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On Proximity Spaces and Topological Hyper Nearrings https://communications.science.ankara.edu.tr Commun.Fac.Sci.Univ.Ank.Ser. A1 Math. Stat. Volume 69, Number 2, Pages 1418—1427 (2020) DOI: 10.31801/cfsuasmas.764635 ISSN 1303—5991 E-ISSN 2618—6470 Received by the editors: July 06, 2020; Accepted: September 21, 2020 ON PROXIMITY SPACES AND TOPOLOGICAL HYPER NEARRINGS Somaye BORHANI-NEJAD and B. DAVVAZ Department of Mathematics, Yazd University, Yazd, IRAN Abstract. In 1934 the concept of algebraic hyperstructures was first intro- duced by a French mathematician, Marty. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyper- structure, the result of this composition is a set. In this paper, we prove some results in topological hyper nearring. Then we present a proximity relation on an arbitrary hyper nearring and show that every hyper nearring with a topol- ogy that is induced by this proximity is a topological hyper nearring. In the following, we prove that every topological hyper nearring can be a proximity space. 1. Introduction In 1934, the concept of hypergroups was first introduced by a French mathemati- cian, Marty [22]. In the following, it was studied and extended by many researchers, namely, Corsini [3], Corsini and Leoreanu [4], Davvaz [6—8],Frenni [12], Koskas [20], Mittas [23], Vougiouklis, and others. The topological hyper nearring notion is de- fined and studied by Borhani and Davvaz in [2]. In the 1950’s, Efremovi¼c[10, 11], a Russian mathematician, gave the definition of proximity space, which he called infinitesimal space in a series of his papers. He axiomatically characterized the proximity relation A is near B for subsets A and B of any set X. The set X, together with this relation, was called an infinitesimal (proximity) space. Defining the closure of a subset A of X to be the collection of all points of X near A, Efremovi¼c[10,11] showed that a topology can be introduced in a proximity space. In this paper, we study some remarks on topological hyper nearring, then we 2020 Mathematics Subject Classification. Primary 16Y99; Secondary 20N20. Keywords and phrases. Hyper nearring, topological hyper nearring, complete part, proximity relation. [email protected]; [email protected] author; 0000-0001-2345-6789; X0000-0003-1941-5372. c 2020 Ankara University Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics 1418 ON PROXIMITY SPACES AND TOPOLOGICAL HYPER NEARRINGS 1419 define a proximity relation on hyper nearring and, we will prove that every hyper nearring with a topology that is induced by this proximity is a topological hyper nearring. In the following, we show that every topological hyper nearring is a proximity space. 2. Preliminaries In this section, we recall some basic classical definitions of topology from [21] and definitions related to hyperstructures that are used in what follows. Definition 1. [6] A hyper nearring is an algebraic structure (R, +, ) which satisfies the following axioms: · (1) (R, +) is a quasi canonical hypergroup, i.e., in (R, +) the following conditions hold: (i) x + (y + z) = (x + y) + z for all x, y, z R; (ii) There is 0 R such that x + 0 = 0 + x2= x, for all x R; 2 2 (iii) For any x R there exists one and only one x0 R such that 0 x + x0 2 2 2 (we shall write x for x0 and we call it the opposite of x); (iv) z x + y implies y x + z and x z y. 2 2 2 If A and B are two non-empty subsets of R and x R, then we define: 2 A + B = a + b, x + A = x + AandA + x = A + x . f g f g a A b[2B 2 (2) (R, ) is a semigroup respect to the multiplication, having 0 as a left absorbing element, i.e.,· x 0 = 0 for all x R. But, in general, 0 x = 0 for some x R. (3) The multiplication· is left2 distributive with respect· to6 the hyperoperation2 +, i.e., x (y + z) = x y + x z for all x, y, z R. · · · 2 Note that for all x, y R, we have ( x) = x, 0 = 0, (x + y) = y x and x( y) = xy. Let R2 and S be two hyper nearrings. The map f : R S is called a homomorphism if for all x, y R, the following conditions hold: f(x!