On weak Γ-(semi)hypergroups

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1 PURE MATHEMATICS RESEARCH ARTICLE On weak Γ-(semi)hypergroups T Zare 1, M Jafarpour 1 * and H Aghabozorgi 1 Received: 29 September 2016 Accepted: 28 December 2016 First Published: 14 February 2017 *Corresponding author: M Jafarpour, Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran mj@vruacir Reviewing editor: Lishan Liu, Qufu Normal University, China Additional information is available at the end of the article Abstract: In this paper, we introduce the classes of weak Γ-(semi)hypergroups as a generalization of the class of Γ-(semi)hypergroups and complementable weak Γ-(semi)hypergroups We show that every non-cover weak Γ-complete hypergroup is complementable Γ-hypergroup Moreover, the associated semihypergroup from weak Γ-(semi)hypergroups and homomorphism of weak Γ-(semi)hypergroups are investigated Subjects: Science; Mathematics & Statistics; Advanced Mathematics; Algebra Keywords: (semi)hypergroup; complementable (semi)hypergroup; weak Γ-(semi)hypergroup AMS subject classification: 20N20 1 Introduction The algebraic hyperstructure notion was introduced in 1934 by a French mathematician Marty (1934), at the 8th Congress of Scandinavian Mathematicians He published some notes on hypergroups, using them in different contexts: algebraic functions, rational fractions, and non-commutative groups Algebraic hyperstructures are a suitable generalization of classical algebraic structures In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set Around the 1940s, the general aspects of the theory, the connections with groups, and various applications in geometry were studied The theory knew an important progress starting with the 1970s, when its research area enlarged A recent book on hyperstructures (Corsini & Leoreanu, 2003) points out to their applications in cryptography, codes, automata, probability, geometry, lattices, binary relations, and graphs and hypergraphs Many authors studied different aspects of semihypergroups, for instance, Bonansinga and Corsini (1982), Corsini (1980), Davvaz (2000), Freni (2004), Leoreanu (2000), and Onipchuk (1993) Sen and Saha (1986) defined the notion of a Γ-semigroup as a generalization of a semigroup One can see that Γ-semigroups are generalizations of semigroups Many classical notions of semigroups have been extended to Γ-semigroups and a lot of results on Γ-semigroups are published by a lot of mathematicians, for instance, Saha (1987), Sen and Chattopadhyay (2004), Sen and Seth (1993), Sen and Saha (1990), Sen and Saha (1986), Zhong and Sen (1997), Seth (1992) and Hila, Davvaz, and Dina (2012) ABOUT THE AUTHOR Morteza Jafarpour is an associate professor in the Department of Mathematics in Vali-e-Asr University of Rafsanjan He has completed his MSc and PhD in pure mathematics He received his PhD at the Vali-e-Asr University of Rafsanjan Jafarpour s current researches are in the field of Algebra, specially hypergroup theory He has a research group with five PhD students who have been working on hypergroup theory Hereby, the findings in this study are the part of the researches about the class of weak Γ-(semi)hypergroups PUBLIC INTEREST STATEMENT Many connections between classical algebraic structures and Γ-structures have been established and investigated In this research, we introduce the classes of weak Γ-(semi)hypergroups as a generalization of the class of Γ-(semi)hypergroups 2017 The Author(s) This open access article is distributed under a Creative Commons Attribution (CC-BY) 40 license Page 1 of 8

