- Email: [email protected]
Relations on Groups, Polygroups and Hypergroups
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Electronic Notes in Discrete Mathematics 45 (2014) 9–16 www.elsevier.com/locate/endm
Relations on Groups, Polygroups and Hypergroups Bijan Davvaz 1,2 Department of Mathematics Yazd University Yazd, Iran
Abstract 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. By using a certain type of equivalence relations, we can connect semihypergroups to semigroups and hypergroups (polygroups) to groups. These equivalence relations are called strongly regular relations. In this paper, we review some strongly regular relations on hyperstructures and we give some their applications. Keywords: Group, polygroup, hypergroup, binary relation, fundamental relation.
1
Regular relations
Hyperstructure theory both extends some well-known group results and introduce new topics leading us to a wide variety of applications, as well as to a broadening of the investigation fields, for example see [8,9,15]. A comprehensive review of the theory of hyperstructures appears in [3,4,11,20]. Let H 1
I would like to thank Professor A. R. Ashrafi for his hospitality during my stay at the Department of Mathematics, University of Kashan 2 Email: [email protected] 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.11.003
10
B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
be a non-empty set and ◦ : H × H −→ P ∗ (H) be a hyperoperation, where P ∗ (H) is the set of all non-empty subsets of H. The couple (H, ◦) is called a hypergroupoid. Forany two non-empty subsets A and B of H and x ∈ H, we define A ◦ B = a∈A,b∈B a ◦ b, A ◦ x = A ◦ {x} and x ◦ B = {x} ◦ B. A hypergroupoid (H, ◦) is called a semihypergroup if for all a, b, c of H we have (a ◦ b) ◦ c = a ◦ (b ◦ c), which means that u∈a◦b u ◦ c = v∈b◦c a ◦ v. A hypergroupoid (H, ◦) is called a quasihypergroup if for all a of H we have a ◦ H = H ◦ a = H. This condition is also called the reproduction axiom. A hypergroupoid (H, ◦) which is both a semihypergroup and a quasihypergroup is called a hypergroup. A regular hypergroup H is a hypergroup which it has at least one identity and every element has at least one inverse. By using a certain type of equivalence relations, we can connect semihypergroups to semigroups and hypergroups to groups. These equivalence relations are called strongly regular relations. More exactly, starting with a (semi)hypergroup and using a strongly regular relation, we can construct a (semi)group structure on the quotient set. Let us define these notions. First, we do some notations. Let (H, ◦) be a semihypergroup and R be an equivalence relation on H. If A and B are non-empty subsets of H, then ARB means that for all a ∈ A there exists b ∈ B such that aRb and for all b ∈ B there exists a ∈ A such that a Rb ; also ARB means that for all a ∈ A and for all b ∈ B, we have aRb. The equivalence relation R is called (1) regular on the right (on the left) if for all x of H, from aRb, it follows that (a ◦ x)R(b ◦ x) ((x ◦ a)R(x ◦ b) respectively); (2) strongly regular on the right (on the left) if for all x of H, from aRb, it follows that (a ◦ x)R(b ◦ x) ((x ◦ a)R(x ◦ b) respectively); (3) R is called regular (strongly regular) if it is regular (strongly regular) on the right and on the left. The reader can find the proofs of the following facts in [3,11]. •
Let (H, ◦) be a semihypergroup and R be an equivalence relation on H. If R is regular, then H/R is a semihypergroup, with respect to the following hyperoperation: x ⊗ y = {z | z ∈ x ◦ y}. If the above hyperoperation is well defined on H/R, then R is regular.
•
If (H, ◦) is a hypergroup and R is an equivalence relation on H, then R is regular if and only if (H/R, ⊗) is a hypergroup.
•
Let (H, ◦) be a semihypergroup and R be an equivalence relation on H. If R is strongly regular, then H/R is a semigroup, with respect to the following operation: x ⊗ y = z for all z ∈ x ◦ y. If the above operation is well defined on H/R, then R is strongly regular.
•
If (H, ◦) is a hypergroup and R is an equivalence relation on H, then R is strongly regular if and only if (H/R, ⊗) is a group.
