Transitivity of The delta^n-Relation in Hypergroups | Mirvakili | Journal of the Indonesian Mathematical Society

Vol. 24, No. 02 (2018), pp. 36–46.

TRANSITIVITY OF THE δ

ⁿ

-RELATION IN HYPERGROUPS

Peyman Ghiasvand and Saeed Mirvakili

Department of Mathematics, Payame Noor University, Tehran, Iran E-mail: mathpnu2015@gmail.com, saeed_mirvakili@pnu.ac.ir

Abstract. Theδⁿ-relation was introduced by Leoreanu-Fotea et. al. [13]. In this article, we introduce the concept ofδⁿ-heart of a hypergroup and we determine necessary and sufficient conditions for the relationδⁿ to be transitive. Moreover, we determine a familyPσ(H) of subsets of a hypergroupH and we give sufficient conditions such that the geometric space (H, Pσ(H)) is strongly transitive and the relationδⁿis transitive.

Key words and Phrases: Geometric spaces, Hypergroup, strongly regular relation.

Abstrak. Konsep relasi-δⁿ telah diperkenalkan oleh Leoreanu-Fotea et. al. [13].

Dalam artikel ini, diperkenalkan konsepδⁿ-heart dari suatu hipergrup dan diten- tukan syarat perlu dan cukup bagi relasi-δⁿyang transitif. Lebih jauh, ditentukan juga suatu famili subsetPσ(H) dari suatu hipergrupHdan diberikan syarat cukup bagigeometric space(H, Pσ(H)) yang transitif kuat dan relasi-δⁿyang transitif.

Kata kunci:Geometric spaces, Hipergrup, relasi reguler kuat

1. Introduction

The concept of a hyperstructure first was introduced by Marty in [14], and then it studied by many authors, for example see [3, 5, 6, 15, 16]. The notion of fundamental relation on hypergroups was introduced by Koskas [11], and then studied by Corsini [2], Freni [7, 9] and Gutan [10], Vougiouklis [18, 19], Davvaz et.

al. [6] and Leoreanu-Fotea et. al. [13]. In [9], Freni firstly proved that the relation β is transitive in every hypergroup. The relationγ andγ^∗ were firstly introduced and analyzed by Freni [7]. He proved that the relationγon hypergroup is transitive andγ=γ^∗. Also, Freni [8] determined a familyPσ(H) of subsets of a hypergroup H such that the geometric space (H, Pσ(H)) is strongly transitive. Anavariyeh and

2000 Mathematics Subject Classification: 20N20, 16Y99.

Received: 13-06-2017, revised: 14-04-2018 accepted: 17-04-2018.

36

Davvaz [1] used the notion of strongly transitive geometric space on hypermodules.

Mirvakili and Davvaz [17] used the notion of strongly transitive geometric space on arbitrary hyperring and obtained new result in this respect.

Let us recall now some basic notions and results of hypergroup theory. A hyperstructure is a set H together with a function · : H ×H −→ ℘^∗(H) called hyperoperation, where ℘^∗(H) denotes the set of all non-empty subsets of H. If A, B⊆H, x∈H then we define

A·B= [

a∈A, b∈B

a·b, x·B ={x} ·B, A·x=A· {x}.

The structure (H,·) is called asemihypergroupifa·(b·c) = (a·b)·cfor alla, b, c∈H, and is called ahypergroup if it is a semihypergroup and a·H =H·a=H for all a∈H. A non-empty subsetK of a hypergroup H is calledleft invertible if for all (a, b)∈H², the implicationy ∈K◦x⇒x∈K◦y holds. K isinvertible ifK is left and right invertible. Suppose that (H,·) and (H⁰,◦) are two semihypergroup.

A function f : H −→ H⁰ 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). A non-empty subset K of a hypergroup (H,·) is called a subhypergroup if it is a hypergroup, that is for all k ∈ K, K·k =k·K = K.

A non-empty subset of a hypergroup (H,·) is called a complete part of H if the following implication holds:

A∩

n

Y

i=1

x_i6=∅ ⇒

n

Y

i=1

x_i⊆A.

