔ is an MV-algebra. Also, they show directly ̃ ,𝑐 , 𝑈𝑃 ) is an MV-algebra and that (Soft𝑃 (𝑈), ⊓ (Soft𝑃 (𝑈), _, 0𝑃 , 𝑈𝑃 ) is a bounded BCK-algebra whose every element is an involution. By [10], if (ϝ1 , ℘1 ) and (ϝ2 , ℘2 ) are two soft sets over the same universe 𝑈, then 𝑐

̃ (ϝ2 , ℘2 )𝑐 ) . ̃ (ϝ2 , ℘2 ) = ((ϝ1 , ℘1 )𝑐 ⊔ (ϝ1 , ℘1 ) ⊓

relation between an element of ℘ and an element of 𝑅; that is R 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) ∈ R}, for all 𝜖 ∈ ℘. Obviously, the pair (ϝ, ℘) is a soft set over 𝑅 which is derived from the relation R. 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 𝑅5 = {0, 𝑥1 , 𝑥2 , 𝑥3 , 1}. Define on 𝑅5 the following operations: →

0

1

⊙

0

x1 x2 x3

1

1 1 1 1 x3

1 1 1 1 1

0 x1 x2 x3 1

0 0 0 0 0

0 0 x1 0 0 x2 x1 x2 x1 x2

0 x1 x2 x3 1

x1 x2 x3

1

∧

0

x1 x2 x3

1

x1 x1 x3 x3 1

1 1 1 1 1

0 x1 x2 x3 1

0 0 0 0 0

0 0 x1 0 0 x2 x1 x2 x1 x2

0 x1 x2 x3 1

x1 x2 x3

0 1 1 1 x1 x2 1 x2 x2 x2 x1 1 x3 0 x2 x1 1 0 x1 x2 ∨

0

0 0 x1 x1 x2 x2 x3 x3 1 1

x2 x3 x2 x3 1

x3 x3 x3 x3 1

0 x1 x2 x3 x3

0 x1 x2 x3 x3

Then, 𝑅5 is a distributive divisible residuated lattice which it is not an MTL-algebra since (𝑥1 → 𝑥2 ) ∨ (𝑥2 → 𝑥1 ) = 𝑥3 ≠ 1. Now, let (ϝ, ℘) be a soft set over residuated lattice 𝑅5 , where ℘ = 𝑅5 and ϝ : ℘ → P(𝑅5 ) is a set valued function defined by ϝ (𝑥) = {𝑦 ∈ 𝑅5 | 𝑦 ≥ 𝑥} ∪ {0}

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for all 𝑥 ∈ ℘. Then, ϝ(0) = 𝑅5 , ϝ(𝑥1 ) = {0, 𝑥1 , 𝑥3 , 1}, ϝ(𝑥2 ) = {0, 𝑥2 , 1}, ϝ(𝑥3 ) = {0, 𝑥3 , 1} and ϝ(1) = {0, 1}. So ϝ(𝜖) is a subalgebra of 𝑅5 , for each 𝜖 ∈ 𝑅5 . Therefore, (ϝ, ℘) is a soft residuated lattice over 𝑅5 .

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Example 24. Consider the standard MV-algebra [0, 1]. Let ℘ = N. We define ϝ : ℘ → P([0, 1]), in which ϝ(𝜖) = S𝜖 . Obviously, the pair (ϝ, ℘) is a soft MV-algebra over [0, 1].

Lemma 20. Let (ϝ1 , ℘1 ) and (ϝ2 , ℘2 ) be two soft sets over the same universe 𝑈. Then,

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: ̃ (ϝ, ℘); (SF1) (ϝ󸀠 , ℘󸀠 )⊆

̃ ,𝑐 , 𝑈𝑃 ) is an MV-algebra. So (Soft𝑃 (𝑈), ⊓

̃ (ϝ2 , ℘2 )𝑐 . (1) (ϝ1 , ℘1 )_(ϝ2 , ℘2 ) = (ϝ1 , ℘1 )⊓ ̃ (ϝ2 , ℘2 )𝑐 . Then, ℘ = ℘1 ∩ ℘2 , and Proof. Let (ϝ, ℘) = (ϝ1 , ℘1 )⊓ for each 𝜖 ∈ ℘ we have ϝ(𝜖) = ϝ1 (𝜖) ∩ ϝ2 (𝜖)𝑐 = ϝ1 (𝜖)_ϝ2 (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, (𝑆𝑜𝑓𝑡𝑃 (𝑈), _, 0𝑃 ) is a bounded commutative BCK-algebra.

(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 (ℸ, 𝑃).

