On some Hypergroups and their Hyperlattice Structures

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1 BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 3(43), 2003, Page ISSN On ome Hypergroup and their Hyperlattice Structure G.A. Moghani, A.R. Ahrafi Abtract. Let G be a hypergroup and L(G) be the et of all ubhypergroup of G. In thi urvey article, we introduce ome hypergroup G from combinatorial tructure and tudy the tructure of the et L(G). We prove that in ome cae L(G) ha a lattice or hyperlattice tructure. Mathematic ubject claification: 20N20. Keyword and phrae: Hypergroup, hyperlattice, integer partition. 1 Introduction Firt of all we will recall ome algebraic definition ued in the paper. A hypertructure i a et H together with a function : H H P (H) called hyperoperation, where P (H) denote the et of all non-empty ubet of H. F.Marty [18] defined a hypergroup a a hypertructure (H,.) uch that the following axiom hold: (i) (x.y).z = x.(y.z) for all x,y,z in H, (ii) x.h = H.x = H for all x in H. The axiom (ii) i called the reproduction axiom. A commutative hypergroup (H, o) i called a join pace if for all a,b,c,d H, the implication a/b c/d = aod boc i valid, in which a/b = {x a xob}. The concept of an H v -group i introduced by T.Vougioukli in [20] and it i a hypertructure (H,.) uch that (i) (x.y).z x.(y.z), for all x,y,z in H, (ii) x.h = H.x = H for all x in H. The firt axiom i called weak aociativity. Following Gionfriddo [12] and Vougioukli [20], we define a generalized permutation on a non-empty et X a a map f : X P (X) uch that the reproductive axiom i valid, i.e. x X f(x) = f(x) = X. The et of all generalized permutation on X i denoted by M X. We now aume that (G, ) i a hypergroup and X i a et. The map : G X P(X) i called a generalized action of G on X if the following axiom hold: 1) For all g,h G and x X, (gh) x g (h x), 2) For all g G, g X = X. Here, for any g G and Y X, g Y i defined a x Y g x, and for any x X and B G, B x i, by definition, equal to b B b x. If the equality hold in the axiom 1) of definition, the generalized action i called trong (ee [17]). Following Kontantinidou and Mitta [15], we define a hyperlattice a a et H on which a hyperoperation and an operation are defined which atify the following c G.A. Mogani, A.R. Ahrafi,

2 16 G.A. MOGHANI, A.R. ASHRAFI axiom: 1. a a a and a a = a, 2. a b = b a and a b = b a, 3. (a b) c = a (b c) and (a b) c = a (b c), 4. a [a (a b)] [a (a b)], 5. a a b implie that b = a b. It i well known [8] that in a lattice the ditributivity of the meet ( ) with repect to the join ( ) implie the ditributivity of the join with repect to the meet and vice vera, the lattice i then called ditributive. But in a hyperlattice a ditinction of everal type of ditributivity i needed. According to Kontantinidou [16], a hyperlattice (H,, ) will be called ditributive if and only if, a (b c) = (a b) (a c), for all a,b,c H. Alo, the hyperlattice (H,, ) i called modular if a b, implie that a (b c) = b (a c), for all c H. The econd author in [2, 3] and [5], tudied the contruction of join pace from ome combinatorial tructure. In [4], he found a new cloed formula for the partition function p(n). We encourage reader to conult thee paper for dicuion and background material. Our notation i tandard and taken mainly from [1, 8 10] and [20]. 2 The Structure of ome Hypergroup Let G be a group, Sym(G) be the group of all permutation on G and Sym e (G) be the tabilizer of the identity e G in Sym(G). Given two permutation φ and ψ from Sym e (G) and an element g G, we define a new permutation φ g ψ = L φ(g) 1φL g ψ, where L φ(g) 1,L g Sym(G) are left multiplication by the element φ(g) 1 and g, repectively. According to [13], a ubgroup H of Sym e (G) cloed under taking product of thi form i called rotary cloed, i.e. H Sym e (G) i called rotary cloed provided that φ g ψ H, for all φ,ψ H and g G. A nice family of rotary cloed ubgroup of Sym e (G), for finite G, come from the theory of Cayley graph and can be obained in the following way. Let Ω be a et of generator for a finite group G not containing the identity element e but containing x 1 together with every x contained in Ω. The ubgroup Rot Ω (G) of Sym e (G) of all permutation preerving e and atifying the condition φ(a) 1 φ(ax) Ω, for every a G and x Ω, i rotary cloed (for detail ee [13] and [14]). In thi ection, firt we introduce a hyperoperation on Sym e (G) and prove that (Sym e (G), ) i a hypergroup. Next, we characterize the ub-hypergroup of thi hypergroup. To do thi, aume that φ,ψ Sym e (G), we define φ ψ = {φ g ψ g G}.