If (H,·) is a hypergroup andR⊆H×H is an equivalence relation, we set ARB¯¯ ⇔a R b, ∀a∈A, ∀b∈B,

for all pairs (A, B) of non-empty subsets of H. The relation R is called strongly regular on the left (on the right) ifx R y ⇒a·x R a¯¯ ·y (x R y⇒ x·a R y¯¯ ·a, respectively), for all (x, y, a) ∈H³. Moreover, R is called strongly regular if it is strongly regular on the right and on the left. Strongly regular equivalence play in semi-hypergroup theory a role analogous to congruences in semigroup theory. IfR is a strongly regular equivalence on a hypergroup H, then we can define a binary operation⊗on the quotient setH/Rsuch that (H/R,⊗) is a group.

Definition 1.1. (See[13]) For any natural numbern, we define the relationδⁿ on the hypergroup(H,·), as follows: δⁿ=∪m≥1δ_mⁿ, where for every integerm≥1,δ_mⁿ is the relation defined as follows:

xδ_mⁿy⇔ ∃(x₁,· · ·, x_m)∈H^m, ∃τ ∈Sm, x∈

n

Y

i=1

xi, y∈

n

Y

i=1

x^j_τ(i)^τ(i) or y∈

n

Y

i=1

x_τ(i), x∈

n

Y

i=1

x^j_iⁱ

where∀i∈ {1,2,· · ·, m}, j_i∈ {1, n+ 1} andx^j_iⁱ =x_i·x_i· · · · ·x_i, (j_i times).

Denote by δ^n∗ the transitive closure of δⁿ. The relation δ^n∗ is a strongly regular relation. The relationδ^n∗ is the smallest equivalence relation on hypergroup

H, such that the quotient H/δ^n∗ is an abelian group. Moreover, for all x ∈ H, [δ^n∗(x)]ⁿ⁺¹ = δ^n∗(x) hold, which means that [δ^n∗(x)]ⁿ = e, the identity of the abelian group H/δ^n∗.

Moreover, we recall the following relation onH, which is included in δⁿ_m: xρⁿ_my⇔ ∃(x1,· · ·, xm)∈H^m: x∈

n

Y

i=1

xi, y∈

n

Y

i=1

x^j_iⁱ or y∈

n

Y

i=1

xi, x∈

n

Y

i=1

x^j_iⁱ where∀i∈ {1,2,· · ·, m}, ji∈ {1, n+ 1} andx^j_iⁱ =xi·xi· · · · ·xi, (ji times).

Setρⁿ =S

m∈N^∗ρⁿ_m and letρ^n∗ be the transitive closure ofρⁿ. The relation ρ^n∗ is the smallest equivalence relation on hypergroup H, such that the quotient H/ρ^n∗ is a group and[ρ^n∗(x)]ⁿ =e which is the identity of the groupH/ρ^n∗. Example 1.2. (See [13]) Letn= 2andH =S3×S3, whereS3 be the permutation group of order3, i.e.,

S3={(1),(12),(13),(23),(123),(132)}.

ThenH/δ^n∗∼=Z2×Z2.

2. Transitivity conditions of δⁿ

Definition 2.1. Let M be a non-empty subset of H. Then, we say that M is a δⁿ-part ofH if for every m∈ N, (x₁,· · · , x_m) ∈H^m and for everyσ ∈ Sm and ji∈ {1, n+ 1}, if

Qm

i=1xiS Qm i=1x^j_iⁱ

TM 6=∅ implies that (F₁) Qm

i=1x_i∩M 6=∅ ⇒ Qm

i=1x^j_σ(i)^σ(i)⊆M, (F2) Qm

i=1x^j_iⁱ∩M 6=∅ ⇒ Qm

i=1x_σ(i)⊆M.

Proposition 2.2. Let M be a non-empty subset of a hypergroup H. Then, the following conditions are equivalent:

(1) M is aδⁿ-part ofH. (2) x∈M, xδⁿy⇒y∈M. (3) x∈M, xδ^n∗y⇒y∈M.

