@ Appl. Gen. Topol. 21, no. 2 (2020), 295-304 doi:10.4995/agt.2020.13156 c© AGT, UPV, 2020 Rough action on topological rough groups Alaa Altassan a, Nof Alharbi a, Hassen Aydi b,c, Cenap Özel a a Department of Mathematics, King Abdulaziz University, Jeddah, Kingdom of Saudi Ara- bia (aaltassan@kau.edu.sa,nof20081900@hotmail.com,cozel@kau.edu.sa) b Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia. ( hassen.aydi@isima.rnu.tn) c China Medical University Hospital, China Medical University, Taichung 40402, Taiwan. ( hassen.aydi@isima.rnu.tn) Communicated by J. Galindo Abstract In this paper we explore the interrelations between rough set theory and group theory. To this end, we first define a topological rough group homomorphism and its kernel. Moreover, we introduce rough action and topological rough group homeomorphisms, providing sev- eral examples. Next, we combine these two notions in order to define topological rough homogeneous spaces, discussing results concerning open subsets in topological rough groups. 2010 MSC: 22A05; 54A05; 03E25. Keywords: rough groups; topological rough groups; topological rough sub- groups; product of topological rough groups; topological rough group homomorphisms; topological rough group homeomor- phisms; topologically rough homogeneous spaces; rough kernel. 1. Introduction Rough Set Theory has many applications in economic, medicine and enge- neering [13, 14, 16, 17]. Such a theory was introduced by Pawlak [21] and deals with uncertainty, impression and vagueness in information systems.The starting point of his analysis is the notion of approximation space, namely a pair(U, R), where U is any arbitrary non-empty set, called universe, and R is Received 18 February 2020 – Accepted 28 August 2020 http://dx.doi.org/10.4995/agt.2020.13156 A. Altassan, N. Alharbi, H. Aydi and C. Özel an equivalence relaon U. The set U/R of all equivalence classes [x]R forms a partition of U . Moreover, for any X ⊂ U, he introduced the notions of lower and upper approximations of X as follows: X = {[x]R : [x]R ∩ X ∕= ∅} and X = {[x]R : [x]R ⊂ X}. Next, he defined the rough set to be the orderd pair X = (X, X). Recently, the interrelations between rough set theory and various branches of mathematics, such as combinatorics [12], monoids [10], matroids [15, 23, 24, 25], groups [7, 18], integral domains [11] and modules [9] has been deeply studied and constitute a research field which is developing rapidly. In our perspective, we are interested in the interrelations between rough set theory and groups. To this regard, let us first recall that in [7] and [18] the notions of rough groups, rough subgroups, rough homomorphisms and rough antihomomorphisms have been analyzed in detail. Moreover, the notion of topological rough groups was introduced by Bagirmaz et al in [6]. In this paper, we present rough actions and rough homogenous spaces, and discuss some of their properties. We also define a rough kernel. We organ- ise the paper as follows. In section 2, we collect the needed material about rough groups and rough homomorphisms. Then the definition of topological rough groups and important properties have been recalled in section 3. Section 4 presents our main results where we introduce rough action and homogenous spaces. This paper is produced from the PhD thesis of Ms. Nof Alharbi registered in King Abdulaziz University. 