On dihedral hypergroups

European Journal of Combinatorics 44 (2015) 242–249

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European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc

On dihedral hypergroups M. Jafarpour a , I. Cristea b , F. Alizadeh a a

Department of Mathematics, Vali-e-Asr University, Rafsanjan, Iran

b

Centre for Systems and Information Technologies, University of Nova Gorica, Slovenia

article

abstract

info

Article history: Available online 27 August 2014

In this paper the notion of ψ -dihedral hypergroups is introduced as a generalization of dihedral groups. Basic properties of these algebraic structures are considered; we particularly insist on the regularity property of this kind of hypergroups. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Hyperstructure theory was born in 1934 at the 8th Congress of Scandinavian Mathematicians, where Marty [13] introduced the hypergroup notion as a generalization of groups and after that, he proved its utility in solving some problems of groups, algebraic functions and rational fractions. Surveys of the theory can be found in the books of Corsini [2], Davvaz and Leoreanu-Fotea [9], Corsini and Leoreanu [5] and Vougiouklis [18,20]. Now this field of modern algebra is widely studied from theoretical and applied viewpoints, focusing on their applications to many subjects of pure and applied mathematics. Several applications are used in the following areas: geometry, graphs, fuzzy sets, cryptography, automata, lattices, binary relations, codes, and artificial intelligence (see for example [1,3,4,6–8,11,14–16,21]). In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, playing an important role in group theory, geometry, and chemistry. An example of dihedral group is that of a group generated by two involutions. There are several important generalizations of the dihedral groups, among them we recall:

• The infinite dihedral group, which is an infinite group with the algebraic structure similar to the finite dihedral groups. It can be viewed as the group of symmetries of the integers.

• The orthogonal group O(2), which is the symmetry group of the circle. • The quasidihedral group, a type of finite group.

E-mail addresses: [email protected] (M. Jafarpour), [email protected] (I. Cristea), [email protected] (F. Alizadeh). http://dx.doi.org/10.1016/j.ejc.2014.08.010 0195-6698/© 2014 Elsevier Ltd. All rights reserved.

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This paper introduces a new generalization of dihedral groups for the class of hypergroups, namely the dihedral hypergroups. It is organized as follows: after a short presentation of some fundamental definitions on hypergroups, we define in Section 3 the notion of ψ -dihedral hypergroups as a generalization of dihedral groups and we give some properties of them. In the end, we prove that these hypergroups are regular and, in a special case, they are also reversible. 2. Preliminaries Let us briefly recall some basic notions and results about groups and hypergroups; for a comprehensive overview of this subject, the reader is refereed to [2,5,9,17]. For a nonempty set H, we denote by P ∗ (H ) the set of all nonempty subsets of H. Definition 2.1. A nonempty set H, endowed with a mapping, called hyperoperation, ◦ : H 2 −→ P ∗ (H ) is named hypergroupoid. A hypergroupoid which verifies the following conditions: (i) (x ◦ y) ◦ z = x ◦ (y ◦ z ), for all x, y, z ∈ H (the associativity) and (ii) x ◦ H = H = H ◦ x, for all x ∈ H (the reproduction axiom), is called hypergroup. In particular, an associative hypergroupoid is called a semihypergroup and a hypergroupoid that verifies the reproduction axiom is  called a quasi-hypergroup. If A and B are nonempty subsets of H, then A ◦ B = a∈A a ◦ b. b∈B

Definition 2.2. Let (H , ◦) be a hypergroup. (i) An element e ∈ H is called a right (left) identity if, for all y ∈ H , y ∈ y ◦ e (y ∈ e ◦ y). (ii) An element e ∈ H is called an identity if, for all y ∈ H , y ∈ y ◦ e ∩ e ◦ y. (iii) An element x′ ∈ H is called an inverse of x ∈ H if, for some identity element e of H, it holds that e ∈ x ◦ x′ ∩ x′ ◦ x. Definition 2.3. A hypergroup (H , ◦) is regular if it has at least one identity and each element has at least one inverse. Definition 2.4. Let (H , ◦) and (H ′ , ◦′ ) be two hypergroups. A function f : H −→ H ′ is called a homomorphism if it satisfies the condition: for any x, y ∈ H, f (x ◦ y) ⊆ f (x) ◦′ f (y). f is a good homomorphism if, for any x, y ∈ H , f (x ◦ y) = f (x) ◦′ f (y). We say that the two hypergroups are isomorphic if there is a good homomorphism between them which is also a bijection. Definition 2.5. If (H , ◦) is a hypergroup and R ⊆ H × H is an equivalence relation, we set =