Proof. (1⇒2) If (x, y)∈H²is a pair such thatx∈M andxδⁿy, then there exist m∈N^∗,(z1, z2,· · · , zm)∈Handσ∈Smsuch that (i)x∈Qm

i=1xi, y∈Qm i=1x^j_σ(i)^σ(i) or (ii)x∈Qm

i=1x^j_iⁱ, y∈Qm

i=1x_σ(i), wherej_i ∈ {1, n+ 1}. SinceM is aδⁿ-part of H, if x∈Qm

i=1xiT

M, then we getQm

i=1x^j_σ(i)^σ(i) ⊆M by Definition 2.1(F1). Thus y ∈M. If x∈Qm

i=1x^j_iⁱTM, then we have Qm

i=1x_σ(i)⊆M by Definition 2.1(F₂).

Thusy∈M.

(2⇒3) Let (x, y)∈Hsuch thatx∈M andxδ^n∗y. Obviously, there existk∈ Nand (w0=x, w1,· · ·, w_k−1, wk=y)∈H^ksuch thatx=w0δⁿw1δⁿ· · ·δⁿw_k−1, wk = y. Sincex∈M, applying (2)k-times, we obtainy∈M.

(3⇒1) Let (Qm

i=1xiS Qm

i=1x^j_iⁱ)T

M 6=∅andx∈(Qm

i=1xiS Qm

i=1x^j_iⁱ)T M. If x ∈ Qm

i=1x_i then for every σ ∈ S_m and for every y ∈ Qm

i=1x^j_σ(i)^σ(i) where

j_i ∈ {1, n+ 1}, we have xδⁿy, thus x ∈ M and xδ^n∗y. We obtain y ∈ M by (3), whenceQm

i=1x^j_σ(i)^σ(i) ⊂M. Ifx∈Qm

i=1x^j_iⁱ then for everyσ∈S_m and for every y ∈Qm

i=1xσ(i) where ji ∈ {1, n+ 1}, we havexδⁿy, thus x ∈M and xδ^n∗y. So y∈M by (3), whence Qm

i=1x_σ(i)⊂M.

Definition 2.3. The intersection of all δⁿ-parts which contain M is called δⁿ- closure of M in H and it will be denoted byK(M).

Before proving the next theorem, we introduce the following notations:

For every elementxof a hypergroupH, set:

Tm(x) =Sn Qm

i=1xσ(i)|σ∈Sm, ji={1, n+ 1}, x∈Qm i=1x^j_iⁱo

; Pm(x) =Sn

Qm

i=1x^j_σ(i)^σ(i) |σ∈Sm, ji={1, n+ 1}, x∈Qm i=1xi

o P_σ(x) =S

m≥1(T_m(x)∪P_m(x)).

For the preceding notations and definitions, it follows at once the following:

Lemma 2.4. For every x∈H,P_σ(x) ={y∈H | x δⁿ y}.

Proof. For every pair (x, y) of elements ofH we have:

xδⁿy⇔ ∃(x₁,· · · , x_m)∈H^m, ∃σ∈Sm, x∈

n

Y

i=1

x_i, y∈

n

Y

i=1

x^j_τ(i)^σ(i) ory∈Qn

i=1xσ(i), x∈Qn

i=1x^j_iⁱ⇔ ∃m∈N^∗:y∈Pm(x) ory∈Tm(x)⇔y∈Pσ(x).

Lemma 2.5. Let (H,◦)be a hypergroup and letM be aδ^n∗-part ofH. Ifx∈M, thenPσ(x)⊆M.

Proof. Ify ∈Pσ(x), then xδⁿy. Thus there existsm≥1 such thatxδ_mⁿy, whence there exists (x1, x2,· · · , xm) ∈ H^m and σ ∈ Sm, such that (i) x∈ Qm

i=1xi, y ∈ Qm

i=1x^j_σ(i)^σ(i) or (ii) x∈Qm

i=1x^j_iⁱ, y∈Qm

i=1xσ(i), where ji∈ {1, n+ 1}. If (i) holds, since x∈Qm

i=1xiT

M and M is a δ^n∗-part, it follows that y ∈ Qm

i=1x^j_σ(i)^σ(i) ⊆M by Definition 2.1(F1), and thusy ∈M. If (ii) holds, since x∈Qm

i=1x^j_iⁱTM and M is aδ^n∗-part, it follows that y∈Qm

i=1x_σ(i)⊆M by Definition 2.1(F2), and so y∈M. Therefore, in any case we have Pσ(x)⊆M. Theorem 2.6. Let H be a hypergroup. The following conditions are equivalent:

(1) δⁿ is transitive;

(2) for every x∈H,δ^n∗(x) =P_σ(x);

(3) for every x∈H,P_σ(x)is aδⁿ-part ofH.