B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
11
Aghabozorgi et al. [1] studied hypergroups determined by lattices introduced by Varlet and Comer, especially they enumerated Varlet and Comer hypergroups of orders less than 50 and 13, respectively. Another paper [10] deals with the class of γn∗ -complete hypergroups and provided a good insight into the study of hypergroup theory under the strongly regular equivalence relations. In [18], Leoreanu-Fotea and Davvaz studied the notion of a partial n-hypergroupoid, associated with a binary relation. Some important results concerning Rosenberg partial hypergroupoids, induced by relations, are generalized to the case of n-hypergroupoids. Then, n-hypergroups associated with union, intersection, products of relations and also mutually associative n-hypergroupoids are analyzed.
2
Rgular relations on polygroups
A polygroup [2] is a system ℘ =< P, ·, e,−1 >, where e ∈ P, −1 is a unitary operation on P , · maps P × P into the non-empty subsets of P , and the following axioms hold for all x, y, z in P : (1) (x · y) · z = x · (y · z), (2) e · x = x · e = x, (3) x ∈ y · z implies y ∈ x · z −1 and z ∈ y −1 · x. The following elementary facts about polygroups follow easily from the axioms: e ∈ x · x−1 ∩ x−1 · x, e−1 = e, (x−1 )−1 = x, and (x · y)−1 = y −1 · x−1 , where A−1 = {a−1 | a ∈ A}. Suppose that H is a subgroup of a group G. Define a system G//H =< {HgH | g ∈ G}, ∗, H,−I >, where (HgH)−I = Hg −1 H and (Hg1 H) ∗ (Hg2 H) = {Hg1 hg2 H | h ∈ H}. The algebra of double cosets G//H is a polygroup which introduced by Dresher and Ore [12]. Suppose that G is a projective geometry with a set P of points and suppose, for p = q, pq denoted the set of all points on the unique line through p and q. Choose an object I ∈ P and form the system PG =< P ∪ {I}, ·, I,−1 >, where x−1 = x and I · x = x · I = x for all x ∈ P ∪ {I} and for p, q ∈ P , ⎧ ⎨ pq − {p, q} if p = q p·q = ⎩ {p, I} if p = q. PG is a polygroup which introduced by Prenowitz [19]. Let P be a polygroup. We define the relation β ∗ as the smallest equivalence relation on P such that the quotient P/β ∗ , the set of all equivalence classes, is a group. In this case β ∗ is called the fundamental equivalence relation on P and P/β ∗ is called the fundamental group. The product ⊗ in P/β ∗ is defined as follows: β ∗ (x) ⊗ β ∗ (y) = β ∗ (z) for all z ∈ β ∗ (x) · β ∗ (y). Let UP be the set of all finite products of elements of P . We define the relation β as follows:
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B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
xβy if and only if {x, y} ⊆ u for some u ∈ UP . We have β ∗ = β for hypergroups [14]. Since polygroups are certain subclasses of hypergroups, we have β ∗ = β. The kernel of the canonical map ϕ : P −→ P/β ∗ is called the core of P and is denoted by ωP . Here we also denote by ωP the unit of P/β ∗ . It is easy to prove that the following statements: ωP = β ∗ (e) and β ∗ (x)−1 = β ∗ (x−1 ) for all x ∈ P . The β ∗ relation is introduced by Koskas [16] and studied by Corsini, Davvaz, Freni, Leoreanu, Vougiouklis and many others. In [5,6], Davvaz considered this relation on polygroups. We recall the following definition from [13]. If H is a semihypergroup, then we set: γ1 = {(x, x) | x ∈ H} and, for every integer n > 1, the relation is defined as follows: n n xγn y ⇐⇒ ∃(z1 , z2 , . . . , zn ) ∈ H n , ∃σ ∈ Sn : x ∈ zi , y ∈ zσ(i) . i=1