3. Soft Residuated Lattices and Soft Filters

Example 27. Consider soft residuated lattice (ϝ, ℘) over 𝑅5 in Example 23. We define ϝ󸀠 : {𝑥1 , 𝑥2 , 𝑥3 , 1} → P(𝐴) in which we have ϝ󸀠 (𝑥) = {𝑦 ∈ 𝑅5 | 𝑦 ≥ 𝑥}. Therefore, one can see that soft set (ϝ󸀠 , ℘󸀠 ) is a soft filter of (ϝ, ℘).

In what follows, let 𝑅 and ℘ be a residuated lattice and a nonempty set, respectively, and R will refer to a arbitrary

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

Proof. By Lemma 20, it is straightforward.

6 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 {(ϝ𝑖󸀠 , ℘󸀠𝑖 ) | 𝑖 ∈ Λ} be a nonempty family of soft filter of (ϝ, ℘). ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ) and the extended Then, the restricted intersection ⊓ 󸀠 󸀠 ̃ 𝑖∈Λ(𝜖) (ϝ𝑖 , ℘𝑖 ) are soft filters of (ϝ, ℘) if they are intersection ∩ nonnull. Proof. Suppose that {(ϝ𝑖󸀠 , ℘󸀠𝑖 ) | 𝑖 ∈ Λ} is a nonempty family of ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ) soft filters of (ϝ, ℘). By Definition 9, let (ϝ󸀠 , ℘󸀠 ) = ⊓ 󸀠 󸀠 󸀠 󸀠 where ℘ = ∩𝑖∈Λ ℘𝑖 and ϝ (𝜖) = ∩𝑖∈Λ ϝ𝑖 (𝜖), for all 𝜖 ∈ ℘󸀠 . Clearly, we have ℘󸀠 ⊆ ℘. Now, let 𝜖 ∈ Supp(ϝ󸀠 , ℘󸀠 ), so 𝜖 ∈ Supp(ϝ𝑖󸀠 , ℘󸀠𝑖 ) for each 𝑖 ∈ Λ. By hypothesis ϝ𝑖󸀠 (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 (ϝ𝑖󸀠 , ℘󸀠𝑖 ) is a soft filter of (ϝ, ℘). ∩ Theorem 29. Let (ϝ, ℘) be a soft residuated lattice over 𝐴 and let {(ϝ𝑖󸀠 , ℘󸀠𝑖 ) | 𝑖 ∈ Λ} be a nonempty family of soft filters of ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ) and the extended (ϝ, ℘). Then, the restricted union ⊔ ̃ 𝑖∈Λ(𝜖) (ϝ𝑖󸀠 , ℘󸀠𝑖 ) are soft filters of (ϝ, ℘) if ϝ𝑖󸀠 (𝜖) ⊆ ϝ𝑗󸀠 (𝜖) or union ∪ 󸀠 ϝ𝑗 (𝜖) ⊆ ϝ𝑖󸀠 (𝜖), for all 𝑖, 𝑗 ∈ Λ(𝜖), and they are nonnull. Proof. Suppose that {(ϝ𝑖󸀠 , ℘󸀠𝑖 ) | 𝑖 ∈ Λ} is a nonempty family of ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ) soft filters of (ϝ, ℘). By Definition 9, let (ϝ󸀠 , ℘󸀠 ) = ⊔ where ℘󸀠 = ∩𝑖∈Λ ℘󸀠𝑖 and ϝ󸀠 (𝜖) = ∪𝑖∈Λ ϝ𝑖󸀠 (𝜖), for all 𝜖 ∈ ℘󸀠 . ̃ (ϝ, ℘). First, note that (ϝ󸀠 , ℘󸀠 ) is Clearly, we have (ϝ󸀠 , ℘󸀠 )⊆ 󸀠 󸀠 nonnull since Supp(ϝ , ℘ ) = ∪𝑖∈Λ Supp(ϝ𝑖󸀠 , ℘󸀠𝑖 ) ≠ 0. Let 𝜖 ∈ Supp(ϝ󸀠 , ℘󸀠 ). Then, ϝ󸀠 (𝜖) = ∪𝑖∈Λ ϝ𝑖󸀠 (𝜖) ≠ 0 and so we have ϝ𝑗󸀠 (𝜖) ≠ 0 for some 𝑗 ∈ Λ(𝜖). Now, let 𝑥, 𝑦 ∈ ϝ󸀠 (𝜖) = ∪𝑖∈Λ ϝ𝑖󸀠 (𝜖). Therefore, 𝑥 ∈ ϝ𝑖󸀠 (𝜖) and 𝑦 ∈ ϝ𝑗󸀠 (𝜖), for some 𝑖, 𝑗 ∈ Λ(𝜖). We can suppose that ϝ𝑖󸀠 (𝜖) ⊆ ϝ𝑗󸀠 (𝜖). So by hypothesis we obtain that 𝑥 ⊙ 𝑦 ∈ ϝ𝑗󸀠 (𝜖) ⊆ ϝ󸀠 (𝜖). On the other hand, let 𝑥 ≤ 𝑦, 𝑥 ∈ ϝ󸀠 (𝜖), and 𝑦 ∈ ϝ(𝜖). Then, 𝑥 ∈ ϝ𝑖󸀠 (𝜖), for some 𝑖 ∈ Λ(𝜖), and it implies that 𝑦 ∈ ϝ𝑖󸀠 (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 {(ϝ𝑖󸀠 , ℘󸀠𝑖 ) | 𝑖 ∈ Λ} be a nonempty family of soft filters of (ϝ, ℘). Assume that ℘𝑖 ∩ ℘𝑗 = 0 for all 𝑖, 𝑗 ∈ Λ, 𝑖 ≠ 𝑗. Therefore, Λ(𝜖) has only one element. Let Λ(𝜖) = {𝑖𝜖 }. So for each 𝜖 ∈ ℘, ϝ(𝜖) = ϝ𝑖󸀠𝜖 (𝜖), and it implies that, for each 𝑖, 𝑗 ∈ Λ and 𝜖 ∈ ℘, either ϝ𝑖 (𝜖) = 0 or ϝ𝑗 (𝜖) = 0. Hence, by Theorem 29 we can ̃ 𝑖∈Λ (ϝ𝑖󸀠 , 𝐼𝑖 ) is a soft filter of (ϝ, ℘). conclude that the union ⊔ 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 = {𝑥2 , 𝑥3 } and ℘2 = {𝑥1 , 𝑥2 , 𝑥3 }. Now, we define ϝ1 : ℘1 → P(𝐷) by ϝ1 (𝜖) = 𝐹3 and ϝ2 : ℘2 → P(𝐷) by ϝ2 (𝜖) = 𝐹4 for each 𝜖 ∈ ℘𝑖 , where 𝑖 ∈ {1, 2}. ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ) and the extended Obviously, the restricted union ⊔