Proof. (1⇒2) By Lemma 2.5, for every pair (x, y) of elements ofH we have:

y∈δ^n∗(x)⇔xδ^n∗y⇔xδⁿy⇔y∈Pσ(x).

(2⇒3) By Proposition 2.2, ifM is a non-empty subset ofH, thenM is aδⁿ- part ofH if and only if it is union of equivalence classes moduloδ^n∗. Particularly, every equivalence class moduloδ^n∗ is aδⁿ-part of H.

(3 ⇒ 1) Let xδⁿy and yδⁿz. Thus, x ∈ Pσ(y) and y ∈ Pσ(z) by Lemma 2.4. Since Pσ(z) is aδ^n∗-part, by Lemma 2.5, we havePσ(y)⊆Pσ(z) and hence x∈Pσ(z). Therefore,xδⁿy by Lemma 2.4 and the proof is complete.

Definition 2.7. Let (H,◦) be a hypergroup andφ:H −→H/δⁿ be the canonical projection. We denote by e = [δ^n∗(x)]ⁿ for all x ∈ H the identity of the group H/δⁿ. The setφ⁻¹(e)is called theδⁿ-heart ofH and it is denoted byDδⁿ. Theorem 2.8. D_δn is the smallest subhypergroup ofH, which is also a δⁿ-part of H.

Proposition 2.9. For every non-empty subset M of a hypergroupH, we have:

(1) ϕ⁻¹(ϕ(M)) =D(H)M =M D(H);

(2) M is aδⁿ-part if and only ifϕ⁻¹(ϕ(M)) =M.

Proof. 1) For everyx∈D(H)M, there exists a pair (a, b)∈D(H)×M such that x∈ab. Then ϕ(x) =ϕ(a)⊗ϕ(b) = e⊗ϕ(b) =ϕ(b). Thereforex∈ϕ⁻¹(ϕ(b))⊂ ϕ⁻¹(ϕ(M)).

Conversely, for every x∈ ϕ⁻¹(ϕ(M)), there exists an element a ∈ M such that ϕ(x) = ϕ(a). By the reproducibility, b ∈ H exists such that x ∈ ba, so ϕ(a) =ϕ(x) = ϕ(b)⊗ϕ(a), henceϕ(b) =e and a ∈ϕ⁻¹(e) =D(H). Therefore x∈ba⊂D(H)M. This proves thatϕ⁻¹(ϕ(M)) =D(H)M.

In the same way, we can prove thatϕ⁻¹(ϕ(M)) =M D(H).

For the proof of the sufficiency suppose that mδ^n∗x and m ∈ M. Thus ϕ(x) =ϕ(m)∈ϕ(M) and sox∈ϕ⁻¹(ϕ(M)) =M. Therefore by Proposition 2.2

it follows that M is aδⁿ-part of H.

Definition 2.10. Letzbe some element ofH. A hypergroupH is calledδ^n∗-strong whenever

(i) For allx, y∈H ifxδ^n∗y, thenxz∩yz6=∅ andzx∩zy6=∅ and (ii) {z} is invertible.

Theorem 2.11. IfH is aδ^n∗-strong hypergroup for some elementz∈H, thenδⁿ is transitive.

Proof. By Theorem 2.6, it is enough to show that for allx∈H, P(x) is aδ^n∗-part ofH. According to Proposition 2.9, we have to check thatϕ⁻¹(ϕ(P(x))) =P(x).

