Normal unisoft filters in R 0-algebras

Abstract: In the present paper, the notions of normal uni-soft filters in R 0 -algebras are introduced, and related properties are investigated. Furthermore, characterizations of a normal uni-soft filter are established, and a new normal uni-soft filter from old one is constructed. Finally, a condition for a uni-soft filter to be normal is given and a condensational property of a normal uni-soft filter is established.


Introduction
To solve complicated problem in economics, engineering, and environment, we can not successfully use classical methods because of various uncertainties typical for those problems. Uncertainties can not be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in Molodtsov (1999). Maji, Roy, and Biswas (2002) and Molodtsov (1999) suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov (1999) introduced

PUBLIC INTEREST STATEMENT
Soft set theory is an important mathematical tool to deal with uncertainties, fuzzy or vague objects and has vast applications in real life situations. In this paper, we presented an application of soft set theory in an algebraic structure, called an R 0 -algebra. In fact, using the notion of uni-soft property, we introduced the notion of normal uni-soft filters in R 0 -algebras, and investigated on several properties of them. We hope that this work will provide a deep impact on the upcoming research in this field and other soft algebraic studies to open up new horizons of interest and innovations. To extend these results, one can further study the union soft substructures of different algebras such as hemirings, MTL-algerbas, hyperalgebras and other mathematical branches. One may also apply this concept to study some applications in many fields like decision making, knowledge base systems, data analysis, etc.
the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji, Roy, and Biswas (2002) described the application of soft set theory to a decision making problem. Maji, Biswas, and Roy (2003) also studied several operations on the theory of soft sets. Chen, Tsang, Yeung, and Wang (2005) presented a new definition of soft set parametrization reduction, and compared this definition to the related concept of attributes reduction in rough set theory.
R 0 -algebras, which are different from BL-algebras, have been introduced by Wang (2000) in order to provide an algebraic proof of the completeness theorem of a formal deductive system (Wang, 1999).
Furthermore, filter theories and some more important related concepts of R 0 -algebras are discussed in Pei and Wang (2002), Hongjun and Guojun (2008), Hongjun and Bin (2011).
In this paper, we apply the notion of normal uni-soft property to the filter theory in R 0 -algebras. We introduced the concept of normal uni-soft filters in R 0 -algebras, and investigate related properties. Further, we establish characterizations of a normal uni-soft filter, and construct a new normal uni-soft filter from old one. Also, we provide a condition for an uni-soft filter to be normal. Finally, we construct a condensational property of a normal uni-soft filter.

Basic results on R 0 -algebras
Definition 2.1 (Wang, 2000) Let L be a bounded distributive lattice with order-reversing involution ¬ and a binary operation → . Then (L, ∧, ∨, ¬, →) is called an R 0 -algebra if it satisfies the following axioms: Let L be an R 0 -algebra. For any x, y ∈ L, we define x ⊙ y = ¬(x → ¬y) and x ⊕ y = ¬x → y. It is proven that ⊙ and ⊕ are commutative, associative and x ⊕ y = ¬(¬x ⊙ ¬y), and (L, ∧, ∨, ⊙, →, 0, 1) is a residuated lattice. In the following, let x n denote x ⊙ x ⊙ ⋯ ⊙ x where x appears n times for n ∈ N.
We refer the reader to the book (Iorgulescu, 2008) for further information regarding R 0 -algebras.

Lemma 2.3 (Pei & Wang, 2002) Let F be a nonempty subset of L. Then F is a filter of L if and only if it satisfies
(2) (∀x, y ∈ F) (x ⊙ y ∈ F). https://doi.org/10.1080/23311835.2017.1383006 Lemma 2.4 (Pei & Wang, 2002) Let L be an R 0 -algebra. Then the following properties hold:

Basic results on soft set theory
Soft set theory was introduced by Molodtsov (1999) and Çaǧman, Çitak, and Enginoğlu (2010).
In what follows, let U be an initial universe set and E be a set of parameters. We say that the pair (U, E) is a soft universe. Let (U) (resp. (E)) denotes the power set of U (resp. E).
By analogy with fuzzy set theory, the notion of soft set is defined as follows: Definition 2.5 (Çaǧman, et al. 2010;Molodtsov, 1999) Definition 2.6 (Çaǧman et al., 2010) Definition 2.7 (Muhiuddin & Abdullah, 2015) For any non-empty subset A of E, a soft set ℱ A of E is said to satisfy the uni-soft property (it is also known as uni-soft set) if it satisfies: where ↪ is a binary operation in E.
In our further discussion, S(U, L) will denote the set of all soft sets of L over U where L is an R 0 -algebra unless otherwise specified. where is a subset of U such that ⊋ ∪ Lemma 2.11 (Muhiuddin & Abdullah, 2015) Let ℱ L ∈ S(U, L). Then ℱ L is a uni-soft filter of L if and only if the following assertion is valid: Lemma 2.12 (Muhiuddin & Abdullah, 2015) Let ℱ L ∈ S(U, L) be a uni-soft filter of L. Then ℱ L is orderreversing, that is,

Normal uni-soft filters
We begin with the following definitions and properties that will be needed in the sequel. Definition 3.2 A uni-soft filter ℱ L of L is said to be normal if the following assertion is valid: 1} be a set with the order 0 < a < b < c < 1, and the following Cayley tables: Then (L, ∧, ∨, ¬, →) is an R 0 -algebra (see Lianzhen & Kaitai, 2005) where x ∧ y = min{x, y} and x ∨ y = max{x, y}.
Let ℱ L ∈ S(U, L) be given as follows: where 1 and 2 are subsets of U with 2 ⊊ 1 . Then ℱ L is a normal uni-soft filter of L.
We first consider characterizations of a normal uni-soft filter.