ISRN Algebra ̃ 𝑖∈Λ(𝜖) (ϝ𝑖󸀠 , ℘󸀠𝑖 ) are not soft filters of 𝑈℘ , where ℘ = union ∪ {𝑥2 , 𝑥3 , 𝑥4 , 1}. Theorem 31. Let (ϝ, ℘) be a soft residuated lattice over 𝐴 and let {(ϝ𝑖󸀠 , ℘󸀠𝑖 ) | 𝑖 ∈ Λ} be a nonempty family of soft filters of (ϝ, ℘). ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ) is a soft filter of ∧ ̃ 𝑖∈Λ (ϝ, ℘) Then, the ∧-intersection ∧ 󸀠 󸀠 ̃ 𝑖∈Λ (ϝ𝑖 , ℘𝑖 ) is nonnull. if ∧ ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ), Proof. By Definition 11, we can let (ϝ󸀠 , ℘󸀠 ) = ∧ 󸀠 󸀠 󸀠 󸀠 where ℘ = ∏𝑖∈Λ ℘𝑖 and ϝ (𝜖) = ∩𝑖∈Λ ϝ𝑖 (𝜖𝑖 ), for all 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ ̃ 𝑖∈Λ (ϝ, ℘), where 𝑝 = ∏𝑖∈Λ ℘ and ℘󸀠 . Also, we let (𝐺, 𝑝) = ∧ 𝐺(𝜖) = ∩𝑖∈Λ ϝ(𝜖𝑖 ), for all 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ 𝑝. Clearly, we have ̃ (𝐺, 𝑝). Now, let (ϝ󸀠 , ℘󸀠 ) be nonnull and 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ (ϝ󸀠 , ℘󸀠 )⊆ Supp(ϝ󸀠 , ℘󸀠 ). We have ϝ󸀠 (𝜖) = ∩𝑖∈Λ ϝ𝑖󸀠 (𝜖𝑖 ), and since each ϝ𝑖󸀠 (𝜖𝑖 ) is a filter of ϝ(𝜖𝑖 ), we obtain that ϝ󸀠 (𝜖) is a filter of 𝐺(𝜖). Theorem 32. Let (ϝ, ℘) be a soft residuated lattice over 𝐴 and let {(ϝ𝑖󸀠 , ℘󸀠𝑖 ) | 𝑖 ∈ Λ} be a nonempty family of soft filters of (ϝ, ℘). If ϝ𝑖󸀠 (𝜖) ⊆ ϝ𝑗󸀠 (𝜖) or ϝ𝑗󸀠 (𝜖) ⊆ ϝ𝑖󸀠 (𝜖), for all 𝑖, 𝑗 ∈ Λ and ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ) is a soft filter 𝜖 ∈ ∪𝑖∈Λ ℘󸀠𝑖 , then the ∨-intersection ∨ 󸀠 󸀠 ̃ 𝑖∈Λ (ϝ, ℘) if (ϝ𝑖 ℘𝑖 ) is nonnull, for some 𝑖 ∈ Λ. of ∨ ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ), in which ℘󸀠 = Proof. Suppose that (ϝ󸀠 , ℘󸀠 ) = ∨ 󸀠 󸀠 󸀠 ∏𝑖∈Λ ℘𝑖 and ϝ (𝜖) = ∪𝑖∈Λ ϝ𝑖 (𝜖𝑖 ), for all 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ ℘󸀠 . ̃ 𝑖∈Λ (ϝ, ℘), where 𝑃 = ∏𝑖∈Λ ℘ and Also, we let (𝐺, 𝑃) = ∨ 𝐺(𝜖) = ∪𝑖∈Λ ϝ(𝜖𝑖 ), for all 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ 𝑃. Clearly, we have ̃ (𝐺, 𝑃). Now, let (ϝ𝑗󸀠 , ℘󸀠𝑗 ) be nonnull, for some 𝑗 ∈ Λ. (ϝ󸀠 , ℘󸀠 )⊆ Assume that 𝜖𝑗 ∈ Supp(ϝ𝑗󸀠 , ℘󸀠𝑗 ). Obviously, (ϝ󸀠 , ℘󸀠 ) is a nonnull soft set because, for each 𝜖 ∈ ℘󸀠 , where 𝜋𝑗 (𝜖) = 𝜖𝑗 , we have 𝜖 ∈ Supp(ϝ󸀠 , ℘󸀠 ). Now, suppose that 𝜖 ∈ Supp(ϝ󸀠 , ℘󸀠 ). We have ϝ󸀠 (𝜖) = ∪𝑖∈Λ ϝ𝑖󸀠 (𝜖𝑖 ) ≠ 0. Similar to the proof of Theorem 29, we can obtain that ϝ󸀠 (𝜖) is a filter of 𝐺(𝜖). Hence, the ∨̃ 𝑖∈Λ (ϝ, ℘). ̃ 𝑖∈Λ (ϝ𝑖󸀠 , ℘󸀠𝑖 ) is a soft filter of ∨ intersection ∨ Theorem 33. Let (ϝ, ℘) be a soft residuated lattice over 𝐴 and let {(ϝ𝑖󸀠 , ℘󸀠𝑗 ) | 𝑖 ∈ Λ} be a nonempty family nonnull soft filters ̃ (ϝ , ℘ ) is a soft filter of (ϝ, ℘). Then, the cartesian product ∏ 𝑖 𝑖∈Λ 𝑖 ̃ of ∏ (ϝ, ℘). 𝑖∈Λ