A R B ⇔ a R b,

∀a ∈ A, ∀b ∈ B,

for all pairs (A, B) of nonempty subsets of H. The relation R is called strongly regular on the left (on the =

=

right) if x R y ⇒ a ◦ x R a ◦ y (x R y ⇒ x ◦ a R y ◦ a respectively), for all (x, y, a) ∈ H 3 . Moreover, R is called strongly regular if it is strongly regular on the right and on the left. Theorem 2.6 ([2]). If (H , ·) is a semihypergroup (hypergroup) and ρ is a strongly regular relation on H, then the quotient H /ρ is a semigroup (group) under the operation:

ρ(x) ⊗ ρ(y) = ρ(z ),

for all z ∈ x · y.

We denote ρ(x) by x¯ and, for notational simplicity, we write x¯ y¯ instead of x¯ ⊗ y¯ .

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For all n > 1, we define the relation βn on a semihypergroup H, as follows: a βn b ⇔ ∃(x1 , . . . , xn ) ∈ H n : {a, b} ⊆

n 

xi ,

i=1

and β = i=1 βn , where β1 = {(x, x) | x ∈ H } is the diagonal relation on H. This relation was introduced by Koskas [10] and studied mainly by Corsini [2]. Denote with β ∗ the transitive closure of β . The relation β ∗ is a strongly regular relation [2] and also, we have the following strong result:

n

Theorem 2.7 ([2]). If (H , ◦) is hypergroup then β = β ∗ . Note that, in general, for a semihypergroup may be β ̸= β ∗ . The relation β ∗ is the least equivalence relation on a hypergroup H, such that the quotient H /β ∗ is a group and it is called the fundamental relation on a semihypergroup. The heart ωH of a hypergroup H is the set of all elements x of H, for which the equivalence class β ∗ (x) is the identity of the group H /β ∗ . 3. Dihedral hypergroups First, we present some particular properties of the automorphisms of hypergroups. Definition 3.1. Let N and H be two groups and θ : H → Aut (N ) be a group homomorphism. The semidirect product N oθ H of N and H with respect to θ is the group (N × H , •) with the operation (n1 , h1 )•(n2 , h2 ) = (n1 θh1 (n2 ), h1 h2 ), for n1 , n2 ∈ N and h1 , h2 ∈ H. Moreover, for any h ∈ H , θ h = θh is the automorphism of N given by conjugation θh (n) = hnh−1 , for any n ∈ N. Proposition 3.2 (See [17]). A group G is a semidirect product of its subgroups N and H if N is normal and G → G/N induces an isomorphism H → G/N. Equivalently, G is a semidirect product of subgroups N and H if N ◃ G (i.e. N is normal), NH = G and N ∩ H = {1}, where 1 denotes the identity of G. Example 3.3. The dihedral group of order 2n, denoted D2n , is generated by two elements, a rotation r of order n and a reflection s of order 2 which satisfy the relation srs = r −1 . The distinct elements of D2n can be written as r i , r i s (i = 0, . . . , n − 1). Notice that we also have r i s = sr −i . In D2n , n ≥ 2, taking Cn = ⟨r ⟩ and C2 = ⟨s⟩, then we write D2n = ⟨r ⟩oθ ⟨s⟩ = C n oθ C2 , where θ (s)(r i ) = r −i . Definition 3.4. Let (Zn , +) be the cyclic group of order n. The map ϕ : Zn → Zn defined by ϕ(x) = −x, for all x ∈ Zn is called the inversion automorphism on Zn . Example 3.5. The map ϕ : Z4 → Z4 by ϕ(0¯ ) = 0¯ , ϕ(1¯ ) = 3¯ , ϕ(2¯ ) = 2¯ , ϕ(3¯ ) = 1¯ is the inversion automorphism on Z4 . Definition 3.6 (See [12]). Let (H , ·) and (K , ∗) be hypergroups. A map f : H → P ∗ (K ) is called a good multihomomorphism if and only if