2 In this research, first, we introduce a generalization of the notion of Γ-(semi)hypergroup by the notion of weak Γ-(semi)hypergroup Using the class of weak Γ-(semi)hypergroup, we construct a semihypergroup Then, we introduce the class of complementable weak Γ-(semi)hypergroup and weak Γ-complete semihypergroup, respectively Moreover, we prove that every non-cover weak Γ-complete hypergroup is complementable weak Γ-hypergroup Finally, we study homomorphism between two weak Γ-(semi)hypergroups 2 Preliminaries We recall here some basic notions of hypergroup theory and we fix the notations used in this note We referee the readers to the following fundamental books Corsini (1993) Let H be a non-empty set and (H) denote the set of all non-empty subsets of H Let be a hyperoperation (or join operation) on H, that is a function from the chartezian product H H into (H) The image of the pair (a, b) H H under the hyperoperation in (H) is denoted by a b The joint operation can be extended in a natural way to subsets of H as follows: for non-empty subsets A, B of H, define A B = {a b a A, b B} The notation a A is used for {a} A and A a for A {a} Generally, the singleton {a} is identified with its element a The hyperstructure (H, ) is called a semihypergroup if a (b c) =(a b) c for all a, b, c H, which means that u z = x v u x y A semihypergroup (H, ) is called complete if, for all natural numbers n, m 2 and all tuples (x 1, x 2,, x n ) H n and (y 1, y 2,, y m ) H m, we have the following implication: n m n m x i y j x i = y j, i=1 where n x i = x 1 x 2 x n and i=1 j=1 more useful v y z i=1 j=1 m y j = y 1 y 2 y m In practice, the next characterization is j=1 Theorem 21 (Corsini, 1993) A (semi)hypergroup (H, ) is complete if it can be written as the union H = s S A s of its subsets, where S and A s satisfy the conditions: (1) (S, ) is a (semi)group; (2) for all (s, t) S 2, where s t, we have A s At = ; (3) if (a, b) A s A t, then a b = A s t Suppose that (H, ) and (H, ) are two semihypergroups A function f :H H is called a homomorphism if f (a b) f (a) f (b), for all a and b in H We say that f is a good homomorphism if for all a and b in H, f (a b) =f (a) f (b) Moreover, (H, ) and (H, ) are isomorphic if f is a good homomorphism which is a bijection A semihypergroup (H, ) is called a hypergroup if the reproduction law holds: a H = H a = H, for all a H 3 Weak Γ-semihypergroups In this section, we introduce a generalization of the notion of Γ-(semi) hypergroup by the notion of weak Γ-(semi)hypergroup Using the class of weak Γ-(semi)hypergroup, we construct a semihypergroup Definition 31 (see Mirvakili, Anvariyeh, & Davvaz, 2013) Let S and Γ be two non-empty sets S is called a Γ-(semi)hypergroup if for every γ Γ is a hyperoperation on S, ie xγy S, for every x, y S, and for every (α, β) Γ 2 and (x, y, z) S 3, we have (xαy)βz = xα(yβz) Page 2 of 8