i=1
Obviously, for every n ≥ 1, the relations γn are symmetric, and the relation γ = n≥1 γn is reflexive and symmetric. Let γ ∗ be the transitive closure of γ. Then, γ ∗ is the smallest strongly regular equivalence such that H/γ ∗ is a commutative semigroup. Also, in every hypergroup, the relation γ is transitive, that is γ ∗ = γ, and in this case, the quotient H/γ ∗ is an abelian group [13]. The γ ∗ -relation is a generalization of the β ∗ -relation. Remark 2.1 Note that we can consider the relation γ ∗ on a non-abelian group G. So, γ ∗ is the smallest equivalence relation on G such that the quotient G/γ ∗ is an abelian group. In [7], Davvaz considered this relation on polygroups. Let P be a polygroup. We consider the relation γ ∗ on P . The product ⊗ in P/γ ∗ is defined as follows: γ ∗ (x) ⊗ γ ∗ (y) = γ ∗ (z) for all z ∈ γ ∗ (x) · γ ∗ (y). Clearly, we have γ ∗ (e) = 1P/γ ∗ and (γ ∗ (x))−1 = γ ∗ (x−1 ) for all x ∈ P . Let φ : P −→ P/γ ∗ be the canonical projection. D(P ) is called the derived hypergroup and we have D(P ) = φ−1 (1P/γ ∗ ). For every non-empty subset M of polygroup P we have φ−1 (φ(M )) = D(P )M = M D(P ). The relational notation A ≈ B is used to assert that the sets A and B have at least one element in common. Theorem 2.2 Let P be a polygroup. Then, x γ ∗ y if and only if there exist A, A ⊆ γ ∗ (a) and B, B ⊆ γ ∗ (b) for some a, b ∈ P such that xA ≈ B and yA ≈ B . Proof. Suppose that there exist A, A ⊆ γ ∗ (a) and B, B ⊆ γ ∗ (b) for some a, b ∈ P such that xA ≈ B and yA ≈ B . Then, we have (γ ∗ (x) ⊗ {γ ∗ (y ) | y ∈ A}) ≈ {γ ∗ (z) | z ∈ B},
B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
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(γ ∗ (y) ⊗ {γ ∗ (y ) | y ∈ A }) ≈ {γ ∗ (z ) | z ∈ B }, Therefore, we obtain γ ∗ (x) ⊗ γ ∗ (a) = γ ∗ (b) and γ ∗ (y) ⊗ γ ∗ (a) = γ ∗ (b), which implies that γ ∗ (x) = γ ∗ (y) = γ ∗ (b) ⊗ γ ∗ (a−1 ). Therefore, x γ ∗ y. For the converse, if x γ ∗ y, then we can take A = A = DP and B = B = ∗ γ (x). This completes the proof. 2 Corollary 2.3 Let P be a polygroup. Then, x, y ∈ γ ∗ (e) if and only if there exist A, A ⊆ γ ∗ (z) and B, B ⊆ γ ∗ (z) for some z ∈ P such that xA ≈ B and yA ≈ B . Theorem 2.4 Let P be a finite polygroup. For every a ∈ P , there exists a power ar , we take the minimal one, which contains an element of a lower power, that is, there exists as such that ar ≈ as , 0 < s < r. Then, ar−s ⊆ γ ∗ (e). Proof. From ar ≈ as we have φ(ar ) = φ(as ), and so (γ ∗ (a))r = (γ ∗ (a))s . Since (γ ∗ (a))r and (γ ∗ (a))s are the elements of abelian group P/γ ∗ , then (γ ∗ (a))r−s = 1P/γ ∗ = γ ∗ (e) which implies that φ(ar−s ) = γ ∗ (e), and so ar−s ⊆ γ ∗ (e).
2
Let M be a non-empty subset of a polygroup P , we say that M is a γ-part of P , if for every n ∈ N, for every (z1 , . . . , zn ) ∈ P n and for every σ ∈ Sn , we have: n n zi ≈ M ⇒ zσ(i) ⊆ M. i=1
i=1
Let A be a non-empty subset of P . The intersection of γ-parts P which contain A is called the γ-closure of A in P . It will be denoted Cγ (A). We have (1) A ∈ P ∗ (P ), one has DP A = ADP = Cγ (A). (2) A ∈ P ∗ (P ), then A is a γ-part of P if and only if ADP = A. (3) If A is a γ-part of P then AB, BA are γ-parts of P for every B ∈ P ∗ (P ). (4) DP is a γ-part of P .