2. Rough groups and rough homomorphisms In this section, we recall rough groups, rough homomorphisms and some of their properties. Let (U, R) be an approximation space, where U is a non-empty set and R is an equivalence relation on U. Let (∗) be a binary operation defined on U. For all x, y ∈ U, we write xy instead of x∗y. In 1994, Biswas and Nanda introduced the definition of rough groups which is given in the following definition. Definition 2.1 ([6]). Let (U, R) be an approximation space. Suppose that G is a subset of U and G and G are respectively its upper and lower approxima- tions. Then the rough set G = (G, G) is called a rough group if the following conditions are satisfied: (1) ∀x, y ∈ G, xy ∈ G. (2) (xy)z = x(yz), ∀x, y, z ∈ G. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 296 Rough action on topological rough groups (3) ∀x ∈ G, ∃e ∈ G such that xe = ex = x. (4) ∀x ∈ G, ∃y ∈ G such that xy = yx = e. Definition 2.2 ([6]). A non-empty rough subset H = (H, H) of a rough group G = (G, G) is called a rough subgroup if it is a rough group itself. Note that G = (G, G) is a trivial rough subgroup of itself. Moreover, if e ∈ G, then e = (e, e) is a trivial rough subgroup of the rough group G. Theorem 2.3 ([6]). Suppose that G is a subset of U and G and G are respec- tively its upper and lower approximations. Then a rough subset H is a rough subgroup of a rough group G if (1) ∀x, y ∈ H, xy ∈ H; (2) ∀y ∈ H, y−1 ∈ H. Let H be a rough subgroup of a rough group G. Then H is said to be a rough normal subgroup of G if xH = Hx, ∀x ∈ G Definition 2.4 ([18]). Let (U1, R1) and (U2, R2) be approximation spaces and ∗, ∗ ′ be binary operations on U1 and U2, respectively. Let G1 ⊆ U1 and G2 ⊆ U2 be two rough groups. If the mapping ϕ : G1 → G2 satisfies that ϕ(x ∗ y) = ϕ(x) ∗ ′ ϕ(y), for all x, y ∈ G1, then ϕ is called a rough homomorphism. Definition 2.5 ([18]). Let G1 and G2 be two rough groups. A rough homo- morphism ϕ : G1 → G2 is said to be : (1) a rough epimorphism (or surjective) if ϕ : G1 → G2 is onto. (2) a rough monomorphism if ϕ : G1 → G2 is one-to-one. (3) a rough isomorphism if ϕ : G1 → G2 is both onto and one-to-one. Example 2.6. Let (R, R) be an approximation space, where R is the set of real numbers under addition. Consider the partition R/R = {Q, Qc}, where Q is the set of rational numbers and Qc is the set of irrational numbers. Let G1 = Q, and G2 = R∗ = R − 0. Then G1 = Q and G2 = R. It is clear that G1 and G2 are rough groups. Define ϕ : Q → R as follow: for every x ∈ Q, ϕ(x) = x. It is not difficult to see that ϕ is a rough monomorphism. Example 2.7. Let U = Z4 and consider the partition U/R = {{1, 2}, {0, 3}}. Let G1 = {0, 2}, and G2 = {1, 2, 3}. Then G1 = Z4, and G2 = Z4. It is clear that G1 and G2 are rough groups. Define ϕ : G1 → G2 as follows: ϕ(x) = x, ∀x ∈ G1. It is not difficult to see that ϕ is a rough epimorphism and a rough monomorphism. Thus ϕ is a rough isomorphism. 3. Topological rough groups Throughout this section, we recall the definition of topological rough groups and we give some examples. For more details and properties of these structures, we refer the reader to [6]. Definition 3.1 ([6]). A rough group G with a topology τ on G is called a topological rough group if the following hold. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 297 A. Altassan, N. Alharbi, H. Aydi and C. Özel (1) f : G×G → G which defined by f(x, y) = xy is continuous with respect to a product topology on G × G and the topology τG on G induced by τ; (2) ι : G → G which defined by ι(x) = x−1 is continuous with respect to the topology τG on G induced by τ. Now, we present three different examples of topological rough groups. Example 3.2. Let U = Z4 be the group of integers modulo 4. Let U/R = {{0, 2, 3}, {1}} be a classification of an equivalence relation and G = {1, 2, 3}. Then G = {1} and G = {0, 1, 2, 3} = Z4. Given a topology τ = {∅, G, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} on G. Then the relative topology on G is τG = {∅, G, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}. The two conditions in Definition 3.1 are satisfied