∀(x, y) ∈ H 2 ,

f (x · y) = f (x) ∗ f (y).  If (H , ·) = (K , ∗) and h∈H f (h) = H, then f is called a generalized automorphism. Moreover, we will denote by GAut (H ) the set of all generalized automorphisms of (H , ·). Example 3.7. The map ϕ : Z4 → P ∗ (Z4 ) defined by ϕ(0¯ ) = ϕ(2¯ ) = {0¯ , 2¯ }, ϕ(1¯ ) = ϕ(3¯ ) = {1¯ , 3¯ } is a generalized automorphism on Z4 . Proposition 3.8. Let (H , ∗) be a hypergroup. Then (GAut (H ), ·) is a monoid, where, for all x ∈ H and (f , g ) ∈ (GAut (H ))2 , the operation ‘‘ ·’’ is defined as follows:

(f · g )(x) = ∪a∈g (x) f (a).

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Proof. Let (x, y) ∈ H 2 and f , g be arbitrary elements in GAut (H ). Then

(f · g )(x ∗ y) = ∪a∈g (x∗y) f (a) = ∪a∈g (x)∗g (y) f (a) = f (g (x) ∗ g (y)) = f (g (x)) ∗ f (g (y)) = (f · g )(x) ∗ (f · g )(y). Now we prove the associativity of ‘‘(·)’’. For this, suppose that f , g and h are elements in GAut (H ) and x ∈ H. If a ∈ [(f · g ) · h](x), then a ∈ (f · g )(u), for some u ∈ h(x). Hence a ∈ f (v), where v ∈ g (u). Thus a ∈ f (v), where v ∈ [g · h](x) and so a ∈ [f ·(g · h)](x) consequently we have [(f · g )· h](x) ⊆ [f · (g · h)](x). With a same argument we have the converse inclusion [(f · g ) · h](x) ⊇ [f · (g · h)](x). Thus (f · g ) · h = f · (g · h) hold and so the hyperoperation ‘‘·’’ is associative.  Definition 3.9. Let ϕ and ψ be two elements of GAut (H ). We say that ψ is an extension of ϕ and we write ϕ ≤ ψ if and only if ϕ(x) ⊆ ψ(x), for all x ∈ H. Every extension of the inversion automorphism is called generalized inversion automorphism. Moreover, we will denote by GIAut (H ) the set of all generalized inversion automorphisms of H. Proposition 3.10. The previous relation ≤ is a partially ordering on GAut (H ). Proposition 3.11. The conjugate of any ϕ ∈ GIAut (Zn ) with respect to Aut (Zn ) is in GIAut (Zn ). Proof. The conjugate of ψ ∈ GIAut (Zn ) with respect to Aut (Zn ) has the form ϕψϕ −1 , where ϕ ∈ Aut (Zn ). Let ϕ ∈ Aut (Zn ) and ψ ∈ GIAut (Zn ); then ϕψϕ −1 ∈ GAut (Zn ). Suppose that ϕ(a) = b, ϕ(a′ ) = b′ . Then (ϕψϕ −1 )(b + b′ ) = (ϕψ)(ϕ −1 (b) + ϕ −1 (b′ )) = (ϕψ)(a + a′ ) = ϕ(ψ(a) ∗ ψ(a′ )) = ϕ(ψ(a)) ∗ (ϕψ(a′ )) = ϕψϕ −1 (b) ∗ ϕψϕ −1 (b′ ). For all (a, b) ∈ Zn such that ϕ(a) = b, we have ϕψϕ −1 (b) ∈ P ∗ (Zn ) because ψ ∈ GIAut (Zn ). Since −a ∈ ψ(a), it follows that ϕ(−a) ∈ ϕ(ψ(a)) and thus −ϕ(a) ∈ ϕ(ψ(a)). Thereby −ϕ(ϕ −1 (b)) ∈ ϕψϕ −1 (b). Consequently −b ∈ ϕψϕ −1 (b) and thus ϕψϕ −1 ∈ GIAut (Zn ).  Definition 3.12. Let G be a group and