3 ON SOME HYPERGROUPS AND THEIR HYPERLATTICE STRUCTURES 17 Propoition 2.1. (Sym e (G), ) i a hypergroup. Proof. Suppoe φ, ψ and η are arbitrary permutation of Sym e (G). Then we have (φ ψ) η = {φ g ψ g G} η = (φ g ψ) η = g G = {(φ g ψ) h η h G} = {(φ g ψ) h η g,h G}. g G Uing imilar argument a in above, we can how that φ (ψ η) = {φ g (ψ h η) g,h G}. We now aume that g,h G, then we have (φ g ψ) h η = L φ gψ(h) 1φ g ψl h η = L φ(gψ(h)) 1 φ(g)φ g ψl h η = = L φ(gψ(h)) 1 φ(g)l φ(g) 1φL g ψl h η = L φ(gψ(h)) 1φL g ψl h η = = L φ(gψ(h)) 1φL gψ(h) L ψ(h) 1ψL h η = φ gψ(h) (ψ h η) φ (ψ η). Therefore, φ (ψ η) (φ ψ) η. Uing imilar argument we have φ (ψ η) (φ ψ) η and the aociativity i valid. Next we aume that φ Sym e (G) and we have φ Sym e (G) = φ ψ = {φ g ψ g G}. ψ Sym e(g) ψ Sym e(g) Suppoe δ Sym e (G) i arbitrary and ψ = L g 1φ 1 L φ(g) δ. Then, φ g ψ = δ and o Sym e (G) = φ Sym e (G). Similarly, Sym e (G) φ = Sym e (G), which complete the proof. In what follow, we characterize the ub-hypergroup of the hypergroup (Sym e (G), ). Propoition 2.2. Let G be a finite group and H be a non-empty ubet of Sym e (G). H i a ub-hypergroup of Sym e (G) if and only if H i a rotary cloed ubgroup of Sym e (G). Proof. ( ) Suppoe H i a ub-hypergroup of Sym e (G). We firt how that H i a cloed ubet of Sym e (G). To do thi, uppoe φ and ψ are element of H. Then φψ = φ e ψ φ ψ H and o φψ H. Next, for φ,ψ H and g G, φ g ψ φ ψ H, a deired. ( ) Suppoe H Sym e (G) i rotary cloed and φ G. Since H i rotary cloed φ H H. Suppoe ψ H. Put η = φ 1 ψ and g = e. Then, φ g η = φφ 1 ψ = ψ φ η and o H = φ H. Similar argument how that H φ = H, proving the reult. It i a well-known fact that the et of all ubgroup of a group G ha a lattice tructure under the ordinary operation of meet and join. In general, it i far from

