Vougiouklis, T.

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Cyclicity in a special class of hypergroups

Institute of Mathematics AS CR, v. v. i.

Acta Universitatis Carolinae. Mathematica et PhysicaThomas N. VougiouklisCyclicity in a special class of hypergroupsActa Universitatis Carolinae. Mathematica et Physica, Vol. 22 (1981), No. 1, 3--6Persistent URL: http://dml.cz/dmlcz/142458Terms of use:© Univerzita Karlova v Praze, 1981Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.cz1981 ACTA UNIVERSITATIS CAROLINAE — MATHEMATICA ET PHYSICA Vol. 22. No . 1 Cyclicity in a Special Class of Hypergroups T. VOUGIOUKLIS Democritus University of Thrace, Mathematics Department. Xanthi (Greece) Received 6 March 1980 Let (H, *> be a multiplicative hypergroup as defined in [1], [2] i.e. the non-empty set H equipped with a non-degenerate hyperoperation * :H x H -> 0>(H) :(x,y)\-+x*yczH, x* y 4= 0 (If A, B c H, we set A * B = (J a * b. If A = {a}, we write A * B = a * B.) which aeA beB is associative: x * (y * z) = (x * y) * z, Vx, y9 z e H9 and the condition a * H = = H * a = H, Va G H, is valid. For every integer v > 0, and Vs e if, we get the powers of s : sl = {s}, sv + 1 = = sv * s c H. Now, using the original definition of cyclic hypergroup as we can see in [3] as well, we give the following definitions. Definitions. A hypergroup H is called cyclic, if H = h1 u h2 u ... u hnu . . . , for some heH. (1) If there exists an integer n > 0, the minimum one with the following property H = h1 U / J 2 U . . . U K9 (2) then we call H cyclic hypergroup with finite period and we call h generator of H with period n. If there is no number n for which (2) is valid, but (1) is valid, then we say that H has infinite period for h. If all generators of H have the same period, then we call H cyclic with period. If there exists an integer n > 0, the minimum one with the following property H = hn , (3) then we call H single-power cyclic hypergroup and h generator of H with period w. If (1) is valid and also Vn e M0 and n ^ n09 for constant n0 e N0, the following con-dition is valid h1 uh2u ...uh"-1 £ h\ (4) then we call H single-power cyclic hypergroup with infinite period for h. Obviously we can prove the following proposition. Proposition 1. Let (H, •) be a commutative group and P a subset of H. Then H, \ is a hypergroup, where the hyperoperation is defined by the relation ' : H x H - &(H) : (x, y) r-> x P^y = xy({e} u P) , (5) where e is the unit element of (H, •). We shall call the above hypergroup P-hypergroup. Proposition 2. Let (Hn, •) be a finite cyclic group #H„ = n and P a Hn. Then P\ P H„, ) , where is defined by (5), is a cyclic hypergroup which we shall call P-cyclic hypergroup. Proof. From now on we denote the powers of the elements of Hn for the hyper-operation in square brackets. We can easily see that: x™ = xiv\{e} u P u ^ u . ^ u P ^ 1 ) , V v e N 0 . (6) So if a e H„ is a generator of (Hn, •), all over in this paper, then a[1]ua[2]u...ua["] = H„, so a is a generator of ( Hn, \ with period at most n. In the following, we shall prove some theorems which are valid in the special case of P-cyclic hypergroups, where P = {p} is a set with only one element. We write it as /HM, P Theorem 1. In the P-cyclic hypergroup / Hni \ the element ax is a generator iff (A, x, n) = 1, i.e. k, x, n are relatively prime. ax Proof. The /*-th power of the element ak under the hyperoperation , using the relation (6), is a™ = {aA*, aA«+*, ..., a ^ * " 1 * } . (7) Therefore the elements of the powers of ak have the form aXs+t* , where seN0 and t = 0, 1, . . . , s - 1 . Also we have As + tx = 1 mod n iff 3g e Z : h + tx - Qn = 1 iff (A, x, n) =- 1 . So if we choose appropriate s, t, Q mod n, as we need above, the relation aXs+t* = = al = a is valid iff (X, x,n)=l. Therefore the element ae Hn belongs to some power of ax iff (X, x, n) = 1. Now, if a belongs to some power of ax, then VveN0 the element av e Hn belongs to some power of ax, because aA(vs) + (vr)x __ a v ^ From the above, we obtain that the element ax is a generator of ( Hn, ) iff (X, x, n) = 1. ď Theorem 2. In the P-cyclic hypergroup / Hn, \ ax 4= an = e, (i) the element ax is a generator with period \x = [n/2] + 1 (where [n/2] = z, when n = 2z or n = 2z + 1), (ii) the element an~x is a generator with period n iff (n, x) = 1. Proof (i) From (7) VX e N0, we get ax[A] = | a x A a x ( A + l) j a x ( 2 A - l ) } and ax[A + l ] = | a x(A + l) ax(A + 2)̂  ^ ax(2A-l)^ flx2A^ a x ( 2 A + l ) | Therefore, increasing the power of ax from X to X + 1, there appear at most two new elements, i.e. ax2X and ax ( 2 A + 1 ) . Since ax [ 1 J = {ax} is a set with only one element, to cover Hn we need at least [n/2] other successive powers of ax. In either case, if n is odd or even, for \i = [n/2] + 1 we get a x [ i ] u a x [ 2 ] UmmmU a*M = | a x ? a x 2 ? ^ a««-D9 e] ( 8 ) and in every higher power of ax the same elements are appearing. If (n, x) = 1, then the elements of the set (8) are different, so ax is a generator with period [n/2] + 1. If (n, x) 4= 1, then (x, x, n) # 1; so from theorem 1 we get that ax is not a generator. (ii) From (7), VXeN0 and ^ < n, we get a(n-x)[A] = ja(rt-x)A a(»i-x)A + x ^ a(rt-x)A + (A-l)x) ^^ a (я-x)[Л + l ] = fa(я-x)(A + l) a(я-x)(Л + l) + x ^ a(я-x)(Л + l) + Axì from where we can see easily that a(«-x)[A+i] = | a ( « - x ) ( A + i n u a(«-x)[A] ^ ^ < n . Let (n, x) = 1, then (я-x)(A + l) a _(я-x)[A] a(я-x)(A + l ) £ a , because, if there exists te{0, 1,. . . , k - 1} such that fl(«-*x*+-> = fl0i-*>*+* t h e n x(t + 1) = 0 mod n, which is a contradiction. Therefore the sequence of sets a 0 . - * ) [ l ] a("-*)[2] fl(n-x)[n] is strictly increasing and also the set a^"x)Cn] has exactly n different elements of Hn, i.e. a<"-*>-"- = #„. So the element an~x is a generator with period n of (Hn, Y when (n, x) = 1. Let now (n, x) + 1, then (x, n — x, n) 4= 1. Hence from theorem 1 we get that an~x is not a generator. Q.E.D. The above theorem states that from n P-cyclic hypergroups (Hn, Y <p(n) elements ax and <p(n) elements an~x are generators, where cp(n) is the Euler's phi-function. Theorem 3. The P-cyclic hypergroup ( Hn, Y ax -# e, is a single-power cyclic « ) • " * hypergroup iff (x, n) = 1 and in this case every element of Hn is a generator of ax Hn, \ with period n. / ax Proof. In the relation (7) we have at most \i different elements, so in order / Hn, to be a P-cyclic hypergroup we must have \i = n. For \x = n, we have aA-w- = {aA", aArt+x,..., <,*•+<»-->*} = {̂ , ax} . . . , a(n"1)x} , while, for every a e N, we get ^A[fi + <r] __ flA<r flA[n] Therefore (Hn, Y ax #= e, is a single-power P-cyclic hypergroup with generator ax iff exactly the n-th power of ax is equal to H„. The n elements of j>ACn] are different iff (x, n) = 1, independently of k, and the period of ax is n. References II] DUBREIL P.: Algebre, Tome I. Gauthier-Villars, Paris 1963. [2] KONGUETSOF L., Bull. Soc. Math. Belgique 25 1973, 211. [3] WALL H. S.: Amer. J. Math. 59 1937, 77. 

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Cyclicity in a special class of hypergroups

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Institute of Mathematics AS CR, v. v. i.