Acta Universitatis Carolinae. Mathematica et Physica
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1 Act Universittis Croline. Mthemtic et Physic Thoms N. Vougiouklis Cyclicity in specil clss of hypergroups Act Universittis Croline. Mthemtic et Physic, Vol. 22 (1981), No. 1, 3--6 Persistent URL: Terms of use: Univerzit Krlov v Prze, 1981 Institute of Mthemtics of the Acdemy of Sciences of the Czech Republic provides ccess to digitized documents strictly for personl use. Ech copy of ny prt of this document must contin these Terms of use. This pper hs been digitized, optimized for electronic delivery nd stmped with digitl signture within the project DML-CZ: The Czech Digitl Mthemtics Librry
2 1981 ACTA UNIVERSITATIS CAROLINAE MATHEMATICA ET PHYSICA Vol. 22. No. 1 Cyclicity in Specil Clss of Hypergroups T. VOUGIOUKLIS Democritus University of Thrce, Mthemtics Deprtment. Xnthi (Greece) Received 6 Mrch 1980 Let (H, *> be multiplictive hypergroup s defined in [1], [2] i.e. the nonempty set H equipped with non-degenerte hyperopertion * :H x H -> 0>(H) :(x,y)\-+x*yczh, x* y 4= 0 (If A, B c H, we set A * B = (J * b. If A = {}, we write A * B = * B.) which ea beb is ssocitive: x * (y * z) = (x * y) * z, Vx, y 9 z e H 9 nd the condition * H = = H * = H, V G H, is vlid. For every integer v > 0, nd Vs e if, we get the powers of s : s l = {s}, s v+1 = = s v * s c H. Now, using the originl definition of cyclic hypergroup s we cn see in [3] s well, we give the following definitions. Definitions. A hypergroup H is clled cyclic, if H = h 1 u h 2 u... u h n u..., for some heh. (1) If there exists n integer n > 0, the minimum one with the following property H = h 1 U/J 2 U...U K 9 (2) then we cll H cyclic hypergroup with finite period nd we cll h genertor of H with period n. If there is no number n for which (2) is vlid, but (1) is vlid, then we sy tht H hs infinite period for h. If ll genertors of H hve the sme period, then we cll H cyclic with period. If there exists n integer n > 0, the minimum one with the following property H = h n, (3) then we cll H single-power cyclic hypergroup nd h genertor of H with period w. If (1) is vlid nd lso Vn e M 0 nd n ^ n 09 for constnt n 0 e N 0, the following condition is vlid h 1 uh 2 u...uh"- 1 h\ (4)
3 then we cll H single-power cyclic hypergroup with infinite period for h. Obviously we cn prove the following proposition. Proposition 1. Let (H, ) be commuttive group nd P subset of H. Then H, \ is hypergroup, where the hyperopertion is defined by the reltion ' : H x H - &(H) : (x, y) r-> x P^y = xy({e} u P), (5) where e is the unit element of (H, ). We shll cll the bove hypergroup P-hypergroup. Proposition 2. Let (H n, ) be finite cyclic group #H = n nd P H n. Then P\ P H, ), where is defined by (5), is cyclic hypergroup which we shll cll P-cyclic hypergroup. Proof. From now on we denote the powers of the elements of H n for the hyperopertion in squre brckets. We cn esily see tht: x = x iv \{e} upu^u.^up^ 1), VveN 0. (6) So if e H is genertor of (H n, ), ll over in this pper, then [1] u [2] u...u [ " ] = H, so is genertor of ( H n, \ with period t most n. In the following, we shll prove some theorems which re vlid in the specil cse of P-cyclic hypergroups, where P = {p} is set with only one element. We write it s /H M, P Theorem 1. In the P-cyclic hypergroup / H ni \ the element x is genertor iff (A, x, n) = 1, i.e. k, x, n re reltively prime. Proof. The /*-th power of the element k under the hyperopertion reltion (6), is x, using the = { A *, A «+ *,..., ^*" 1 *}. (7) Therefore the elements of the powers of k hve the form Xs+t *, where sen 0 nd t = 0, 1,..., s - 1. Also we hve As + tx = 1 mod n iff 3g e Z : h + tx - Q n = 1 iff (A, x, n) =- 1.