ψ : G → GAut (Zn ), be a map. ψ is called an extension homomorphism if ψ(xy) ≤ ψ(x)ψ(y), for every (x, y) ∈ G2 . Remark 3.13. Every homomorphism from a group G into GAut (H ) is an extension homomorphism. The previous definition gives us the possibility to generalize the notion of multi-direct hyperproduct of hypergroups which was introduced in [12]. Definition 3.14. Let (K , ·) and (H , ∗) be two hypergroups. Consider the monoid GAut (K ) and the group βH∗ , where β ∗ is the fundamental relation on H. Let

ψ:

H

→ GAut (K ), β∗ β ∗ (x) → ψβ ∗ (x) ,

be an extension homomorphism. Then one can define a hyperoperation in K × H as follows:

(x1 , y1 ) ◦ (x2 , y2 ) = {(x, y) | x ∈ x1 · ψβ ∗ (y ) (x2 ), y ∈ y1 ∗ y2 }. 1

This hyperoperation is called the extension direct hyperproduct of hypergroups K and H through ψ and denoted by K oψ H.

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Considering K = Zn and βH∗ = Z2 in the previous definition, we obtain the following new concept. Definition 3.15. Let ψ be a generalized inversion automorphism of Zn . The extension direct hyperproduct of cyclic groups Zn and Z2 with Z2 acting on Zn by generalized inversion automorphism ¯ : Z2 → GAut (Zn ), for which 0¯ → id (the identity map) and 1¯ → ψ ), is called the ψ -dihedral (i.e. ψ hyperproduct.

¯ is an extension homomorphism. Notice that ψ The Hv -structures introduced in [19] are hyperstructures where the equality is replaced by the nonempty intersection. The fact that this class of hyperstructures is very large, one can use it in order to define several notions that they are not possible to be defined in the classical hypergroup theory. The hyperstructure (H , ◦) is called Hv -semigroup if it satisfies the weak associative condition, that is (x ◦ y) ◦ z ∩ x ◦ (y ◦ z ) ̸= ∅, for all x, y, z ∈ H. The Hv -semigroup is called Hv -group if it also satisfies the reproduction law. Proposition 3.16. Zn oψ Z2 equipped with the ψ -dihedral hyperproduct is an Hv -group. Proof. Let (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) be arbitrary elements in Zn × Z2 and let us denote {(x, y) | x ∈ A, y ∈ B} = (A, B), for all nonempty subsets A and B in H. Now we have

[(x1 , y1 ) ◦ (x2 , y2 )] ◦ (x3 , y3 ) = (x1 + ψ¯ y1 (x2 ), y1 + y2 ) ◦ (x3 , y3 ) = (x1 + ψ¯ y1 (x2 ) + ψ¯ y1 +y2 (x3 ), (y1 + y2 ) + y3 ). On the other hand

(x1 , y1 ) ◦ [(x2 , y2 ) ◦ (x3 , y3 )] = (x1 , y1 ) ◦ (x2 + ψ¯ y2 (x3 ), y2 + y3 ) = (x1 + ψ¯ y1 (x2 + ψ¯ y2 (x3 )), y1 + (y2 + y3 )) = (x1 + ψ¯ y1 (x2 ) + ψ¯ y1 (ψ¯ y2 (x3 )), y1 + (y2 + y3 )). ¯ y +y (x3 ) ⊆ ψ¯ y (ψ¯ y (x3 )) we conclude that Because ψ 1 2 1 2 [(x1 , y1 ) ◦ (x2 , y2 )] ◦ (x3 , y3 ) ⊆ (x1 , y1 ) ◦ [(x2 , y2 ) ◦ (x3 , y3 )]. Hence ◦ satisfies the weak associativity condition. It is easy to see that the reproduction axiom holds too.  From now on, we call Zn oψ Z2 a ψ -dihedral Hv -group and if the ψ -dihedral Hv -group is a hypergroup we call it ψ -dihedral hypergroup. Example 3.17. Let ψ : Z4 → P ∗ (Z4 ) defined by ψ(0¯ ) = ψ(2¯ ) = {0¯ , 2¯ }, ψ(1¯ ) = ψ(3¯ ) = {1¯ , 3¯ } be a generalized inversion automorphism. Then Z4 oψ Z2 is a ψ -dihedral hypergroup.