Lett∈ϕ⁻¹(ϕ(P(x))), thus there existsh∈P(x) such that ϕ(t) =ϕ(h) and hence δ^n∗(t) =δ^n∗(h). Sinceh∈P(x),hδⁿxby Lemma 2.4. Thusδ^n∗(x) =δ^n∗(h) and so δ^n∗(t) =δ^n∗(x). Since H is aδ^n∗-strong hypergroup, we have xz∩tz 6=∅ and

hence there exists s ∈ xz ∩tz. Therefore x ∈ tzz and t ∈ xzz, because {z} is invertible and so t∈tzzzz. Since

(tzz, t zz· · ·z

| {z } j2times

zz· · ·z

| {z } j3 times

)∈δⁿ

where j₁ = 1, j₂ =n+ 1 and j₃ = n+ 1, we have xδⁿt and hence t ∈P(x). So we haveϕ⁻¹(ϕ(P(x)))⊆P(x); it is obvious thatP(x)⊆ϕ⁻¹(ϕ(P(x))). Therefore

ϕ⁻¹(ϕ(P(x))) =P(x) and the proof is complete.

3. Strongly transitive geometric spaces associated to hypergroups According to [8], a geometric space is a pair (S,B) such thatSis a non-empty set, whose elements we call points, and B is a non-empty family of subsets of S, whose elements we call blocks. B is a covering ofS if for every pointy∈S, there exists a block B ∈ B such that y ∈B. If C is a subset of S, we say that C is a B-part orB-subset ofS if for everyB ∈ B,

B∩C6=∅ ⇒B⊆C.

IfB1, B2,· · ·, Bn arenblocks of geometric space (S,B) such thatBi∩Bi+16=∅, for anyi∈ {1,2,· · · , n−1}, then then-tupleB1, B2,· · · , Bn is called a polygonal of (S,B). The concept of polygonal allows us to define onS the following relation.

x≈y⇔x=yor a polygonal (B1, B2,· · · , Bn) exists such thatx∈B1andy∈Bn. The relation ≈ is an equivalence and it is easy to see that it coincides with the transitive closure of the following relation:

x≈y⇔x=y or there exists B∈ Bsuch that{x, y} ∈B, so ≈is equal toS

n≥1∼ⁿ, where∼ⁿ=∼ ◦ ∼ ◦ · · · ◦ ∼ntimes.

Theorem 3.1. [8]For every pair(A, B)of blocks of a geometric space (S,B)and for any integern∈N, the following conditions are equivalent:

(1) A∩B6=∅, x∈B⇒ ∃C∈ B: (A∪ {x})⊆C.

(2) A∩B6=∅, x∈Γ(B)⇒ ∃C∈ B: (A∪ {x})⊆C.

(3) A∩Γ(B)6=∅, x∈Γ(B)⇒ ∃C∈ B: (A∪ {x})⊆C.

Theorem 3.2. If(S,B) is a strongly transitive geometric space, then the relation

∼on S is transitive. Hence ≈=∼.

LetH be a hypergroup and let Pδ(H) be the family of subsets ofH defined as follows: for every integer m >1 and for everym-tuple (z1, z2,· · ·, zm)∈H^m, we set

(1) Bδ(z1) ={z1, z₁ⁿ⁺¹}.

(2) B_δ(z₁, z₂,· · · , z_m) =Sn Qm

i=1z_τ(i)^jτ(i)|τ∈S_m, j_i ∈ {1, n+ 1}o

,ifm≥2.

whereS_mis the symmetric group of all permutations of the set{1,2,· · ·, m}.

Also, we can consider another geometric space (H, P_ρ(H)) that defined as follows: for every integerm>1 and for every m-tuple (z₁, z₂,· · ·, z_m)∈H^m, we set

(1) Bρ(z1) ={z1, z₁ⁿ⁺¹}.

(2) Bρ(z1, z2,· · ·, zm) =Sn Qm

i=1z_i^ji |ji∈ {1, n+ 1}o

,ifm≥2.

Note that ifz₁=z₂=· · ·=z_m=z then B_δ(z₁, z₂,· · · , z_m) =B_ρ(z₁, z₂,· · · , z_m) =[

z^{j1+j2+···+jm} |j_i∈ {1, n+ 1} =B^z_m Corollary 3.3. If for allx∈H, xⁿ⁺¹=xthen

Bδ(z1, z2,· · · , zm) =[ (_m

Y

i=1

z_τ(i)|τ∈Sm

)

Corollary 3.4. If (H,◦) is a commutative hypergroup then two geometric spaces (H, P_ρ(H))and(H, P_δ(H))are equal.