as follows: (1) The product mapping f : G × G → G = Z4 is continuous with respect to product topology on G × G and the topology τG on G induced by the topology τ on G = Z4. For instance, the open set {1, 2} in τG has inverse {{3} × {2} ∪ {3} × {3}} which is open in the product topology. (2) The inverse mapping i : G → G is continuous with respect to the topology τG on G induced by the topology τ. For instance the open set {1} has inverse {3} which is open in τG. Hence G is a topological rough group. Example 3.3. Let U = R and U/R = {{x : x ≥ 0}, {x : x < 0}} be a partition of U. Consider G = R∗ = R − 0. Then G = {x : x < 0}, G = R. And G is a rough additive group. Let D be the discrete topology on G = R, then (1) the product mapping f : R∗ × R∗ → R is continuous with respect to product topology on R∗ × R∗ and the topology DG on R∗ induced by the discrete topology D on R. (2) The inverse mapping i : R∗ → R∗ is continuous with respect to topology DG on R∗ induced by the discrete topology D. Therefore the rough group G is a topological rough group with the discrete topology D on G = R. Example 3.4 ([6]). Let U = S4 be the set of all permutations of four objects and (∗) be the multiplication operation of permutations. Consider U/R = {E1, E2, E3, E4}, to be a partition of U, where E1 = {1, (12), (13), (14), (23), (24), (34)} E2 = {(123), (132), (142), (124), (134), (143), (234), (243)} E3 = {(1234), (1243), (1342), (1324), (1423), (1432)} E4 = {(12)(34), (13)(24), (14)(23)}. For G = {(12), (123), (132)}, G = E1 ∪ E2. It is not difficult to see that G is a rough group. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 298 Rough action on topological rough groups For a given topology τ = {∅, G, {(12)}, {1, (132), (123)}, {1, (12), (132), (123)}} on G, the relative topology on G is τG = {∅, G, {(12)}, {(132), (123)}}. More- over by examine the two conditions in Definition 3.1, we can see that G is a topological rough group. Definition 3.5 ([6]). Let G be a topological rough group. For a fixed element a in G, we define the following: (1) A mapping La : G → G which is defined by La(x) = ax, is called a left transformation from G into G. (2) A mapping Ra : G → G which is defined by Ra(x) = xa, is called a right transformation from G into G. Proposition 3.6 ([6]). Let G be a topological rough group. Then (1) The left transformation map La : G → G is continuous and one-to-one. (2) The right transformation map Ra : G → G is continuous and one-to- one. (3) The inverse mapping ι : G → G is a homeomorphism for all x ∈ G. 4. Rough action and rough homogenous spaces in classical set topology In this section, we discuss our main results. We introduce rough action and rough homogenous spaces in classical set topology using rough groups. First, we recall cartesian product of topological rough groups. Let (U, R1) and (V, R2) be approximation spaces with binary operations ∗1 and ∗2, respec- tively. For x, x ′ ∈ U and y, y ′ ∈ V, we have (x, y), (x ′ , y ′ ) ∈ U × V . Define ∗ as (x, y) ∗ (x ′ , y ′ ) = (x ∗1 x ′ , y ∗2 y ′ ). Then ∗ is a binary operation on U × V . In deed, that the product of equivalence relations R1 and R2 is also an equivalence relation on U × V (see [3] ). Moreover, we have the following result. Theorem 4.1 ([4]). Let G1 ⊆ U and G2 ⊆ V be two rough groups. Then the cartesian product G1 × G2 is also a rough group. Now, let (U, R1) and (V, R2) be approximation spaces. Let G1 ⊆ U and G2 ⊆ V be topological rough groups such that τ1 and τ2 are topologies on G1 and G2, respectively inducing τG1 and τG2 on G1 and G2, respectively. A mapping ϕ : G1 → G2 is called a topological rough group homo- morphism, if ϕ is a rough homomorphism and continuous with respect to the topology τ2 on G2 inducing τG2 on G2 and the topology τ1 on G1 inducing τG1 on G1. A topological rough group homomorphism ϕ : G1 → G2 is called a topo- logical rough group homeomorphism, if there exists a topological rough group homomorphism ϕ−1 : G2 → G1 such that ϕ−1 ◦ ϕ = 1G1. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 299 A. Altassan, N. Alharbi, H. Aydi and C. Özel The next definition is equivalent to the definition of rough kernel in rough groups that is given in [18]. Definition 4.2. Let G1 and G2 be topological rough groups, ϕ : G1 → G2 be a topological rough group homomorphism and let e2 be the rough identity element in G2. Then ker(ϕ) = {g ∈ G1 : ϕ(g) = e2}. is called the rough kernel associated with the map ϕ. In the next theorem, we prove that, the kernel in Definition 4.2 is a rough normal subgroup of G1. Theorem 4.3. Let ϕ be a topological rough group homomorphism from G1 to G2. Then the rough kernel is a rough normal subgroup of G1. Proof. For every x, y ∈ ker(ϕ), we have ϕ(x) = e2, and ϕ(y) = e2. (1) Since ϕ(x ∗ y) = ϕ(x) ∗ ′ ϕ(y) = e2, we have x ∗ y ∈ ker(ϕ). (2) We have ϕ(x−1) = (ϕ(x))−1 = (e2) −1. Hence ker(ϕ) is a rough sub- group of G1. (3) For every x ∈ G1 and r ∈ ker(ϕ), we have ϕ(x ∗ r ∗ x−1) = ϕ(x) ∗ ′ ϕ(r) ∗ ′ ϕ(x−1) = e2. Therefore, x ∗ r ∗ x−1 ∈ ker(ϕ). Thus ker(ϕ) is a rough normal subgroup of G1. □ Note that, the rough kernel is always a subset of the upper approximation of G1. Indeed, if G1 is a group then the kernel is a normal subgroup of G1. Example 4.4. Consider the map ϕ : Z4 → R, where G = {1, 2, 3} and R∗ are the rough groups in Example 3.2 and Example 3.3, respectively. Define ϕ as follows: ϕ(0) = 0, ϕ(1) = 0, ϕ(2) = 0, ϕ(3) = 0. Clearly, ϕ is continuous and homomorphism. Hence ϕ is a topological rough group homomorphism. From Definition 4.2, it is easy to see that ker(ϕ) = {1, 2, 3} is a subset of G. Moreover, ker(ϕ) is a rough normal subgroup of G. Let (U, R) be an approximation space. Assume moreover that G is a topo- logical rough group and X is a subset of U. Denote by X† the topological space inducing the topological space X; where X is a rough set with ordinary topology. Now, we are ready to give the definition of the action of a rough group G on a rough space. Definition 4.5. A continuous map ϕ : G×X† → X† (resp. ϕ : X† ×G → X†) is called a left (resp. right) rough action of G on X, if it satisfies the following conditions: (1) g(g ′ x) = (gg ′ )x (resp. ((xg)g ′ = x(gg ′ )), for all g, g ′ ∈ G and x ∈ X†. (2) ex = x (resp. xe = x), for every x ∈ X†, where e ∈ G is the rough identity. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 300 Rough action on topological rough groups Then the rough set X is called a rough G-space. The action ϕ is said to be effective if gx = g ′ x, for every x ∈ X† implies g = g ′ . In addition, the action ϕ is said to be transitive, if for every x, x ′ ∈ X†, there exists g ∈ G such that gx = x ′ . Definition 4.6. Let X be a rough G-space. Then X is said to be topo- logically rough homogeneous if for all x, y ∈ X†, there is a topological homeomorphism ϕ : X† → X† such that ϕ(x) = y. The action of a topological rough group on itself is discussed in the following proposition. Proposition 4.7. Let H be a rough subgroup of the topological rough group G, and let G be a group. Then H acts on G. Moreover, G acts roughly on itself. Proof. Since H is a rough subgroup of G, H is a subset of the group G. There- fore the continuous map ϕ : H × G → G is a left rough action of H on G. Also since G is a group, the continuous map ϕ : G × G → G is a left rough action of G on G. □ Theorem 4.8. Let G be a topological rough group and X be a rough G- space. Then for every g ∈ G, the left transformation map Lg : X† → X†, (resp. right transformation map