+y) = f(x) + f(y), f(x y) = f(x) f(y) and2 f(0) = 0. It is easy to see that if f is a homomorphism, then· f( x)· = f(x), for all x R. A nonempty subset H of a hyper nearring R is called a subhyper nearring 2if (H, +) is a subhypergroup of (R, +), i.e., (1) a, b H implies a + b H; (2) a H implies a H; and (3) (H, ) is a subsemigroup2 of (R, ). A subhypergroup 2A of the hypergroup 2 (R, +) is called· normal if for all x R, we· have x + A x A. Let H be a normal hyper R-subgroup of hyper nearrring2 R. In [14], Heidari et al. defined the relation x y(mod H) if and only if (x y) H = , for all x, y H. \ 6 ; 2 This relation is a regular equivalence relation on R. Let (x) be the equivalence class of the element x H and denote the quotient set by R/H. Define the 2 1420 S.BORHANI-NEJAD,B.DAVVAZ hyperoperation and multiplication on R/H by (a) (b) = (c): c (a) + (b) , f 2 g (a) (b) = (a b), · for all a, b R. Let (R, +, ) be a hyper nearring and a topology on R. Then, 2 · we consider a topology on (R) which is generated by = SV : V , P B f 2 g where SV = U (R): U V, U , V . In the following we consider f 2 P 2 g 2 the product topology on R R and the topology on (R) [2]. P Definition 2. [2] Let (R, +, ) be a hyper nearring and (R, ) be a topological space. Then, the system (R, +·, , ) is called a topological hyper nearring if · (1) the mapping (x, y) x + y, from R R to (R), (2) the mapping x 7!x, from R to R, P (3) the mapping (x,7! y) x.y, from R R to R, 7! are continuous. Example 1. [2] The hyper nearring R = ( 0, a, b, c , +, ) defined as follows: f g · + 0 a b c 0 a b c · 0 0 a b c 0 0 a b c a fag f0, ag fbg fcg a 0 a b c b fbg f b g0 f, a,g c fb,g c b 0 a b c c fcg fcg f b, c g0 f, a,g b c 0 a b c f g f g f g f g Let = ?, R, 0, a . Then(R, +, , ) is a topological hyper nearrring. f f gg · Lemma 1. [2] Let (R, +, , ) be a topological hyper nearring. If U is an open set and a complete part of R,· then for every c R, c + U and U + c are open sets. 2 Definition 3. [24] A binary relation on P (X) is called a proximity on X if and only if satisfies the following conditions: (P1) AB implies BA, (P2) AB implies A = , (P3) A B = implies6 ;AB, (P4) A\(B 6 C;) if and only if AB or AC, (P5) A B[implies there exists E X such that A E and B Ec. 6 6 6 The pair (X, ) is called a proximity space. If the sets A, B X are -related, then we write AB, otherwise we write A B. 6 Example 2. Let A, B X and AB if and only if A = and B = . Then is a proximity on X. 6 ; 6 ; The following theorem shows a proximity relation on X induces a topology on X. ON PROXIMITY SPACES AND TOPOLOGICAL HYPER NEARRINGS 1421 Theorem 1. [24] If a subset A of a proximity space (X, ) is defined to be closed if and only if xA implies x A, then the collection of complements of all closed sets so defined yields a topology2 = () on X. 3. Some results on topological hyper nearrings In this section, we present some results and properties in topological hyper near- ring. Lemma 2. Let (R, +, , ) be a topological hyper nearring. Then, 0 U. · 2 R=U 6 S2 Proof. If 0 U, then for every R = U , 0 / U. Let U , U = 62 R=U 6 2 2 2 6 ; and 0 = x U.6 S By2 the continuity of the mapping +, there exist neighborhoods 6 2 V1,V2 of x and 0, respectively, such that V1 + V2 U. Hence, we conclude that 2 V2 = R and V1 + R U. Hence, we have 0 x + ( x) V1 + R U and it is a 2 contradiction. Therefore, we have 0 U. 2 R=U 6 S2 Lemma 3. Let (R, +, , ) be a topological hyper nearring such that every open subset of it is a complete· part of R. Let be the system of all neighborhoods of 0, then for any subset A of R, U A = (A + U). U 2U T Proof. Suppose that x A and U . x U is an open neighborhood of x, hence we have x U A2 = . Thus2 there U exists a A such that a x U. So, x a+U A+U, for\ all U6 ; . Therefore, A 2(A+U). Now, let2 x A+U, 2 2 U U 2 2U for every U and let V be a neighborhood of xT.
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