Lemma 3.5. Let (H,◦)be a hypergroup. Then (1) Bρ(z1, z2,· · ·, zm)⊆Bδ(z1, z2,· · · , zm).

(2) Bδ(z1, z2,· · · , zm) =S Bρ(z_τ(1), z_τ(2),· · ·, z_τ(m))|τ∈Sm .

Proof. It is straightforward.

Lemma 3.6. If (z₁, z₂,· · ·, z_m) is a m-tuple of elements of a hypergroup (H,◦), Then:

(1) For every σ∈S_m we have

B_δ(z₁, z₂,· · · , z_m) =B_δ(z_σ(1), z_σ(2),· · ·, z_σ(m)).

(2) For every z∈H, we have

[Bδ(z1, z2,· · ·, zm)]◦z⊂Bδ(z1, z2,· · ·, zm, z).

z◦[Bδ(z1, z2,· · · , zm)]⊂Bδ(z, z1, z2,· · ·, zm).

(3) For every (m+k)-tuple of elements of a hypergroup (H,◦), we have Bδ(z1, z2,· · · , zm)◦Bδ(x1, x2,· · ·, xk)⊂Bδ(z1, z2,· · ·zm, x1, x2,· · ·, xk) Proof. (1) For every permutationσ∈Sm, we have

x∈Bδ(zσ₍₁₎, zσ₍₂₎,· · · , zσ_(m))⇔ ∃τ ∈Sm:x∈Qm

i=1z_τ(σ(i))^jτ(σ(i))

⇔ ∃τ ∈S_m:x∈Qm

i=1z_τ◦σ(i)^jτ◦σ(i)

⇔x∈Bδ(z1, z2,· · ·, zm).

(2) Ifw∈[Bδ(z1, z2,· · ·, zm)]◦z, then an elementy∈Bδ(z1, z2,· · ·, zm) and a τ ∈ Sm exist such that w ∈ y◦z and y ∈ Qm

i=1z_τ(i)^jτ(i). Setting z = zm+1 and jm+1= 1, ifσis the permutation ofSm+1 such that:

(σ(i) =τ(i), ∀i∈ {1,2,· · · , m};

σ(m+ 1) =m+ 1

we havew∈ Qm

i=1z_σ(i)^jσ(i)

◦z_m+1=Qm+1 i=1 z_σ(i)^jσ(i).

(3) For everyy∈Bδ(z1, z2,· · · , zm)◦Bδ(x1, x2,· · ·, xk), there exist elements a ∈ Bδ(z1, z2,· · · , zm) and b ∈ Bδ(x1, x2,· · ·, xk) such that y ∈ a◦b. If a ∈ Bδ(z1, z2,· · · , zm), a permutation σ ∈Sm exists such that a∈ Qm

i=1z^j_σ(i)^σ(i) where j_i∈ {1, n+ 1}and ifb∈B_δ(x₁, x₂,· · · , x_k), a permutationθ∈S_k exists such that b∈Qk

i=1x^j_θ(i)^θ(i) whereji∈ {1, n+ 1}. Thus y∈a◦b⊂

m

Y

i=1

z_σ(i)^jσ(i)

!

◦

k

Y

i=1

x^j_θ(i)^θ(i)

! .

Supposing that x1 =zm+1, x2 =zm+2,· · · , xk =zm+k, a permutationτ ∈Sm+k

exists such that

y∈

m

Y

i=1

z_σ(i)^jσ(i)

!

◦

k

Y

i=1

x^j_θ(i)^θ(i)

!

=

m+k

Y

i=1

z_τ(i)^jτ(i), thus

y ∈ B_δ(z₁, z₂,· · · , z_m, z_m+1,· · · , z_m+k) = B_δ(z₁, z₂,· · ·, z_m, x₁, x₂,· · ·, x_k).