Let P be a polygroup and (P ) be the set of hyperproducts of elements of P . Let X =< (P ), > be the set of non-empty subsets of P endowed with the hyperoperation defined as follows: A B = {C ∈ (P ) | C ⊆ AB} , for all A, B ∈ (P ) with A, B = {e}, and A {e} = {e} A = A. Then, similar to [17], we have
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B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
Theorem 2.5 If P is a polygroup, then < (P ), > is a regular hypergroup. Proof. First, we show that on (P ) is associative. Let A1 , A2 , A3 ∈ (P ). Then, we have A1 (A2 A3 ) = A1 {C ∈ (P ) | C ⊆ A2 A3 } = {D ∈ (P ) | D ⊆ A1 (A2 A3 )} = {D ∈ (P ) | D ⊆ (A1 A2 )A3 } = {C ∈ (P ) | C ⊆ A1 A2 } A3 = (A1 A2 ) A3 . p q Let’s prove now the reproducibility. Let A = ai and B = bi be elements i=1 i=1 of (P ). By reproducibility of ·, there exists y1 ∈ P such that ap ∈ y1 · bq . Similarly, there is y2 such that y1 ∈ y2 · bq−1 , whence ap ∈ y2 · bq−1 · bq . Going up in the same way, one obtains yq such that yq−1 ∈ yq · b1 . Hence, p−1 ap ∈ yq · b1 · b2 · . . . · bq . Therefore, if we let X = ai · yq , we have A ∈ X B. i=1
Similarly, we can find z1 , z2 , . . . , zq such that a1 ∈ b1 ·z1 , z1 ∈ b2 ·z2 , . . . , zq−1 ∈ bq · zq , whence A = a1 · a2 · . . . · ap ⊆ b1 · b2 · . . . · bq · zq · a1 · a2 · . . . · ap . Now, let E = {e}, then for all A ∈ (P ) we have A E = E A = A. We define the unary operation −I as follows: −I
:
(P ) −→
(P )
−1 (x1 . . . xn )−I = x−1 n . . . x1 .
2 Note that the following example shows that, in general, by Theorem 2.5, we do not obtain a polygroup structure. Example 2.6 Let P = {a, b} with a−1 = a, b−1 = b and the following hyperoperation · a
b
a a
b
b b {a, b} If X = a · a, Y = b · b and Z = a · a, then X ∈ Y Z but Y ∈ X z −I .
B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
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Indeed, Theorem 6 in [7] holds, if we add the following condition in the definition of : (∗) X ∈ A B, if for every a ∈ A and b ∈ B, there exists x ∈ X such that x ∈ a · b. Theorem 2.7 Let P is a polygroup. Then < (P ), > is a polygroup, if satisfies in the condition (∗).
References [1] H. Aghabozorgi, M. Jafarpour, and B. Davvaz, Enumeration of Varlet and Comer hypergroups, Electron. J. Combin. 18 (2011), Paper 31, 9 pp. [2] S. D. Comer, Polygroups derived from cogroups, J. Algebra 89 (1984), 397–405. [3] P. Corsini, “Prolegomena of Hypergroup Theory,” Second Edition, Aviani Editore, 1993. [4] P. Corsini, and V. Leoreanu-Fotea, “Applications of Hyperstructures Theory,” Advances in Mathematics, Kluwer Academic Publisher, 2003. [5] B. Davvaz, A survey on polygroups and their properties, Proceedings of the International Conference on Algebra 2010, 148–156, World Sci. Publ., Hackensack, NJ, 2012. [6] B. Davvaz, Isomorphism theorems of polygroups, Bulletin of the Malaysian Mathematical Sciences Society (2) 33 (2010), 385–392. [7] B. Davvaz, Applications of the γ ∗ -relation to polygroups, Comm. Algebra 35 (2007), 2698–2706. [8] B. Davvaz, A. Dehghan Nezhad, and A. Benvidi, Chemical hyperalgebra: Dismutation reactions, MATCH Communications in Mathematical and in Computer Chemistry 67 (2012), 55–63. [9] B. Davvaz, A. Dehghan Nezhad, and A. Benvidi, Chain reactions as experimental examples of ternary algebraic hyperstructures, MATCH Communications in Mathematical and in Computer Chemistry 65 (2011), 491– 499. [10] B. Davvaz, and M. Karimian, On the γn∗ -complete hypergroups, European J. Combin.28 (2007) 86-93. [11] B. Davvaz, and V. Leoreanu-Fotea, “Hyperring Theory and Applications,” International Academic Press, Palm Harbour, USA, 2007.
