Topological Structures of Lower and Upper Rough Subsets in a Hyperring

In this paper, we study the connection between topological spaces, hyperrings (semi-hypergroups), and rough sets. We concentrate here on the topological parts of the lower and upper approximations of hyperideals in hyperrings and semi-hypergroups. We provide the conditions for the boundary of hyp-ideals of a hyp-ring to become the hyp-ideals of hyp-ring.


Introduction
Algebraic hyp-structure (hyperstructure) represents a real extension of classical algebraic structure. Algebraic hypstructures depend on hyperoperations and their properties. Sm-hyp-group (semi-hypergroup) was first introduced by French Mathematician Marty [1] in 1934. e sm-hyp-group concept is the generalization of sm-group (semigroup) concept, likewise the hyp-ring (hyperring) concept is the generalization of ring concept. In [2,3], authors provided many applications of hyp-structures.
ere are several creators who added numerous outcomes to the hypothesis of algebraic hyp-structures, for instance, Hila and Dine [4] studied the hyperideals of left almost semi-hypergroups. Tang et al. [5] introduced the idea of hyperfilters in ordered semi-hypergroups, also see [6,7].
In 1982, Pawlak [8] introduced R-sets (rough sets) for the very first time. R-set theory has been a knowledge discovery in rational databases. Set approximation is divided into two parts, i.e., lower approximation and upper approximation. e applications of R-sets are considered in finance, pattern recognition, industries, information processing, and business. It provides a mathematical tool to find out pattern hidden in data. e major advantages of R-set approach is that it does not need any primary/ secondary information about the data like the theory of probability in statistics and the grade of membership in the theory of fuzzy set. It gives systematic procedures, tools, and algorithms to find out hidden patterns in data, and it permits generating in mechanized way the sets of decision rules from data. ivagar and Devi [9] introduced the concept of nanotopology via ring structure. R-set theory has been studied by several authors in algebraic structures and also in algebraic hyperstructures. Ahn and Kim applied R-set theory to BE-algebras [10]. Ali et al. [11] studied generalized roughness in (ε, ε∨q k )-fuzzy filters of ordered semigroups. Biswas and S. Nanda [12] applied R-set theory to groups. Shabir and Irshad [13] applied roughness in ordered semigroups. In [14][15][16][17][18][19][20][21][22], authors studied roughness in different hyperstructures. Fuzzy sets were also considered by many authors, for instance, Fotea and Davvaz [23] studied fuzzy hyperrings. Ameri and Motameni [24] applied fuzzy set theory to the hyperideals of fuzzy hyperrings. Bayrak and Yamak [25] introduced some results on the lattice of fuzzy hyperideals of a hyperring. Davvaz [26] studied fuzzy Krasner (m, n)hyperrings. Connections between fuzzy sets and topology are considered in [27][28][29].

Preliminaries and Notations
Definition 1. A topological space refers to a pair (5, τ), where 5 is a nonempty set and τ is a topology on 5. (1) A left and right hyp-ideal I of F is known as hyp-ideal of F.

T-Structures of R-Sets Based on Sm-Hyp-Groups
In this section, we develop some concepts related to topology of R-sets based on sm-hyp-groups.
Definition 6. Let F be a sm-hyp-group, Υ ⊆ F, and ξ be a REG-relation (regular relation) on F. en, the (l−) u-approximations and boundary of Υ with respect to the REG-relation ξ are given as follows: forms a topology on F.
under the binary hyperoperation "°" defined in Cayley (Table 1). Let be a REG-relation on the sm-hyp-group F with the following regular classes: As ξ(x) is a regular class of x, so xεξ(x). However, as ξ(x) ⊆ Υ, thus xεΥ. Now, we prove that Υ ⊆ ξ Upper (Υ). Let y εΥ. As ξ(y) is a regular class of y, so y εξ(y). us, us, yεξ Upper (Υ). (ii) e proof of this part is straightforward. (iii) e proof of this part is straightforward.

Theorem 2. Let F be a sm-hyp-group and ξ be a REG-relation on
Proof. Since Υ 1 ⊆ Υ 2 ⊆ F, the approximations with respect to the sm-hyp-group satisfy which implies that ξ τ (Υ 1 ) ⊆ ξ τ (Υ 2 ). □ Proposition 2. Suppose ξ and c are two REG-relations on F such that ξ ⊆ c, and let Υ 1 be the nonempty subset of F. en, Proof. Suppose ξ and c are two REG-relations on F such that ξ ⊆ c, and let Υ 1 be the nonempty subset of F.
(iii) e proof of this part implies from (i) and (ii). □ Theorem 3. Let F be a sm-hyp-group and ξ and c be the REG-relations on F such that ξ ⊆ c, and let Υ 1 be the nonempty subset of F. en, ξ τ (Υ 1 ) ≠ c τ (Υ 1 ).

T-Structures of R-Sets Based on Hyp-Rings
In this section, we develop some concepts related to topology of R-sets based on hyp-rings.

Definition 7.
Let R be a hyp-ring, Υ ⊆ R, and F be a hyperideal of R. en, the (l-) u-approximations and boundary of Υ with respect to the hyp-ideal F are given as follows: (14) forms a topology on R with respect to F.  (Tables 2 and 3).
Remark 2. Let R be a hyp-ring, F be a hyp-ideal of R, and Υ ⊆ R.
Theorem 4. Let R be a hyp-ring, F be a hyp-ideal of R, and Υ ⊆ R. en, Proposition 3. Let R be a hyp-ring, F be a hyp-ideal of R, and Υ 1 and Υ 2 two subsets of R such that Υ 1 ⊆ Υ 2 . en, Theorem 5. Let R be a hyp-ring and F be a hyp-ideal of R, and Υ 1 , Proposition 4. Suppose F, W are two hyp-ideals of R such that F ⊆ W, and let Υ 1 be the nonempty subset of R. en, Theorem 6. Let R be a hyp-ring and F, W be the hyp-ideals of R such that F ⊆ W and let Υ 1 be the non-empty subset of R. en F τ (Υ 1 ) ≠ W τ (Υ 1 ).
e following theorem can also be seen in [17].

Conclusion and Future Work
Relations between R-sets, hyp-rings, and topological structures are considered in this paper. In place of universal set, we added sm-hyp-groups and hyp-rings. In future, this work can be extended to soft set theory [30], bipolar fuzzy sets [31], intuitionistic fuzzy sets [32], or neutrosophic sets [33].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.