Rg : X † → X†) which is defined by Lg(x) = gx (Rg(x) = xg), is a topological homeomorphism. Proof. Indeed, the continuity of the action ϕ implies the continuity of Lg. The conditions 1 and 2 in Definition 4.5 are respectively equivalent to (1) Lg ◦ Lg′ = Lgg′ . (2) Le = 1X. Therefore, the maps Lg and Lg−1 are inverses of each other. Thus, Lg is a topological homeomorphism from X to X†. □ Note that, the left (resp. right) transformation map Lg(Rg) from X † into X†, is not in general a topological homeomorphism for every g ∈ G. Indeed, this is only true in the case where G is a group. From now on, we will focus on studying open subsets in topological rough groups. Corollary 4.9. Let G be topological rough group. Then for every open set O in X† and g ∈ G, Lg(O) = gO is open in X†. Proof. By Theorem 4.8, Lg(O) = X † → X† is a topological homeomorphism. Thus gO is open set in X†. □ Theorem 4.10. Let G be a topological rough group such that G is a group. For any open subset O of G, if A is a subset of G, then AO (respectively OA) is open in G. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 301 A. Altassan, N. Alharbi, H. Aydi and C. Özel Proof. The fact that G is a group implies that G acts on itself. Thus for every g ∈ G, Lg is a topological homeomorphism. The rest of proof follows immediately from left transformation definition. Therefore AO = ∪a∈ALa(O). Similarly OA = ∪a∈ARa(O) is open in G. □ Theorem 4.11. Let G be a topological rough group such that G is a group. Let H be a rough subgroup of G such that H is closed under multiplication. If there is an open set O in G such that e ∈ O and O ⊆ H, then H is open set in G. Proof. Let O be a non-empty open set in G such that O ⊆ H and e ∈ O. Then for every h ∈ H, Lh(O) = hO is open in G. Hence H = ! h∈H hO is open in G. □ Theorem 4.12. Let G be a topological rough group such that G is a group and let H be a rough subgroup of G. Let O be an open set in G such that O ⊆ H. Then for every h ∈ H, hO is an open set in H. Proof. Since H ⊆ G, and G is a group, Lh is a topological homeomorphism. By the definition of left transformation, Lh(O) = hO is open in G. The fact that O ⊆ H implies hO ⊆ H. Hence, hO is open in H. □ Using the notion of open subsets in topological rough groups, we define the following set. Definition 4.13. Let G be a topological rough group and let B ⊆ τ be a base for τ. For g ∈ G, the family Bg = {O ∩ G : O ∈ B, g ∈ O} ⊆ B is called a base at g in τG. Example 4.14. In Example 3.3 the family B = {{x} : x ∈ R} is a base for D. For every g ∈ G the collection B = {{g} : g ∈ R∗} is a base for τG. Theorem 4.15. Let G be a topological rough group such that the identity element e ∈ G and G is closed under multiplication. Let G be an open set in G. For g ∈ G the base of g in G is equal to Bg = {gO : O ∈ Be}, where Be is the base of the identity e in τG. Proof. Since g ∈ G, we have g ∈ G. Let O1 be an open set in G and let g ∈ U. Since e ∈ G, and G is a topological rough group, there are two open sets O2 and O3 such that g ∈ O2, e ∈ O3 and ϕ(O2 ×O3) ⊆ O1. We have G is an open set in τ. Then O3 is a neighbourhood of e in τ. Then there is a basic open set O ∈ Be such that e ∈ O ⊆ O3. Hence Lg(O) = gO ⊆ ϕ(O2×O) ⊆ ϕ(O2×O3) ⊆ O1. □ c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 302 Rough action on topological rough groups Acknowledgement The authors wish to thank the Deanship for Scien- tific Research (DSR) at King Abdulaziz University for financially funding this project under grant no. KEP-PhD-2-130-39. Also, we would like to thank the editor and referees for their valuable suggestions which have improved the presentation of the paper. References [1] S. Akduman, E. Zeliha, A. Zemci and S. Narli, Rough topology on covering based rough