For simplicity, denote the elements of Z4 oψ Z2 by e = (0¯ , 0¯ ), a = (0¯ , 1¯ ), b = (1¯ , 0¯ ), c = ¯ (1, 1¯ ), d = (2¯ , 0¯ ), f = (2¯ , 1¯ ), g = (3¯ , 0¯ ), h = (3¯ , 1¯ ). Moreover let ψ¯ : Z2 → GAut (Z4 ), such ¯ 0¯ ) = id and ψ( ¯ 1¯ ) = ψ , where id is the identity map, be the ψ -dihedral hyperproduct. The that ψ( hyperoperation in Z4 oψ Z2 is represented by the following table:

· e a b c d f g h

e e

a, f b c, h d a, f g c, h

a a e, d c b, g f e, d h b, g

b b c, h d a, f g c, h e a, f

c c

b, g f e, d h b, g a e, d

d d a, f g c, h e a, f b c, h

f f

e, d h b, g a e, d c b, g

g g

c, h e a, f b c, h d a, f

h h b, g a e, d c b, g f e, d

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We present here the entire computation only for two elements, as an example:

¯ 1¯ (0¯ ), y ∈ 1¯ + 0¯ } a · e = (0¯ , 1¯ ) · (0¯ , 0¯ ) = {(x, y) ∈ Z4 × Z2 | x ∈ 0¯ + ψ = {(x, y) ∈ Z4 × Z2 | x ∈ 0¯ + ψ(0¯ ), y = 1¯ } = {(0¯ , 1¯ ), (2¯ , 1¯ )} = {a, f } and

¯ 1¯ (3¯ ), y ∈ 1¯ + 0¯ } f · g = (2¯ , 1¯ ) · (3¯ , 0¯ ) = {(x, y) ∈ Z4 × Z2 | x ∈ 2¯ + ψ = {(x, y) ∈ Z4 × Z2 | x ∈ 0¯ + ψ(3¯ ), y = 1¯ } = {(1¯ , 1¯ ), (3¯ , 1¯ )} = {c , h}. Furthermore it is not difficult to see that the associativity property for (·) holds and hence

(Z4 oψ Z2 , ·) is a ψ -dihedral hypergroup.

Remark 3.18. In the previous example if ψ is an inversion automorphism, then Z4 oψ Z2 is the dihedral group D8 . Proposition 3.19. Let (G, ·) be a group and ϕ be a good multihomomorphism on G. Then ϕ is an automorphism if and only if ϕ(e) = e, where e is the identity element of G. Proof. If ϕ is an automorphism then it is obvious that ϕ(e) = e. Conversely, for all x ∈ G we have ϕ(x · x−1 ) = ϕ(e) = e and so |ϕ(x) · ϕ(x−1 )| = 1. Therefore |ϕ(x)| = 1 and hence ϕ is an automorphism.  Now we can say when a ψ -dihedral hypergroup is a dihedral group.

¯ Corollary 3.20. The ψ -dihedral hypergroup Zn oψ Z2 is a dihedral group if and only if ψ(0¯ ) = 0. The following result gives us a sufficient condition such that two dihedral hypergroups are isomorphic. Theorem 3.21. Let ψ and θ be two elements in GIAut (Zn ). If there exists ϕ ∈ Aut (Zn ) such that θ(x) = ϕψϕ −1 (x), for all x ∈ H, then ∼ Zn oθ Z2 . Zn oψ Z2 = Proof. Consider the map

τ : Zn oψ Z2 → Zn oθ Z2 ,

(a, b) → (ϕ(a), b).

Then

τ (a, b)τ (c , d) = = = =

(ϕ(a), b)(ϕ(c ), d) {(x, y) | x ∈ ϕ(a) + θb (ϕ(c )), y ∈ b + d} {(x, y) | x ∈ ϕ(a) + ϕ(ψb (c )), y ∈ b + d} {(x, y) | x ∈ ϕ(a + ψb (c )), y ∈ b + d}.

On the other hand

τ ((a, b)(c , d)) = {τ (x, y) | x ∈ a + ψb (c ), y ∈ b + d} = {(ϕ(x), y) | x ∈ a + ψb (c ), y ∈ b + d} = {(z , y) | z ∈ ϕ(a + ψb (c )), y ∈ b + d}. Therefore τ is a good homomorphism. It is easy to see that τ is an isomorphism.