Notice that the permutationτ is defined as follows:

(τ(i) =σ(i), if 1≤i≤m;

τ(i) =θ(i). ifm+ 1≤i≤m+k

Lemma 3.7. Let (H,◦)be a hypergroup. Then

(1) If zk∈a·b then

B_δ(z₁, z₂,· · ·, z_m)⊆B_δ(z₁, z₂,· · ·, z_k−1, a, a,· · · , a

| {z } j_k times

, b, b,· · ·, b

| {z } j_k times

, z_k+1· · ·, z_m).

(2) If z^j_k^k ⊂a^jk·b^jk then

Bδ(z1, z2,· · · , zm)⊆Bδ(z1, z2,· · ·, zk−1, a, b, zk+1· · · , zm).

Proof. (1) Letzk∈a·bandy∈Bδ(z1, z2,· · · , zm). Then there existsτ ∈Smsuch that y∈Qm

i=1z_τ(i)^jτ(i) and ji∈ {1, n+ 1}. Settingτ(h) =k, we have y∈

m

Y

i=1

z^j_τ(i)^τ(i) =

h−1

Y

i=1

z_τ(i)^jτ(i)◦z_k^jk◦

m

Y

i=h+1

z_τ(i)^jτ(i) ⊂

h−1

Y

i=1

z_τ(i)^jτ(i)◦(a◦b)^jk◦

m

Y

i=h+1

z_τ(i)^jτ(i)

⊂

h−1

Y

i=1

z_τ(i)^jτ(i)◦(a◦b)◦(a◦b)◦ · · · ◦(a◦b)

| {z }

jk times

◦

m

Y

h+1

z_τ(i)^jτ(i).

Setting thatz_k⁰ =aandz_m+1⁰ =b, z_m+2⁰ =a, z⁰_m+3 =b, z⁰_m+3 =a,· · · , z_m+2j⁰

k−2= a, z⁰_m+2j

k−1=b, and a permutationσ∈Sm+1exists such that y∈

h−1

Y

i=1

z^j_σ(i)^σ(i)

!

◦z_σ(h)⁰ ◦z⁰_σ(h+1)◦z_σ(h+2)⁰ ◦ · · · ◦z_σ(h+2j⁰ _i−1)◦





m+2j_i−1

Y

i=h+2ji

z_σ(i)^jσ(i)



, thusy∈B_δ(z₁,· · · , z_k−1, z_k⁰, z_k+1,· · · , z_m, z_m+1⁰ , z⁰_m+2,· · ·, z⁰_m+2j

i−1). Moreover, Bδ(z1,· · ·, zk−1, z_k⁰, zk+1,· · · , zm, z_m+1⁰ , z_m+2⁰ ,· · · , z_m+2j⁰ _i−1)

=Bδ(z1,· · ·, z_k−1, z_k⁰, z_m+1⁰ , z_m+2⁰ ,· · · , z_m+2j⁰ _i−1, zk+1,· · ·, zm)

=Bδ(z1,· · ·, zk−1, a, b, a, b,· · ·, a, b, zk+1,· · · , zm)

=Bδ(z1,· · ·, zk−1, a, a,· · · , a

| {z } jk times

, b, b,· · · , b

| {z } jk times

, zk+1,· · · , zm).

Therefore, we have

B_δ(z₁, z₂,· · ·, z_m)⊆B_δ(z₁, z₂,· · ·, z_k−1, a, a,· · · , a

| {z } jk times

, b, b,· · ·, b

| {z } jk times

, z_k+1· · ·, z_m).

We notice that, ifh=mthen the permutationσis defined as follows:

(σ(i) =τ(i), ∀i∈ {1,2,· · ·, m};

σ(m+ 1) =m+ 1, σ(m+ 2) =m+ 2,· · · , σ(m+ 2ji−1) =m+ 2ji−1.

while, if 16h < m, thenσis such that:







σ(i) =τ(i), if 1≤i≤h;

σ(m+ 1) =m+ 1;

σ(i) =τ(i−2j_i). ifh+ 1≤i≤m+ 2j_i−1.

(2) The proof follows the same argument exploited in Lemma 3.1 of [8].