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[12] M. Dresher and O. Ore, Theory of multigroups, Amer. J. Math. 60 (1938), 705–733. [13] D. Freni, A new characterization of the derived hypergroup via strongly regular equivalences, Comm. Algebra 30 (2002), 3977–3989. [14] D. Freni, Une note sur le cur d’un hypergroupe et sur la clˆ oture transitive β ∗ de β. (French) [A note on the core of a hypergroup and the transitive closure β ∗ of β], Riv. Mat. Pura Appl. 8 (1991), 153–156. [15] M. Ghadiri, B. Davvaz and R. Nekouian, Hv -Semigroup structure on F2 offspring of a gene pool, International Journal of Biomathematics 5 (2012), 1250011 (13 pages). [16] M. Koskas, Groupoides, demi-hypergroupes et hypergroupes. (French), J. Math. Pures Appl. (9) 49 (1970), 155–192. [17] V. Leoreanu, The heart of some important classes of hypergroups, Pure Math. Appl. 9 (1998), 351–360. [18] V. Leoreanu-Fotea, and B. Davvaz, n-hypergroups and binary relations, European J. Combin. 29 (2008), 1207-1218. [19] W. Prenowitz, Projective geometries as multigroups, Amer. J. Math. 65 (1943), 235–256. [20] T. Vougiouklis, “Hyperstructures and Their Representations,” Hadronic Press, Inc., 115, Palm Harber, USA, 1994.
Electronic Notes in Discrete Mathematics 45 (2014) 9–16 www.elsevier.com/locate/endm
Relations on Groups, Polygroups and Hypergroups Bijan Davvaz 1,2 Department of Mathematics Yazd University Yazd, Iran
Abstract 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. By using a certain type of equivalence relations, we can connect semihypergroups to semigroups and hypergroups (polygroups) to groups. These equivalence relations are called strongly regular relations. In this paper, we review some strongly regular relations on hyperstructures and we give some their applications. Keywords: Group, polygroup, hypergroup, binary relation, fundamental relation.
1
Regular relations
Hyperstructure theory both extends some well-known group results and introduce new topics leading us to a wide variety of applications, as well as to a broadening of the investigation fields, for example see [8,9,15]. A comprehensive review of the theory of hyperstructures appears in [3,4,11,20]. Let H 1
I would like to thank Professor A. R. Ashrafi for his hospitality during my stay at the Department of Mathematics, University of Kashan 2 Email: [email protected] 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.11.003
10
B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
be a non-empty set and ◦ : H × H −→ P ∗ (H) be a hyperoperation, where P ∗ (H) is the set of all non-empty subsets of H. The couple (H, ◦) is called a hypergroupoid. Forany two non-empty subsets A and B of H and x ∈ H, we define A ◦ B = a∈A,b∈B a ◦ b, A ◦ x = A ◦ {x} and x ◦ B = {x} ◦ B. A hypergroupoid (H, ◦) is called a semihypergroup if for all a, b, c of H we have (a ◦ b) ◦ c = a ◦ (b ◦ c), which means that u∈a◦b u ◦ c = v∈b◦c a ◦ v. A hypergroupoid (H, ◦) is called a quasihypergroup if for all a of H we have a ◦ H = H ◦ a = H. This condition is also called the reproduction axiom. A hypergroupoid (H, ◦) which is both a semihypergroup and a quasihypergroup is called a hypergroup. A regular hypergroup H is a hypergroup which it has at least one identity and every element has at least one inverse. By using a certain type of equivalence relations, we can connect semihypergroups to semigroups and hypergroups to groups. These equivalence relations are called strongly regular relations. More exactly, starting with a (semi)hypergroup and using a strongly regular relation, we can construct a (semi)group structure on the quotient set. Let us define these notions. First, we do some notations. Let (H, ◦) be a semihypergroup and R be an equivalence relation on H. If A and B are non-empty subsets of H, then ARB means that for all a ∈ A there exists b ∈ B such that aRb and for all b ∈ B there exists a ∈ A such that a Rb ; also ARB means that for all a ∈ A and for all b ∈ B, we have aRb. The equivalence relation R is called (1) regular on the right (on the left) if for all x of H, from aRb, it follows that (a ◦ x)R(b ◦ x) ((x ◦ a)R(x ◦ b) respectively); (2) strongly regular on the right (on the left) if for all x of H, from aRb, it follows that (a ◦ x)R(b ◦ x) ((x ◦ a)R(x ◦ b) respectively); (3) R is called regular (strongly regular) if it is regular (strongly regular) on the right and on the left. The reader can find the proofs of the following facts in [3,11]. •
Let (H, ◦) be a semihypergroup and R be an equivalence relation on H. If R is regular, then H/R is a semihypergroup, with respect to the following hyperoperation: x ⊗ y = {z | z ∈ x ◦ y}. If the above hyperoperation is well defined on H/R, then R is regular.