sets, 1st International Eurasian Conference on Mathematical Sciences and Applications (IECMSA), Prishtine, Kosovo, 3ÔÇô7 September 2012. [2] S. Akduman, A. Zemci and C. Özel, Rough topology on covering-based rough sets, Int. J. Computational Systems Engineering 2, no. 2 (2015),107–111. [3] N. Alharbi, H. Aydi and C. Özel, Rough spaces on covering based rough sets, European Journal of Pure And Applied Mathematics (EJPAM) 12, no. 2 (2019). [4] N. Alharbi, H. Aydi, C. Park and C. Özel, On topological rough groups, J. Computa- tional Analysis and Applications 29, no. 1 (2021), 117 –122. [5] A. Arhangel’skii and M. Tkachenko, Topological groups and related structures, Atlantis press/ World Scientific, Amsterdam-Paris, 2008. [6] N. Bagirmaz, I. Icen and A. F. Ozcan, Topological rough groups, Topol. Algebra Appl. 4 (2016), 31–38. [7] R. Biswas and S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math. 42 (1994), 251–254. [8] E. Brynairski, A calculus of rough sets of the first order, Bull. of the Polish Academy Sciences: Mathematics 37, no. 1-6 (1989), 71–78. [9] G. Chiaselotti and F. Infusino, Some classes of abstract simplicial complexes motivated by module theory, Journal of Pure and Applied Algebra 225 (2020), 106471, [10] G. Chiaselotti and F. Infusino, Alexandroff topologies and monoid actions, Forum Math- ematicum 32, no. 3 (2020), 795–826. [11] G. Chiaselotti, F. Infusino and P. A. Oliverio, Set relations and set systems induced by some families of integral domains, Advances in Mathematics 363 (2020), 106999, [12] G. Chiaselotti, T. Gentile and F. Infusino, Lattice representation with algebraic granular computing methods, Electronic Journal of Combinatorics 27, no. 1 (2020), P1.19. [13] S. Hallan, A. Asberg and T. H. Edna, Additional value of biochemical tests in suspected acute appendicitis, European Journal of Surgery 163, no. 7 (1997), 533–538. [14] R. R. Hashemi, F. R. Jelovsek and M. Razzaghi, Developmental toxicity risk assessment: A rough sets approach, Methods of Information in Medicine 32, no. 1 (1993), 47–54. [15] A. Huang, H. Zhao and W. Zhu, Nullity-based matroid of rough sets and its application to attribute reduction, Information Sciences 263 (2014), 153–165. [16] A. Kusiak, Decomposition in data mining: An industrial case study, IEEE Transactions on Electronics Packaging Manufacturing 23 (2000), 345–353. [17] A. Kusiak, Rough set theory: A data mining tool for semiconductor manufacturing, IEEE Transactions on Electronics Packaging Manufacturing 24, no. 1(2001), 44–50. [18] C. A. Neelima and P. Isaac, Rough anti-homomorphism on a rough group, Global Journal of Mathematical Sciences: Theory and Practical 6, no. 2, (2014), 79–80. [19] M. Novotny and Z. Pawlak, On rough equalities, Bulletin of the Polish Academy of Sciences, Mathematics 33, no. 1-2 (1985), 99–104. [20] N. Paul, Decision making in an information system via new topology, Annals of fuzzy Mathematics and Informatics 12, no. 5 (2016), 591–600. [21] Z. Pawlak,Rough sets, Int. J. Comput. Inform. Sci. 11, no. 5 (1982), 341–356. [22] J. Pomykala, The stone algebra of rough sets, Bulletin of the Polish Academy of Sciences, Mathematics 36, no. 7-8 (1988), 495–508. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 303 A. Altassan, N. Alharbi, H. Aydi and C. Özel [23] J. Tanga, K. Shea, F. Min and W. Zhu, A matroidal approach to rough set theory, Theoretical Computer Science 471 (2013), 1–11. [24] S. Wang, Q. Zhu, W. Zhu and F. Min, Graph and matrix approaches to rough sets through matroids, Information Sciences 288 (2014), 1–11. [25] S. Wang, Q. Zhu, W. Zhu and F. Min, Rough set characterization for 2-circuit matroid, Fundamenta Informaticae 129 (2014), 377–393. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 304