3 The Γ-semihypergroup S Γ is called non-cover if for every x, y S and γ Γ we have xγy S From now on, we show the (semi)hypergroup (S, γ) by S γ If every γ Γ is an operation, then S is a Γ-semigroup If S γ is a hypergroup for every γ Γ, then S is called a Γ-hypergroup Definition 32 Let S and Γ be two non-empty sets S is called a weak Γ-(semi)hypergroup if for every γ Γ is a hyperoperation on S, ie xγy S, for every x, y S, and for every (α, β) Γ 2 and (x, y, z) S 3, we have [(xαy)βz] [(xβy)αz] =[xα(yβz)] [xβ(yαz)] Moreover, S is called a weak Γ-hypergroup if for every γ Γ, we have aγs γ = S γ γa = S γ, for all a S γ Example 33 Suppose that S ={e, a, b, c} and Γ ={α, β} Consider the semihypergroups S α and S β, where the hyperoperations α and β are defined on S by the following tables: α e a b c e e a b c a a b c e b b c e a c c e a b β e a b c e e a b c a a e c b b b c e a c c b a e This case (S, Γ) is a weak Γ-hypergroup, which is not a Γ-hypergroup Notice that e = cβc =(aαb)βc aα(bβc) =aαa = b and b = cαc =(aβb)αc aβ(bαc) =aβa = e But we have [(aαb)βc] [(aβb)αc] =[aα(bβc)] [aβ(bαc)] Example 34 Let n, t N, 2 n, and (Z n, +) be the cyclic group of order n Consider the operation ā + t b = a + b + t, for all ā, b Zn Now let Γ = {+ k k K N}; then, (Z n, Γ) is a weak Γ-group Let (S, Γ) be a weak Γ-hypergroup We define the hyperoperation Γ on S as follows: x Γ y = γ Γ xγy for every (x, y) S 2 Proposition 35 If (S, Γ) is a weak Γ-semihypergroup, then (S, Γ ) is a semihypergroup Proof Let (x, y, z) S 3 and a (x Γ y) Γ z Then, there exists (α, β) Γ 2 such that a (xαy)βz so a xα(yβz) or a xβ(yαz) Hence, a x Γ (y Γ z) Therefore, (x Γ y) Γ z x Γ (y Γ z) Similarly, we have x Γ (y Γ z) (x Γ y) Γ z Thus, (S, Γ ) is a semihypergroup From now on, we call (S, Γ ) the (semi)hypergroup associated from (S, Γ) Page 3 of 8

4 Example 36 The associated hypergroup of (S, Γ) in Example 33 is as follows: Γ e a b c e e a b c a a {e, b} c {e, b} b b c e a c c {e, b} a {e, b} Example 37 as follows: Suppose that in Example 34, n = 3 and K ={1, 3} Then, the associated hypergroup is Γ , 1 1, 2 0, 2 1 1, 2 0, 2 0, 1 2 0, 2 0, 1 1, 2 4 Complementable weak Γ-semihypergroups In this section, we introduce the class of complementable weak Γ-(semi)hypergroup and weak Γ-complete semihypergroup, respectively Moreover, we prove that every non-cover weak Γ-complete hypergroup is complementable weak Γ-hypergroup Definition 41 (see Aghabozorgi, Jafarpour, & Cristea, 2016) Let (S, ) be a non-cover semihypergroup We call the complement of (S, ) the hypergroupoid (S, c ) endowed with the complement hyperoperation: x c y = S x y We say that the semihypergroup (S, ) is complementable if its complement (S, c ) is a semihypergroup too, and in this case, (S, c ) is called the complement semihypergroup of (S, ) Definition 42 A non-cover weak Γ-(semi)hypergroup S is called complementable if and only if for every (α, β) Γ 2 and (x, y, z) S 3, [(xα c y)β c z] [(xβ c y)α c z]=[xα c (yβ c z)] [xβ c (yα c z)] Example 43 The weak Γ-hypergroup in Example 33 is a complementable Γ-hypergroup, where their complements, defined as follows α c e a b c e a, b, c e, b, c e, a, c e, a, b a e, b, c e, a, c e, a, b a, b, c b e, a, c e, a, b a, b, c e, b, c c e, a, b a, b, c e, b, c e, a, c β c e a b c e a, b, c e, b, c e, a, c e, a, b a e, b, c a, b, c e, a, b e, a, c b e, a, c e, a, b a, b, c e, b, c c e, a, b e, a, c e, b, c a, b, c are hypergroups, too Example 44 Suppose that S ={e, a, b, c} and Γ ={β} Consider the semihypergroup (S, β) endowed with the hyperoperation β defined as follows: β e a b c e c a, b a, b c a a, b c c a, b b a, b c c a, b c c a, b a, b c In this case, (S, Γ) is not complementable Γ-semihypergroup since (S, β c ) is not a semihypergroup Page 4 of 8