•
If (H, ◦) is a hypergroup and R is an equivalence relation on H, then R is regular if and only if (H/R, ⊗) is a hypergroup.
•
Let (H, ◦) be a semihypergroup and R be an equivalence relation on H. If R is strongly regular, then H/R is a semigroup, with respect to the following operation: x ⊗ y = z for all z ∈ x ◦ y. If the above operation is well defined on H/R, then R is strongly regular.
•
If (H, ◦) is a hypergroup and R is an equivalence relation on H, then R is strongly regular if and only if (H/R, ⊗) is a group.
B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
11
Aghabozorgi et al. [1] studied hypergroups determined by lattices introduced by Varlet and Comer, especially they enumerated Varlet and Comer hypergroups of orders less than 50 and 13, respectively. Another paper [10] deals with the class of γn∗ -complete hypergroups and provided a good insight into the study of hypergroup theory under the strongly regular equivalence relations. In [18], Leoreanu-Fotea and Davvaz studied the notion of a partial n-hypergroupoid, associated with a binary relation. Some important results concerning Rosenberg partial hypergroupoids, induced by relations, are generalized to the case of n-hypergroupoids. Then, n-hypergroups associated with union, intersection, products of relations and also mutually associative n-hypergroupoids are analyzed.
2
Rgular relations on polygroups
A polygroup [2] is a system ℘ =< P, ·, e,−1 >, where e ∈ P, −1 is a unitary operation on P , · maps P × P into the non-empty subsets of P , and the following axioms hold for all x, y, z in P : (1) (x · y) · z = x · (y · z), (2) e · x = x · e = x, (3) x ∈ y · z implies y ∈ x · z −1 and z ∈ y −1 · x. The following elementary facts about polygroups follow easily from the axioms: e ∈ x · x−1 ∩ x−1 · x, e−1 = e, (x−1 )−1 = x, and (x · y)−1 = y −1 · x−1 , where A−1 = {a−1 | a ∈ A}. Suppose that H is a subgroup of a group G. Define a system G//H =< {HgH | g ∈ G}, ∗, H,−I >, where (HgH)−I = Hg −1 H and (Hg1 H) ∗ (Hg2 H) = {Hg1 hg2 H | h ∈ H}. The algebra of double cosets G//H is a polygroup which introduced by Dresher and Ore [12]. Suppose that G is a projective geometry with a set P of points and suppose, for p = q, pq denoted the set of all points on the unique line through p and q. Choose an object I ∈ P and form the system PG =< P ∪ {I}, ·, I,−1 >, where x−1 = x and I · x = x · I = x for all x ∈ P ∪ {I} and for p, q ∈ P , ⎧ ⎨ pq − {p, q} if p = q p·q = ⎩ {p, I} if p = q. PG is a polygroup which introduced by Prenowitz [19]. Let P be a polygroup. We define the relation β ∗ as the smallest equivalence relation on P such that the quotient P/β ∗ , the set of all equivalence classes, is a group. In this case β ∗ is called the fundamental equivalence relation on P and P/β ∗ is called the fundamental group. The product ⊗ in P/β ∗ is defined as follows: β ∗ (x) ⊗ β ∗ (y) = β ∗ (z) for all z ∈ β ∗ (x) · β ∗ (y). Let UP be the set of all finite products of elements of P . We define the relation β as follows:
12
B. Davvaz / Electronic Notes in Discrete Mathematics 45 (2014) 9–16
xβy if and only if {x, y} ⊆ u for some u ∈ UP . We have β ∗ = β for hypergroups [14]. Since polygroups are certain subclasses of hypergroups, we have β ∗ = β. The kernel of the canonical map ϕ : P −→ P/β ∗ is called the core of P and is denoted by ωP . Here we also denote by ωP the unit of P/β ∗ . It is easy to prove that the following statements: ωP = β ∗ (e) and β ∗ (x)−1 = β ∗ (x−1 ) for all x ∈ P . The β ∗ relation is introduced by Koskas [16] and studied by Corsini, Davvaz, Freni, Leoreanu, Vougiouklis and many others. In [5,6], Davvaz considered this relation on polygroups. We recall the following definition from [13]. If H is a semihypergroup, then we set: γ1 = {(x, x) | x ∈ H} and, for every integer n > 1, the relation is defined as follows: n n xγn y ⇐⇒ ∃(z1 , z2 , . . . , zn ) ∈ H n , ∃σ ∈ Sn : x ∈ zi , y ∈ zσ(i) . i=1