4 18 G.A. MOGHANI, A.R. ASHRAFI true that the et of all ub-hypergroup of a hypergroup ha a lattice tructure under thee operation. In fact, the interection of two ub-hypergroup of a hypergroup i not necearily non-empty. Let L(G) be the et of all ub-hypergroup of the hypergroup G. In what follow, we how that Sym e (G) ha a lattice tructure under the ordinary operation of join and meet. Propoition 2.3. L(Sym e (G)) ha a lattice tructure under the ordinary operation of meet and join. Proof. It i an eay fact that {1 G } and Sym e (G) are rotary cloed. Suppoe that H and K are two rotary cloed ubgroup of Sym e (G). It i clear that H K i rotary cloed. We claim that H,K i alo rotary cloed. To do thi, we aume that ψ H, φ K and g G. Then we have: ψ g φ = L ψ(g) 1ψL g φ = L ψ(g) 1ψL g ψψ 1 φ = ψ g ψψ 1 φ H,K. Alo, for ψ 1,ψ 2 H, φ 1,φ 2 K and g G, we have ψ 1 φ 1 g ψ 2 φ 2 = L ψ1 φ 1 (g) 1ψ 1φ 1 L g ψ 2 φ 2 = = L (ψ1 (φ 1 (g))) 1ψ 1L φ1 (g)ψ 1 ψ 1 1 L φ 1 (g) 1φ 1Lgψ 2 φ 2 = = (ψ 1 φ1 (g) ψ 1 )ψ 1 1 (φ 1 g ψ 2 φ 2 ) HK H,K. Uing imilar argument a in above, we can how that H,K i a rotary cloed ubgroup of Sym e (G). Thi how that L(Sym e (G)) ha a lattice tructure under ordinary operation of join and meet. Let G be a et, B an algebraic Boolean algebra and a function from G into B. We define the hyperoperation a follow: a b = {x G (x) (a) (b)}. Since for all x,y G {x,y} x y, (G, ) i an H v -group. It i alo clear that the hyperoperation i commutative. In what follow, we tudy the ub-hypergroup tructure of the hypergroup (G, ). In ome pecial cae we will how that the et L(G) ha a hyperlattice tructure. We alo aume that G a = {g G (g) a}. It i eay to ee that if a B and G a then G a i an H v -ubgroup of G. In what follow, when we write G a, we aume that G a.

5 ON SOME HYPERGROUPS AND THEIR HYPERLATTICE STRUCTURES 19 Propoition 2.4. Let B be a complete Boolean algebra and : G B be a function uch that (G, ) contitute a hypergroup. Alo, we aume that that a 1 a2 an = {g G (g) (a 1 ) (a n )}, and H i a ub-hypergroup of G. Then there exit an element a B uch that H = G a. Proof. Let H be a ub-hypergroup of G and a = b H (b). We claim that H = G a. To ee thi, aume x H. Then (x) b H (b) = a and o x G a, i.e., H G a. We now aume that x G a. Then (x) a = b H (b). Since B i algebraic, there are the element b 1,b 2,,b r of H uch that (x) (b 1 ) (b r ). Now by aumption, x {g G (g) (b 1 ) (b r )} = b 1 b2 br and H i a ub-hypergroup of G, o x H, proving the reult. It i clear that G a b = G a G b, for all a,b B. It i far from true that G a b = G a G b. To ee thi, we contruct an example a follow: Example 2.5. Suppoe G = B = P(X), i the identity function, X 3 and a,b,c ditinct element of X. Set R = {a,b} and S = {c}. Then G R = P(R),G S = P(S) and G R S = P(R S). Now it i eay to ee that G R S G R G S. By the reult of [3] and [4], if the image of G i a -ub-emilattice or contitute a partition of 1, then L(G) = {G a a B&G a }. In thi cae, we define a hyperoperation on L(G) uch that (L(G),, ) contitute a hyperlattice. To do thi, we aume that G a G b = {G x a b x}. In the following lemma we invetigate the condition of a hyperlattice. Lemma 2.6. G a G a G a,g a G a = G a, G a G b = G b G a and G a G b = G a b = G b G a. Proof. Obviou. Lemma 2.7. (G a G b ) G c = G a (G b G c ) and (G a G b ) G c = G a (G b G c ). Proof. The aociativity of i obviou. We will how the aociativity of. Suppoe a,b,c B. Then (G a G b ) G c = {G x a b x} G c = G x G c = = a b x a b x {G t x c t} = {G u a b c u} Similar argument how that G a (G b G c ) = {G u a b c u}, and the reult follow.