Theorem 3.4 Let ℱ L ∈ S(U, L). Then ℱ L is a normal uni-soft filter of L if and only if it satisfies the condition (2.16) and
Proof Suppose that ℱ L is a normal uni-soft filter of L. Obviously, (2.16) is valid. For every x, y, z ∈ L, we have by (2.17) and (3.2).
Conversely, assume that ℱ L satisfies two conditions (2.16) and (3.3). If we take y = 1 in (3.3), then Hence ℱ L is a uni-soft filter of L. Now if we put z = 1 in (3.3), then by (R2) and (2.16). Therefore ℱ L is a normal uni-soft filter of L. ✷

Theorem 3.5 A soft set ℱ L ∈ S(U, L) is a normal uni-soft filter of L if and only if every nonempty -exclusive set of ℱ L is a normal filter of L for all ∈ (U).
If ℱ L is a normal uni-soft filter of L, every -exclusive set e(ℱ L ; ) is called a normal exclusive filter of L.
Proof Assume that ℱ L is a normal uni-soft filter of L and let ∈ (U) be such that e(ℱ L ; ) ≠ �.
Conversely, suppose that every nonempty -exclusive set e(ℱ L ; ) of ℱ L is a normal filter of L for all ∈ (U). For any x ∈ L, let f L (x) = . Then 1 ∈ e(ℱ L ; ), which implies that f L (1) ⊆ = f L (x) for all x ∈ L. For any x, y, z ∈ L, let Then z ∈ e(ℱ L ; ) and z → (y → x) ∈ e(ℱ L ; ). Since e(ℱ L ; ) is a normal filter of L, it follows from (3.1) that ((x → y) → y) → x ∈ e(ℱ L ; ) and so that Using Theorem 3.4, we know that ℱ L is a normal uni-soft filter of L. ✷ Obviously, every normal uni-soft filter is a uni-soft filter, but the converse may not be true as seen in the following example.
Example 3.6 Let L = [0, 1]. For any a, b ∈ L, we define Then (L, ∧, ∨, ¬, →) is an R 0 -algebra (see Wang, 2000). Let ℱ L ∈ S(U, L) be given as follows: where 1 and 2 are subsets of U with 2 ⊊ 1 . Then ℱ L is a uni-soft filter of L. But and so ℱ L is not a normal uni-soft filter of L.
We provide a condition for a uni-soft filter to be normal.
Theorem 3.7 Let L be an R 0 -algebra satisfying the following inequality: Then every uni-soft filter of L is normal .
Proof Let ℱ L be a uni-soft filter of L. Using (2.5), (2.6) and (3.4), we have for all x, y ∈ L. It follows from Lemma 2.12 that for all x, y ∈ L. Therefore ℱ L is a normal uni-soft filter of L. ✷ We make a new normal uni-soft filter from old one. Proof Assume that ℱ L is a normal uni-soft filter of L. Then ℱ L is a uni-soft filter of L, and so ℱ * L is a uni-soft filter of L by Lemma 2.10. Let x, y, z ∈ L. If z ∈ e(ℱ L ; ) and z → (y → x) ∈ e(ℱ L ; ), then ((x → y) → y) → x ∈ e(ℱ L ; ) because e(ℱ L ; ) is a normal filter of L by Theorem 3.5. Hence Therefore ℱ * L is a normal uni-soft filter of L by Theorem 3.4. ✷ Corollary 3.9 Let ℱ L ∈ S(U, L). For a subset of U, define a soft set ℱ * L of L by If ℱ L is a normal uni-soft filter of L, then so is ℱ * L .
Proof Straightforward. ✷ We consider a condensational property of a normal uni-soft filter.
Theorem 3.10 Let ℱ L and L be two uni-soft filters of L such that L⊆ ℱ L and f L (1) = g L (1). If ℱ L is normal, then so is L .
Proof Assume that ℱ L is a normal uni-soft filter of L. For any x, y ∈ L, let a = y → x. Since ℱ L is a normal uni-soft filter of L, it follows from (R4), (3.2) and assumption that and so that Since L is a uni-soft filter of L, we have Using (R4) and (2.14), we have It follows from (3.7) and Lemma 2.11 that (3.5) (3.8) (3.9)

Applications
Soft set theory, introduced by Molodtsov (1999), is an important mathematical tool to deal with uncertainties, fuzzy or vague objects and has vast applications in real life situations. Several possible applications of soft set theory in various directions are given in Molodtsov (1999).
In this paper, we presented an application of soft set theory in an algebraic structure, called an R 0 -algebra. In fact, using the notion of uni-soft property, we introduced the notion of normal uni-soft filters in R 0 -algebras, and investigated on several properties of them. We hope that this work will provide a deep impact on the upcoming research in this field and other soft algebraic studies to open up new horizons of interest and innovations. Indeed, this work may serve as a foundation for further study of soft R 0 -algebras. To extend these results, one can further study the union soft substructures of different algebras such as hemirings, MTL-algerbas, hyperalgebras and other mathematical branches. One may also apply this concept to study some applications in many fields like decision making, knowledge base systems, data analysis, etc.

Conclusion
Using the notion of uni-soft property, we have introduced the concept of normal uni-soft filters in R 0 -algebras, and investigated related properties. We have established characterizations of a normal uni-soft filter, and made a new normal uni-soft filter from old one. We have provided a condition for a uni-soft filter to be normal. Furthermore, we constructed a condensational property of a normal uni-soft filter. We can extend these results as follows: (1) to develop strategies for obtaining more valuable results, (2) to apply these notions and results for studying related notions in other (soft) algebraic structures, (3) to study the notions of implicative uni-soft filters and Boolean uni-soft filters.