i=1
Obviously, for every n ≥ 1, the relations γn are symmetric, and the relation γ = n≥1 γn is reflexive and symmetric. Let γ ∗ be the transitive closure of γ. Then, γ ∗ is the smallest strongly regular equivalence such that H/γ ∗ is a commutative semigroup. Also, in every hypergroup, the relation γ is transitive, that is γ ∗ = γ, and in this case, the quotient H/γ ∗ is an abelian group [13]. The γ ∗ -relation is a generalization of the β ∗ -relation. Remark 2.1 Note that we can consider the relation γ ∗ on a non-abelian group G. So, γ ∗ is the smallest equivalence relation on G such that the quotient G/γ ∗ is an abelian group. In [7], Davvaz considered this relation on polygroups. Let P be a polygroup. We consider the relation γ ∗ on P . The product ⊗ in P/γ ∗ is defined as follows: γ ∗ (x) ⊗ γ ∗ (y) = γ ∗ (z) for all z ∈ γ ∗ (x) · γ ∗ (y). Clearly, we have γ ∗ (e) = 1P/γ ∗ and (γ ∗ (x))−1 = γ ∗ (x−1 ) for all x ∈ P . Let φ : P −→ P/γ ∗ be the canonical projection. D(P ) is called the derived hypergroup and we have D(P ) = φ−1 (1P/γ ∗ ). For every non-empty subset M of polygroup P we have φ−1 (φ(M )) = D(P )M = M D(P ). The relational notation A ≈ B is used to assert that the sets A and B have at least one element in common. Theorem 2.2 Let P be a polygroup. Then, x γ ∗ y if and only if there exist A, A ⊆ γ ∗ (a) and B, B ⊆ γ ∗ (b) for some a, b ∈ P such that xA ≈ B and yA ≈ B . Proof. Suppose that there exist A, A ⊆ γ ∗ (a) and B, B ⊆ γ ∗ (b) for some a, b ∈ P such that xA ≈ B and yA ≈ B . Then, we have (γ ∗ (x) ⊗ {γ ∗ (y ) | y ∈ A}) ≈ {γ ∗ (z) | z ∈ B},
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(γ ∗ (y) ⊗ {γ ∗ (y ) | y ∈ A }) ≈ {γ ∗ (z ) | z ∈ B }, Therefore, we obtain γ ∗ (x) ⊗ γ ∗ (a) = γ ∗ (b) and γ ∗ (y) ⊗ γ ∗ (a) = γ ∗ (b), which implies that γ ∗ (x) = γ ∗ (y) = γ ∗ (b) ⊗ γ ∗ (a−1 ). Therefore, x γ ∗ y. For the converse, if x γ ∗ y, then we can take A = A = DP and B = B = ∗ γ (x). This completes the proof. 2 Corollary 2.3 Let P be a polygroup. Then, x, y ∈ γ ∗ (e) if and only if there exist A, A ⊆ γ ∗ (z) and B, B ⊆ γ ∗ (z) for some z ∈ P such that xA ≈ B and yA ≈ B . Theorem 2.4 Let P be a finite polygroup. For every a ∈ P , there exists a power ar , we take the minimal one, which contains an element of a lower power, that is, there exists as such that ar ≈ as , 0 < s < r. Then, ar−s ⊆ γ ∗ (e). Proof. From ar ≈ as we have φ(ar ) = φ(as ), and so (γ ∗ (a))r = (γ ∗ (a))s . Since (γ ∗ (a))r and (γ ∗ (a))s are the elements of abelian group P/γ ∗ , then (γ ∗ (a))r−s = 1P/γ ∗ = γ ∗ (e) which implies that φ(ar−s ) = γ ∗ (e), and so ar−s ⊆ γ ∗ (e).
2
Let M be a non-empty subset of a polygroup P , we say that M is a γ-part of P , if for every n ∈ N, for every (z1 , . . . , zn ) ∈ P n and for every σ ∈ Sn , we have: n n zi ≈ M ⇒ zσ(i) ⊆ M. i=1
i=1
Let A be a non-empty subset of P . The intersection of γ-parts P which contain A is called the γ-closure of A in P . It will be denoted Cγ (A). We have (1) A ∈ P ∗ (P ), one has DP A = ADP = Cγ (A). (2) A ∈ P ∗ (P ), then A is a γ-part of P if and only if ADP = A. (3) If A is a γ-part of P then AB, BA are γ-parts of P for every B ∈ P ∗ (P ). (4) DP is a γ-part of P .