6 20 G.A. MOGHANI, A.R. ASHRAFI Lemma 2.8. G a [G a (G a G b )] [(G a (G a G b )], for all a,b B. Proof. Suppoe a,b are arbitrary element of B, then we have G a (G a G b ) = G a G a b = = {G t a (a b) t} = {G t a t}. Therefore, G a G a (G a G b ). On the other hand, G a = G a (a b) = G a G a b G a (G a G b ), a required. Lemma 2.9. G a G a G b implie that G b = G a G b. Proof. Suppoe G a G a G b, then there exit t B uch that G a = G t and a b t. Thu, b = b (a b) b t and o G b G b t = G b G t = G a G b. Therefore, G b = G a G b and the lemma i proved. We ummarize the above lemma in the following theorem: Theorem Let : G B be a function uch that (G, ) contitute a hypergroup. Alo, we aume that for all poitive integer n and the element a 1,,a n of G, we have a 1 a2 an = {g G (g) (a 1 ) (a n )}. Then (L(G),, ) i a hyperlattice. We now invetigate the ditributivity of L(G) and how thi hyperlattice i not ditributive, in general. In fact, we have the following example. Example There exit a function : G B uch that (G, ) i a hypergroup which atifie the condition of Theorem 2.10, but L(G) i not ditributive. To ee thi, we aume that H i a finite group, Π e (H) = {o(x) x H} and : P(H) P(Π e (H)) defined by (A) = {o(x) x A}. It i eay to ee that the function i onto, o by Theorem 3.6 L(P(H)) i a hyperlattice. Suppoe, H = Z 4 = {e,a,a 2,a 3 }, the cyclic group of order four, and G = P(H). Then Π e (Z 4 ) = {1,2,4}. Set A = {1,2},B = {1},C = {2,4} and D = {2}. It i clear that G A (G B G C ) = G A G Πe(G) = G A and (G A G B ) (G A G C ) = G B G D = {G A,G Πe(G)}. Thi how that G A (G B G C ) (G A G B ) (G A G C ). Therefore, L(P(Z 4 )) i a hyperlattice which i not ditributive. It i natural to ak about modularity of L(G). Here, we obtain an example uch that it ub-hypergroup hyperlattice i not modular. Example Aume that X = {1,2,3,4,5}, G = B = P(X) and i the identity function on G. Set A = {1,2},B = {1,2,3} and C = {4,5}. Then G A G B, G A (G B G C ) = 8 and G B (G A G C ) = 2. Thi how that the hyperlattice L(G) i not modular.