5 Proposition 45 Let (S, Γ) be a complementable weak Γ-semihypergroup Then, for every (x, y) S 2, x c Γ y = γ Γ xγ c y Proof The proof is straightforward Proposition 46 Every non-trivial Γ-group is complementable Proof Let (G, Γ) be a Γ-group If G = 2, then Γ = 1 or Γ = 2 If Γ = 1, then (G, Γ) is a group and hence it is complementable It is to see that if Γ = 2, then Γ c =Γ Hence, (G, Γ) is a complementable group Now suppose that G 3 In this case, we have (xα c y)β c z = u xα c y where u 1 u 2 and u 1, u 2 xα c y, for every (α, β) Γ 2 and (x, y, z) G 3 On the other hand, xα c (yβ c z)= v yβ c z uβ c z u 1 β c z u 2 β c z = G xα c v xα c v 1 xα c v 2 = G where v 1 v 2 and v 1, v 2 xβ c y, for every (α, β) Γ 2 and (x, y, z) G 3 Thus, (xα c y)β c z = xα c (yβ c z), for every (α, β) Γ 2 and (x, y, z) G 3 Theorem 47 Every non-cover weak Γ-complete (semi)hypergroup is complementable Proof Let (H, Γ) be a weak Γ-complete (semi)hypergroup Then, for every γ Γ, H γ is a complete (semi)hypergroup So by Theorem 21, there exist a group (G γ, γ ) and disjoint family {A γ, i } i Gγ such that xγy = A, where x A, y A Now we have i γ j i j xγc y = H A γ,i γ j = A γ,t, for every (x, y) H 2 Thus, (xγ c y)β c z = H = xγ c (yβ c z), for every (γ, β) Γ 2 and (x, y, z) H 3 t G γ i γ j Proposition 48 If (G, α) is a group, then (G, Γ) is a Γ-hypergroup, where Γ ={α, α c } Proof Let (x, y, z) G 3 Then, we have (xα c y)αz =(G xαy)αz = G [(xαy)αz] =G [xα(yαz)] = xα c (yαz) Similarly, we have (xαy)α c z = G [(xαy)αz] =G [xα(yαz)] = xα[g (yαz)] = xα(yα c z) Theorem 49 If (H, β) is a non-cover complete hypergroup, then (H, Γ) is a Γ-hypergroup, where Γ={β, β c } Proof If (H, β) is a non-cover complete hypergroup, then there exist a group (G, α) and a disjoint family {A g } g G such that H = A g, and uβv = A aαb, where u A a, v A b Let x A a, y A b, z A c g G Then, (xβ c y)βz =(H (xβy))βz =(H A aαb )βz = H A (aαb)αc = g G, On the other hand, g (aαb)αc xβ c (yβz) =H xβ(yβz) = H xβa bαc A g = H A aα(bαc) = A g g G, g aα(bαc) Page 5 of 8

6 Thus, (xβ c y)βz = xβ c (yβz); similarly, we have xβ(yβ c z)=(xβy)β c c Moreover, (H, β c ) is also a hypergroup Thus, the aim is valid 5 Homomorphism of weak Γ-semihypergroups In this section, we study homomorphism between two weak Γ-(semi)hypergroups Definition 51 Let (S 1 ) and (S 2 ) be two weak Γ-semihypergroups An ordered pair (θ, φ) of mapping is called a homomorphism of S 1 into S 2 if it satisfies the following properties: (x, y) S 2, γ Γ 1 1, θ(xγy) =θ(x)φ(γ)θ(y) Moreover, if mapping θ and φ are one-to-one correspondence, then (θ, φ) is called an isomorphism; in this case, we write (S 1 ) (S 2 ) Proposition 52 Let (S 1 ) and (S 2 ) be two non-cover isomorphic weak Γ-semihypergroups Then, (S 1, Γ c ) and (S, 