We conclude this section with some results regarding the reversibility and regularity properties of the dihedral hypergroups.

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Proposition 3.22. Every ψ -dihedral hypergroup Zn oψ Z2 is regular. Proof. Let x = (x1 , x2 ) be an arbitrary element of the hypergroup Zn oψ Z2 . We have e · x = (0, 0) · (x1 , x2 ) = {(w1 , w2 )|w1 ∈ 0 + ψ¯ 0 (x1 ), w2 ∈ 0 + x2 } = {(w1 , w2 )|w1 = x1 , w2 = x2 } = (x1 , x2 ) = x. ¯ x2 (0), w2 ∈ 0 + x2 }. There are two On the other side x · e = (x1 , x2 ) · (0, 0) = {(w1 , w2 )|w1 ∈ x1 + ψ cases.

¯ 0 (0) = id(0) = 0, and thus x · e = x. (i) If x2 = 0, then ψ ¯ 1 (0) = ψ(0). Because 0 ∈ ψ(0) we have x ∈ x · e. (ii) If x2 = 1, then ψ Thereby, for all x ∈ Zn oψ Z2 , x ∈ x · e ∩ e · x. So, the ψ -dihedral hypergroup Zn oψ Z2 has the identity e = (0, 0). ¯ −x2 (−x1 ), −x2 ). We notice that x′ = (−x1 , −x2 ) ∈ X ′ For any x = (x1 , x2 ) consider the set X ′ = (ψ and therefore e ∈ x′ · x ∩ x · x′ , meaning that x′ is an inverse for x.  Definition 3.23. A regular hypergroup (H , ·) is called reversible if for any (x, y, z ) ∈ H 3 , it satisfies the following conditions: (i) if x ∈ y · z, then there exists an inverse z ′ of z, such that y ∈ x · z ′ ; (ii) if x ∈ y · z, then there exists an inverse y′ of y, such that z ∈ y′ · x. Proposition 3.24. Let (a, b) be an arbitrary element in the ψ -dihedral hypergroup D = Zn oψ Z2 . If

(ψ¯ −b (−a), −b) is the set of inverses of the element (a, b), then D is a reversible hypergroup. ¯ −y2 (−y1 ), −y2 ) and Proof. Let x = (x1 , x2 ), y = (y1 , y2 ), z = (z1 , z2 ) be elements of H and y′ = (ψ ¯ −z2 (−z1 ), −z2 ) be the set of inverses of y and z respectively, then the relation x ∈ y · z means z ′ = (ψ (x1 , x2 ) ∈ (y1 , y2 ) · (z1 , z2 ) so x1 ∈ y1 + ψ¯ y2 (z1 ) and x2 ∈ y2 + z2 . Therefore z1 ∈ ψ¯ −y2 (−y1 + x1 ) = ψ¯ −y2 (−y1 ) + ψ¯ −y2 (x1 ) and z2 ∈ −y2 + x2 hence we have (z1 , z2 ) ∈ (ψ¯ −y2 (−y1 ), −y2 ) · (x1 , x2 ). Now we want to prove that y ∈ x · z ′ or equivalently

(y1 , y2 ) ∈ (x1 , x2 ) · (ψ¯ −z2 (−z1 ), −z2 ). ¯ y2 (z1 ) and x2 ∈ y2 + z2 we have x1 + ψ¯ x2 (ψ¯ −z2 (−z1 )) ⊆ y1 + ψ¯ y2 (z1 ) + Because x1 ∈ y1 + ψ ¯ ¯ ψy2 +z2 (ψ−z2 (−z1 )) = y1 + ψ¯ y2 (z1 ) + ψ¯ y2 (−z1 ) = y1 + ψ¯ y2 (z1 + −z1 ) = y1 + ψ¯ y2 (0). Thus ¯ y2 (0) and so our claim holds.  y1 ∈ y1 + ψ 4. Conclusions Once again, a strong connection between groups and hypergroups has been addressed. In this paper we have obtained a natural generalization of the dihedral groups, called ψ -dihedral hypergroups, studying their basic properties, like weakly associativity, regularity, reversibility. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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