Let P be a polygroup and (P ) be the set of hyperproducts of elements of P . Let X =< (P ), > be the set of non-empty subsets of P endowed with the hyperoperation defined as follows: A B = {C ∈ (P ) | C ⊆ AB} , for all A, B ∈ (P ) with A, B = {e}, and A {e} = {e} A = A. Then, similar to [17], we have
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Theorem 2.5 If P is a polygroup, then < (P ), > is a regular hypergroup. Proof. First, we show that on (P ) is associative. Let A1 , A2 , A3 ∈ (P ). Then, we have A1 (A2 A3 ) = A1 {C ∈ (P ) | C ⊆ A2 A3 } = {D ∈ (P ) | D ⊆ A1 (A2 A3 )} = {D ∈ (P ) | D ⊆ (A1 A2 )A3 } = {C ∈ (P ) | C ⊆ A1 A2 } A3 = (A1 A2 ) A3 . p q Let’s prove now the reproducibility. Let A = ai and B = bi be elements i=1 i=1 of (P ). By reproducibility of ·, there exists y1 ∈ P such that ap ∈ y1 · bq . Similarly, there is y2 such that y1 ∈ y2 · bq−1 , whence ap ∈ y2 · bq−1 · bq . Going up in the same way, one obtains yq such that yq−1 ∈ yq · b1 . Hence, p−1 ap ∈ yq · b1 · b2 · . . . · bq . Therefore, if we let X = ai · yq , we have A ∈ X B. i=1
Similarly, we can find z1 , z2 , . . . , zq such that a1 ∈ b1 ·z1 , z1 ∈ b2 ·z2 , . . . , zq−1 ∈ bq · zq , whence A = a1 · a2 · . . . · ap ⊆ b1 · b2 · . . . · bq · zq · a1 · a2 · . . . · ap . Now, let E = {e}, then for all A ∈ (P ) we have A E = E A = A. We define the unary operation −I as follows: −I
:
(P ) −→
(P )
−1 (x1 . . . xn )−I = x−1 n . . . x1 .
2 Note that the following example shows that, in general, by Theorem 2.5, we do not obtain a polygroup structure. Example 2.6 Let P = {a, b} with a−1 = a, b−1 = b and the following hyperoperation · a
b
a a
b
b b {a, b} If X = a · a, Y = b · b and Z = a · a, then X ∈ Y Z but Y ∈ X z −I .
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Indeed, Theorem 6 in [7] holds, if we add the following condition in the definition of : (∗) X ∈ A B, if for every a ∈ A and b ∈ B, there exists x ∈ X such that x ∈ a · b. Theorem 2.7 Let P is a polygroup. Then < (P ), > is a polygroup, if satisfies in the condition (∗).
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[12] M. Dresher and O. Ore, Theory of multigroups, Amer. J. Math. 60 (1938), 705–733. [13] D. Freni, A new characterization of the derived hypergroup via strongly regular equivalences, Comm. Algebra 30 (2002), 3977–3989. [14] D. Freni, Une note sur le cur d’un hypergroupe et sur la clˆ oture transitive β ∗ de β. (French) [A note on the core of a hypergroup and the transitive closure β ∗ of β], Riv. Mat. Pura Appl. 8 (1991), 153–156. [15] M. Ghadiri, B. Davvaz and R. Nekouian, Hv -Semigroup structure on F2 offspring of a gene pool, International Journal of Biomathematics 5 (2012), 1250011 (13 pages). [16] M. Koskas, Groupoides, demi-hypergroupes et hypergroupes. (French), J. Math. Pures Appl. (9) 49 (1970), 155–192. [17] V. Leoreanu, The heart of some important classes of hypergroups, Pure Math. Appl. 9 (1998), 351–360. [18] V. Leoreanu-Fotea, and B. Davvaz, n-hypergroups and binary relations, European J. Combin. 29 (2008), 1207-1218. [19] W. Prenowitz, Projective geometries as multigroups, Amer. J. Math. 65 (1943), 235–256. [20] T. Vougiouklis, “Hyperstructures and Their Representations,” Hadronic Press, Inc., 115, Palm Harber, USA, 1994.