1 2 Γc) are isomorphic, where 2 Γc i ={γ c γ Γ i }, for i = 1, 2 Proof have Let (θ, φ) be an isomorphism from (S 1 ) into (S 2 ) Suppose that ψ(γ c )=φ(γ) c ; then, we θ(xγ c y)=θ(s 1 xγy) = θ(s 1 ) θ(xγy) = S 2 θ(x)φ(γ)θ(y) = θ(x)φ(γ) c θ(y) = θ(x)ψ(γ c )θ(y) Therefore, (θ, ψ) is an isomorphism Theorem 53 Let (S 1 ) and (S 2 ) be two weak Γ-semihypergroups and (θ, φ) be a homomorphism from (S 1 ) into (S 2 ) Then, θ is a homomorphism from (S 1, Γ1 ) into (S 2, Γ2 ) Moreover, if (θ, φ) is an isomorphism, then their associated semihypergroups are isomorphic Proof Let (x, y) S 2 1, θ(x Γ1 y)=θ( γ Γ1 xγy) = γ Γ1 θ(xγy) = θ(x)φ(γ)θ(y) γ Γ1 θ(x)γ θ(y) γ Γ 1 = θ(x) Γ2 θ(y) If (θ, φ) is an isomorphism, then θ(x Γ1 y)=θ(x) Γ2 θ(y), for every (x, y) S 2 Thus, (S, Γ ) and (S, Γ ) are isomorphism semihypergroups Page 6 of 8

7 6 Weak Γ-Rosenberg semihypergroups In Rosenberg (1998), the Rosenberg partial hypergroupoid H ρ =(H, ρ ) associated with a binary relation ρ defined on a nonempty set H is constructed as follows For any x, y H, x ρ x ={z H (x, z) ρ} =ρ(x) and x ρ y = x ρ x y ρ y = ρ(x, y) (61) The set D(ρ) ={x H y H:(x, y) ρ} is called the domain of ρ, while R(ρ) ={y H x H:(x, y) ρ} is the range of the relation ρ An element x H is called outer element of ρ if there exists h H such that (h, x) ρ 2 The next theorem gives necessary and sufficient conditions, obtained by Rosenberg, under which the partial hypergroupoid H ρ is a hypergroup Theorem 61 (Rosenberg, 1998) H ρ is a hypergroup if and only if (i) ρ has full domain: D(ρ) =H; (ii) ρ has full range: R(ρ) =H; (iii) ρ ρ 2 ; (iv) If (a, x) ρ 2, then (a, x) ρ, whenever x is an outer element of ρ Let ρ, σ be two relations on H; we define ρσ = {(x, y) (x, u) ρ, (u, y) σ, for some u H} Definition 62 The Γ-semihypergroup (S, Γ) is called Γ-Rosenberg semihypergroup if S γ is a Rosenberg semihypergroup, for every γ Γ Lemma 63 Let ρ, σ be two relations on H such that H ρ and H σ are hypergroups Then, (x ρ z = ρσ(x, y) σ(z) and x ρ (y σ z)=σρ(y, z) ρ(x), for all (x, y, z) H 3 Proof Let (x, y, z) H 3 and a (x ρ z Then, we have a b σ z and b x ρ y, so [a b σ b or a z σ z] and [b x ρ x or b y ρ y] Case1 If a b σ b and [b x ρ x or b y ρ y], we have (b, a) σ and (x, b) ρ; therefore, (x, a) ρσ or (y, a) ρσ; thus, a ρσ(x, y) Case2 If a z σ z, then a σ(z); thus, in each cases, we have (x ρ z ρσ(x, y) σ(z) For the converse, if a ρσ(x, y) σ(z), then a σ(z) or a ρσ(x, y) From a σ(z), we conclude that a (x ρ z If a ρσ(x, y), then (x, a) ρσ or (y, a) ρσ Hence, there exists u H such that (x, u) ρ and (u, a) σ or (y, u) ρ and (u, a) σ Therefore, a (x ρ z Similarly, we have x ρ (y σ z)=σρ(y, z) ρ(x) Theorem 64 Let ρ, σ be two relations on H such that H ρ and H σ are hypergroups and Γ ={ ρ, σ } Then, (H, Γ) is a weak Γ-Rosenberg semihypergroup if and only if [ρσ(x, y) σ(z)] [σρ(x, y) ρ(z)] = [σρ(y, z) ρ(x)] [ρσ(y, z) σ(x)] for all (x, y, z) H 3 Proof The aim follows from previous Lemma, directly Page 7 of 8