̃ (ϝ󸀠 , ℘󸀠 ), in which ℘󸀠 = Proof. Suppose that (ϝ󸀠 , ℘󸀠 ) = ∏ 𝑖∈Λ 𝑖 𝑖 󸀠 󸀠 󸀠 ∏𝑖∈Λ ℘𝑖 and ϝ (𝜖) = ∏𝑖∈Λ ϝ𝑖 (𝜖𝑖 ), for all 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ ℘󸀠 . ̃ (ϝ, ℘), where 𝑃 = ∏ ℘ and Also, let (𝐻, 𝑃) = ∏ 𝑖∈Λ 𝑖∈Λ 𝐻(𝜖) = ∏𝑖∈Λ ϝ(𝜖𝑖 ), for all 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ 𝑃. Clearly, we have ̃ (𝐻, 𝑃). By [20], for each 𝜖 ∈ 𝑃, 𝐻(𝜖) is a residuated (ϝ󸀠 , ℘󸀠 )⊆ lattice so (𝐻, 𝑃) is a soft residuated lattice over ∏𝑖∈Λ 𝐴. Now, let 𝜖 = (𝜖𝑖 )𝑖∈Λ ∈ ∏𝑖∈Λ Supp(ϝ𝑖󸀠 , ℘󸀠𝑖 ). Therefore, by [20] ϝ󸀠 (𝜖) is a ̃ (ϝ󸀠 , ℘󸀠 ) is a soft filter of ∏ ̃ (ϝ, ℘) filter of 𝐻(𝜖). Hence, ∏ 𝑖∈Λ 𝑖 𝑖∈Λ 𝑖 󸀠 󸀠 on ∏𝑖∈Λ Supp(ϝ𝑖 , ℘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

ISRN Algebra (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)) ≠ 0. 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. (i) If ϝ(𝜖) = ker 𝑓 for all 𝜖 ∈ ℘, then (𝑓(ϝ), ℘) is the trivial soft filter over 𝐴󸀠 . (ii) 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)) = 1𝐴󸀠 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.

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