4 Xs+t So if we choose pproprite s, t, Q mod n, s we need bove, the reltion * = = is vlid iff (X, x,n)=l. Therefore the element e H n belongs to some l = power of x iff (X, x, n) = 1. Now, if belongs to some power of x, then VveN 0 the element v e H n belongs to some power of x, becuse A(vs) + (vr)x v ^ From the bove, we obtin tht the element x is genertor of ( H n, ) iff (X, x, n) = 1. ď Theorem 2. In the P-cyclic hypergroup / H n, \ x 4= n = e, (i) the element x is genertor with period \x = [n/2] + 1 (where [n/2] = z, when n = 2z or n = 2z + 1), (ii) the element n ~ x is genertor with period n iff (n, x) = 1. Proof (i) From (7) VX e N 0, we get x[a] = xa x(a + l) j x(2a-l)} nd x[a + l] = x(a + l) x(a + 2)^ ^ x(2a-l)^ fl x2a^ x(2a+l) Therefore, incresing the power of x from X to X + 1, there pper t most two new elements, i.e. x2x x(2a+1) nd. Since x[1j = { x } is set with only one element, to cover H n we need t lest [n/2] other successive powers of x. In either cse, if n is odd or even, for \i = [n/2] + 1 we get x[i] u x[2] UmmmU *M = x? x2? ^ ««-D 9 e ] ( 8 ) nd in every higher power of x the sme elements re ppering. If (n, x) = 1, then the elements of the set (8) re different, so x is genertor with period [n/2] + 1. If (n, x) 4= 1, then (x, x, n) # 1; so from theorem 1 we get tht x is not genertor. (ii) From (7), VXeN 0 nd ^ < n, we get (n-x)[a] = j (rt-x)a (»i-x)a + x ^ (rt-x)a + (A-l)x) ^^ (я-x)[л + l] = f (я-x)(a + l) (я-x)(л + l) + x ^ (я-x)(л + l) + Axì from where we cn see esily tht («-x)[a+i] = («-x)(a+in u («-x)[a] ^ ^ < n. Let (n, x) = 1, then (я-x)(a + l) _(я-x)[a] (я-x)(a + l),
5 becuse, if there exists te{0, 1,..., k - 1} such tht fl(«-*x* + -> = fl0i-*>*+* then x(t + 1) = 0 mod n, which is contrdiction. Therefore the sequence of sets 0.-*)[l] ("-*)[2] fl(n-x)[n] is strictly incresing nd lso the set ^" x)cn] hs exctly n different elements of H n, i.e. <"-*>-"- = #. So the element n ~ x is genertor with period n of (H n, Y when (n, x) = 1. Let now (n, x) + 1, then (x, n x, n) 4= 1. Hence from theorem 1 we get tht n ~ x is not genertor. Q.E.D. The bove theorem sttes tht from n P-cyclic hypergroups (H n, Y <p(n) elements x nd <p(n) elements n ~ x re genertors, where cp(n) is the Euler's phifunction. Theorem 3. The P-cyclic hypergroup ( H n, Y x -# e, is single-power cyclic «) " * hypergroup iff (x, n) = 1 nd in this cse every element of H n is genertor of x H n, \ with period n. / x Proof. In the reltion (7) we hve t most \i different elements, so in order / H n, to be P-cyclic hypergroup we must hve \i = n. For \x = n, we hve A - w - = { A ", Art+x,..., <,* +<»-->*} = {^, x }..., (n " 1)x }, while, for every e N, we get ^A[fi + <r] fla<r fla[n] Therefore (H n, Y x #= e, is single-power P-cyclic hypergroup with genertor x iff exctly the n-th power of x is equl to H. The n elements of j> ACn] re different iff (x, n) = 1, independently of k, nd the period of x is n. References II] DUBREIL P.: Algebre, Tome I. Guthier-Villrs, Pris [2] KONGUETSOF L., Bull. Soc. Mth. Belgique , 211. [3] WALL H. S.: Amer. J. Mth , 77.
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