8 7 Conclusions Since 1986, when Sen and Saha (1986) defined the notion of Γ-semigroup, many connections between classical algebraic structures and Γ-structures have been established and investigated In this research, we introduce the classes of weak Γ-(semi)hypergroups as a generalization of the class of Γ-(semi)hypergroups and complementable weak Γ-(semi)hypergroups We intend to continue this study in order to obtain a connection between lattices and weak Γ-semihypergroup Funding The authors received no direct funding for this research Author details T Zare 1 Tahere_zare@yahoocom ORCID ID: M Jafarpour 1 h_aghabozorgi1@yahoocom ORCID ID: H Aghabozorgi 1 mj@mailvruacir ORCID ID: 1 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Citation information Cite this article as: On weak Γ-(semi)hypergroups, T Zare, M Jafarpour & H Aghabozorgi, Cogent Mathematics (2017), 4: References Aghabozorgi, H, Jafarpour, M, & Cristea, I (2016) On complementable semihypergroups Communications in Algebra, 44, Bonansinga, P, & Corsini, P (1982) On semihypergroup and hypergroup homomorphisms Bollettino dell Unione Matematica Italiana B, 6, Corsini, P (1980) Sur les, semi-hypergroupes (French) Atti della Societa Peloritana di Scienze Fisiche Matematiche e Naturali, 26, Corsini, P (1993) Prolegomena of hypergroup theory Tricesimo: Aviani Editore Corsini, P, & Leoreanu, V (2003) Applications of hyperstructure theory Advances in Mathematics Dordrecht: Kluwer Academical Publications Davvaz, B (2000) Some results on congruences on semihypergroups Bulletin of the Malaysian Mathematical Sciences Society, 23, Freni, D (2004) Strongly transitive geometric spaces: Applications to hypergroups and semigroups theory Communications in Algebra, 32, Hila, K, Davvaz, B, & Dine, J (2012) Study on the structure of Γ-semihypergroups Communications in Algebra, 40, Leoreanu, V (2000) About the simplifiable cyclic semihypergroups Italian Journal of Pure and Applied Mathematics, 7, Marty, F (1934) Sur une generalization de la notion de groupe In 8th Congress Math Scandinaves (pp 45 49), Stockholm Mirvakili, S, Anvariyeh, SM, & Davvaz, B (2013) Relations on Γ-semihypergroups Journal of Mathematics, 2013, 7 pages Article ID Onipchuk, S V (1993) Regular semihypergroups (Russian) Russian Academy of Sciences Sbornik Mathematics, 76, Rosenberg, I G (1998) Hypergroups and join spaces determined by relations Italian Journal of Pure and Applied Mathematics, 4, Saha, N K (1987) On Γ-semigroup II Bulletin of the Calcutta Mathematical Society, 79, Sen, M K, & Chattopadhyay, S (2004) Semidirect product of a monoid and a Γ-semigroup East West Journal of Mathematics, 6, Sen, M K, & Saha, N K (1986) On Γ-semigroup I Bulletin of the Calcutta Mathematical Society, 78, Sen, M K, & Saha, N K (1990) Orthodox Γ-semigroups International Journal of Mathematics and Mathematical Sciences, 13, Sen, M K, & Seth, A (1993) On po-γ-semigroups Bulletin of the Calcutta Mathematical Society, 85, Seth, A (1992) Γ-group congruences on regular Γ-semigroups International Journal of Mathematics and Mathematical Sciences, 15, Zhong, Z X, & Sen, M K (1997) On several classes of orthodox Γ-semigroups Journal of Pure Mathematics, 14, The Author(s) This open access article is distributed under a Creative Commons Attribution (CC-BY) 40 license You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially The licensor cannot revoke these freedoms as long as you follow the license terms Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits Page 8 of 8