Corollary 3.8. If (H,◦) is a hypergroup andzk∈a·b then B_δ(z₁, z₂,· · ·, z_m)⊆B_δ(z₁, z₂,· · · , z_k−1, B_j^a

k, B_j^b

k, z_k+1· · ·, z_m).

Corollary 3.9. Let (z1, z2,· · · , zm) be a m-tuple of elements of a hypergroup (H,◦). If an integer k ≥ 1, a k-tuple (x1, x2,· · ·, xk) ∈ H^k and an element k⁰ ∈ {1,2,· · ·, m} exist such thatz_k^jk0⁰ ∈Bδ(x1, x2,· · ·, xk), then

B_δ(z₁, z₂,· · · , z_m)⊆B_δ(z₁, z₂,· · ·, z_k⁰₋₁, x₁, x₂,· · ·, x_k, z_k⁰₊₁· · · , z_m).

Lemma 3.10. Let (H,◦) be a commutative hypergroup. If there exists an integer k ≥1, ak-tuple (x₁, x₂,· · ·, x_k)∈ H^m and element k⁰ ∈ {1,2,· · ·, m} such that z_k⁰ ∈Qk

i=1x_i, then

Bρ(z1, z2,· · · , zm)⊆Bρ(z1, z2,· · ·, zk⁰−1, x1, x2,· · · , xk, zk⁰+1· · ·, zm).

Theorem 3.11. If(H,◦)is a hypergroup and for everyt_k,(x₁, x₂,· · · , x_k⁰)∈H^k0 andtk∈Qk⁰

i=1xiwe have t^j_k^k⊂Qk⁰

i=1x^j_i^k wherejk∈ {1, n+ 1}, Then the geometric space(H, Pδ(H))is strongly transitive.

Proof. Let Bδ(z1, z2,· · ·, zm) and Bδ(y1, y2,· · ·, ys) be two block of Pδ(H) such that

Bδ(z1, z2,· · ·, zm)∩Bδ(y1, y2,· · · , ys)6=∅ and y∈Bδ(y1, y2,· · ·, ys).

Letb∈Bδ(z1, z2,· · · , zm)∩Bδ(y1, y2,· · · , ys). A pair (a, c)∈H of elements of H exists such that zm∈a◦y andy ∈b◦c. Since y∈Bδ(y1, y2,· · ·, ys), by Lemma 3.6 and Lemma 3.7, we have

y∈b◦c⊂[B_δ(z₁, z₂,· · · , z_m)]◦c⊂B_δ(z₁, z₂,· · · , z_m, c)

⊂B_δ(z₁, z₂,· · · , z_m−1, a, y, c)

⊂Bδ(z1, z2,· · · , zm−1, a, y₁^jy, y₂^jy,· · · , y^js^y, c) wherejy∈ {1, n+ 1}.

Moreover, since b∈B_δ(y₁, y₂,· · · , y_s), we obtain B_δ(z₁, z₂,· · ·, z_m)⊂B_δ(z₁, z₂,· · · , z_m−1, a, y)

⊂B_δ(z₁, z₂,· · · , z_m−1, a, b, c)

⊂Bδ(z1, z2,· · · , z_m−1, a, y₁^jy, y₂^jy,· · · , y^js^y, c).

ThereforeBδ(z1, z2,· · ·, zm)∪{y} ⊂Bδ(z1, z2,· · ·, z_m−1, a, y₁^jy, y^j₂^y,· · · , y^js^y, c) and the geometric space (H, Pδ(H)) is strongly transitive.

Corollary 3.12. If(H,◦)is a hypergroup and for everyx∈H, xⁿ⁺¹=x,then the geometric space (H, Pδ(H))is strongly transitive.

Proof. Since for everyx∈H, xⁿ⁺¹=x,thus by Corollary 3.3 we have Bδ(z1, z2,· · ·, zm) =[

(_m Y

i=1

z_τ(i)|τ ∈Sm

) .

Hence the proof follows the same argument exploited in Theorem 3.4 of [8].

Theorem 3.13. If (H,◦) is a commutative hypergroup, Then the geometric space (H, Pρ(H))is strongly transitive.

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