7 ON SOME HYPERGROUPS AND THEIR HYPERLATTICE STRUCTURES 21 3 About ome Generalized Action Suppoe : G B i a function uch that (G, ) i a hypergroup and A = Atom(B). Define the map : G A P (A) by g x = {a A a x (g)}. In thi ection, we obtain a condition on uch that i a generalized action and prove that under thi condition the hypergroup (G, ) i iomorphic to a ub-hypergroup of M A. Finally, we define a generalized action of an H v -group on a et X a in hypergroup. We will apply the elementary propertie of a generalized action and prove an inequality between the partition function po(n) and the order of the hypergroup M I(n). Lemma 3.1. Let B be a Boolean algebra and A = Atom(B). If : G B i a function uch that the image of G i a partition of 1, then the map : G A P (A) defined by g a = {x A x a (g)} i a trong generalized action of G on A. Proof. Suppoe g G. Then it i obviou that for all x A, we have x g x a A g a, i.e., A = a A g a. Thu, g A = A and the condition (i) i atified. We now aume that T = {a A a x (g) (h)} and prove that gh x = g (h x) = T. It i eay to ee that gh x g (h x) T. Suppoe a T. Then a x (g) (h) and we have a = (a (h)) [a (x (g)]. We firt aume that a (h), then a (h) = 0 and o a x (g). Thi how that a g x t gh t x = gh x. Next we aume that a = (h). Then a h x gh x and o T = gh x. Uing imilar argument a above, we have T = g (h x), proving the lemma. Lemma 3.2. Let B be a Boolean algebra and A = Atom(B). If : G B i a one-to-one function uch that the image of G i a partition of 1 and that for all g G, there exit an atom x uch that g x 2, then (G, ) i iomorphic to a ub-hypergroup of the hypergroup M A. Proof. By Lemma 3.1, : G A P (A) i a trong generalized action of G on A and by Propoition 3.1 of [17] thi action induced a good homomorphim ξ : G M A defined by ξ(g)(a) = g a. It i enough to how that thi homomorphim i oneto-one. To do thi, uppoe g x = h x, for all x A. By aumption, there exit an atom x uch that g x 2. If a x and a g x then a (g) x = (h) x, and o a x ((g) (h)). Thu, a = (a x) (a (g) (h)) = a (g) (h), i.e., a (g) (h). But, the image of G i a partition of 1, hence (g) = (h) and by injectivity of, g = h. Suppoe : Π d (n) P (I(n)) i defined by (λ) = Part(λ). Then (Π d (n), ) i an H v -group. Define the map : Π d (n) I(n) P (I(n)) by λ k = Part(λ) {k}. In the following imple lemma we how that the map i a trong generalized action of Π d (n) on the et I(n).

8 22 G.A. MOGHANI, A.R. ASHRAFI Lemma 3.3. The map : Π d (n) I(n) P (I(n)) defined by λ k = Part(λ) {k} i a trong generalized action of Π d (n) on the et I(n). Proof. We firt aume that n i an odd poitive integer, we define the partition µ i, 0 i [ n 2 ], by the following table: µ 0 µ 1 µ 2 m [ n 2 ] n = n n = 1 + (n 1) n = 2 + (n 2) n = [ n 2 ] + (n [n 2 ]) Next we aume that n i even, then we define the partition ξ i, 0 i n 2, by ξ i = µ i, for all i < n 2 and ξ n i the partition n = 1 + (n 2 2 1) + n 2. Then it i clear that [ n 2 ] i=1 Part(µ i ) = [ n 2 ] i=1 Part(ξ i ) = I(n), and o the reproduction axiom i valid. We now aume that λ,µ are arbitrary partition and m I(n). Then we have = On the other hand, δ λ µ = k λ m (λ µ) m = δ λ µ δ m = Part(δ) {m} = Part(λ) Part(µ) {m}. λ (µ m) = k λ m λ k = (Part(λ) {k}) = Part(λ) Part(µ) {m}. Therefore, the map i a trong generalized action of the H v -group Π d (n) on the et I(n). n 1 Lemma 3.4. po(n) ( 1) i ( n i )(2n i 1) n. i=0 Proof. By Euler partition theorem [1], po(n) = Π d (n) and by Propoition 4.1 of [17], the right hand of thi inequality i the order of M I(n). Therefore, it i enough to how that Π d (n) M I(n). To do thi, we now prove that the induced homomorphim η : Π d (n) M I(n) by η(µ)(x) = µ x i one-to-one. Aume that η(µ) = η(ξ), then µ x = ξ x, for all x I(n). Thu, Part(µ) {x} = Part(ξ) {x}, for all x I(n). Chooe x Part(µ). We have Part(ξ) Part(ξ) {x} = Part(µ). Similarly, Part(µ) Part(ξ) and o Part(µ) = Part(ξ). Now ince µ and ξ have ditinct part, µ = ξ.

9 ON SOME HYPERGROUPS AND THEIR HYPERLATTICE STRUCTURES 23 In the end of thi paper, we define a generalized action of Sym e (G) on the group G. Suppoe : Sym e (G) G P (G) end (φ,g) to φ( g ). Then we have φ (ψ g) = φ ψ( g ) = i Z φ ψ(g i ) = = i Z φ( ψ(g i ) ) = {φ(ψ(g i )) j i,j Z} and φψ g = φψ( g ) = {φ(ψ(g i )) i Z}. Thi how that φψ g φ (ψ g). On the other hand, φ G = g G φ g = g G φ( g ) = G, which how that i a generalized action of Sym e (G) on the group G. Quetion 3.5. When thi generalized action i trong? Reference [1] Andrew G.E., Number Theory. Hindutan Publihing Corporation (India) Delhi, [2] Ahrafi A.R., About ome Join Space and Hyperlattice. To appear in Italian J. of Pure and Appl. Math., [3] Ahrafi A.R., Contruction of ome Join Space from Boolean Algebra. Iranian Int. J. Sci., 2000, no. 1(2), p [4] Ahrafi A.R., An Exact Expreion for the partition function p(n). Far Eat J. Math. Sci., (FJMS), 2000, no. 1(2), p [5] Ahrafi A.R., Hoein-Zadeh A., Yavari M., Hypergraph and Join Space To appear in Italian J. Pure Appl. Math., [6] Ahrafi A.R., Elami-Harandi A.R., Contruction of ome Hypergroup from Combinatorial Structure. To appear in J. Zhejian Univerity Science. [7] Bigg N., White A.T., Permutation Group and Combinatorial Structure. Mathematical Society Lecture Note 33, Cambridge Univerity Pre, Cambridge, [8] Birkhoff G., Lattice Theory. 3rd. ed., AMS Coll. Publ. Providence, [9] Corini P., Prolegomena of Hypergroup Theory. Second Edition, Aviani Edittore, [10] P. Corini, Space J., Set P., Set F., Algebraic Hypertructure and Application. Hadronic Pre, Inc., 1994, p [11] Corini P., Hypergraph and Hypergroup. Algebra Univerali, 1996, 35, p [12] Gionfriddo M., Hypergroup aociated with multihomomorphim between generalized graph. Convegno u itemi binari e loro applicazioni, Edited by P. Corini. Taormina, 1978, p [13] Jajcay R., On a new product of group. Europ. J. Combinatoric, 1994, 15, p [14] Jajcay R., Automorphim group of Cayley map. J. of Comb. Theory, Ser. B, 1993, 59, p [15] Kontantinidou M., Mitta J., An introduction to the theory of hyperlattice. Mathematica Balcanica, 1977, no. 7(23), p [16] Kontantinidou-Serafimidou M., Ditributive and complemented hyperlattice. Πρακτ ικα Aκαδηµιαζ Aθηνων, T oµoζ 53oζ, 1978.

10 24 G.A. MOGHANI, A.R. ASHRAFI [17] Madanhekaf A., Ahrafi A.R., Generalized Action of a hypergroup on a et. Italian J. of Pure and Appl. Math., 1998, no. 3, p [18] Marty F., Sur une generalization de la notion de groupe. 8 iem Congre Math Scandinave, Stockholm, 1934, p [19] Sikorki R., Boolean Algebra. Springer-Verlag, Berlin, 2nd Edition, [20] Vougioukli T., Hypertructure and their Repreentation. Hadronic Pre, Inc., G.A. Mogani Department of Mathematic Univerity of Mazandaran Babolar, Iran moghani@umz.ac.ir Received January 20, 2003 A.R. Ahrafi Department of Mathematic Univerity of Kahan, Kahan ahrafi@kahanu.ac.ir

Flag-transitive non-symmetric 2-designs with (r, λ) = 1 and alternating socle

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