Abelian groups derived from hypergroups

Abstract: We introduce a new strongly regular relation αon a given group Gand show that αis a congruence

relation on G, with respect to module the commutator subgroup of G. Then we show that the composition of this

relation with the fundamental relation β∗is equal to the fundamental and γare is equal to the relation α. and we

conclude that if ρis an arbitrary strongly regular relation on the hypergroup H, then the effect of αon ρ, results in

a strongly regular relation such that its quotients is an abelian group.

Key–Words: Hypergroup, fundamental relation, fundamental group, Strongly regular relation.

Received: December 23, 2022. Revised: August 19, 2023. Accepted: September 14, 2023. Published: October 6, 2023.

1 Introduction

The hyperstructure theory, born in 1934 with Marty’s

paper at the V iii Congress of Scandinavian Mathe-

maticians , was subsequently developed around the

400swith the contribution of various authors espe-

cially in France and in the United States [3]. Marty

showed that the characteristics of hypergroups can be

used in solving some problems of groups, algebraic

functions, and rational functions. Surveys of the the-

ory can be found in [8]. A special equivalence relation

which is called fundamental relations play important

roles in the theory of algebraic hyperstructures. The

fundamental relations are one of the most important

and interesting concepts in algebraic hyperstructures

that ordinary algebraic structures are derived from al-

gebraic hyperstructures by them. The fundamental re-

lation β∗on hypergroups was deﬁned by Koskas[7]

and studied by many of authors( for more details see

[2, 3] [4, 5, 6], [9] and Vogiouklis[10]).

2 Preliminaries

In this section, we provide the basic deﬁnitions of hy-

pergroup and hyperring theory. For a complete intro-

duction, we refer the readers to [3].

Let Hbe a set, elements of which will be de-

noted a, b, ..., and subsets of which will be denoted

A, B, ..... Let P∗(H)be the family of nonempty sub-

sets of Hand ◦ahyperoperation or join operation in

H, that is, ◦is a function from H×Hinto P∗(H).

If (a, b)∈H×H, its image under ◦in P∗(H), is

denoted by a◦bor ab. The join operation is extended

to subsets of Hin a natural way, so that A◦Bor AB

is given by AB =T{ab|a∈A, b ∈B}. The notation

aA and Aa is used for {a}Aand A{a}, respectively.

Generally, the singleton {a}is deﬁned by its member

a. A non-empty set Htogether with a hyperoperation

·is called a hypergroupoid or a hyperstructures, and it

is denoted by the pair (H, ·). A hypergroupoid (H, ·)

is called a semihypergroup if for all x, y, z of H, the

associativity is hold: (x·y)·z=x·(y·z), which

means that Su∈x·yu·z=Sv∈y·zx·v. An element

eof His called an identity (resp. scalar identity) of

(H, ·)if for all a∈H, one has a∈(e·a)∩(a·e),

({a}= (e·a)∩(a·e)).

Deﬁnition 2.1 A semihypergroup (H, ·)is a hyper-

group if x·H=H·x=H, for all x∈H(Re-

production axiom).

A hypergroup (H, ·)is commutative if a.b =b◦a

for all a, b ∈H.

Deﬁnition 2.2 Let (H, ◦)be a semihypergroup and ρ

be an equivalence relation on H. Then ρis said to be:

(i) regular on the right (resp. on the left) if for all x

of H, from aρb, it follows that:

(a◦x)ρ(b◦x)(resp.(x◦a)ρ(x◦b));

(ii) strongly regular on the right (resp. on the left)

if for all xof H, from aρb, it follows that (a◦

x)ρ(b◦x)(resp. (x◦a)ρ(x◦b));

(iii) ρis called (resp. strongly regular)regular if it is

(resp. strongly regular) regular both on the right

and on the left.

Theorem 2.3 Let (H, ◦)be a semihypergroup(resp.

hypergroup) and ρbe an equivalence relation on H.

REZA AMERI, B. AFSHAR

School of Mathematics and Computer Sciences, College of Science, University of Tehran, Tehran, IRAN

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The ρis strongly regular if and only if (H/ρ, ⊗)is a

semigroup(resp. group), with respect to operation:

x⊗y={z|z∈x◦y}.

Deﬁnition 2.4 Let (H, ◦)be a semihypergroup and n

be a nonzero natural number. We say that

xβny⇔ ∃(a1, a2, . . . , an)∈Hn,{x, y} ⊆

i=1

ai.

Let β=S

n≥1

βn. Clearly, βis reﬂexive and symmetric.

Denote by β∗the transitive closure of β.

Theorem 2.5 [5] [?]β∗is the smallest strongly reg-

ular relation on H.

The smallest equivalence relation β∗in above

is called the fundamental relation on (resp.

semi)hypergroup (H, ◦), and the derived(resp.

semi)group H/β∗is called the fundamental(resp.

semigroup) group of H. R. Ameri in [?] shown

that this relation is fanctorial, that is, the rela-

tion β∗induced a functor from category of (resp.

semi)hypergroups to category of (resp. semi)groups.

Theorem 2.6 [6] If (H, ◦)is a hypergroup, then β∗=

β.

Example 2.7

Let (H, ◦)be a very thin hypergroup, such that for all,

only a pair of the element of Hthe hypercomposition

is a singleton set, that is there exists a unique pair

(a, b)∈H2,|a◦b|>1, and |x◦y|= 1 for all

x, y ∈H, (x, y)6= (a, b). Then β∗(x) = {x}, for all

x /∈a◦b, and β∗(y) = β∗(a◦b), for all y∈a◦b.

Remark 2.8 Freni in [4] introduced a new relation γ

on a hypergroup Has follows:

γ=Sn≥1γn, where γ1={(x, x); x∈H}, and

for positive integer n > 1,γnis deﬁned by

xγny⇐⇒ ∃ai∈H, ∃σ∈Sn,1≤i≤n;

x∈Qn

i=1 ai, y ∈Qn

i=1 aσ(i).

Evidently, for every n∈N,the relations γnhave

symmetric and reﬂexive properties, and hence the re-

lation γ=Sn≥1γnhas reﬂexive and symmetric prop-

erties. Assume γ∗be the transitive closure of γ. Also,

the class of H/γ∗is considered γ∗(z) = {w|zγ∗w},

for z, w ∈H. It was proved that the relation γis

transitive, and also γ∗has the smallest strongly reg-

ular equivalence property so that H/γ∗is an abelian

group.

Theorem 2.9 [4] The relation γ∗is the small-

est strongly regular relation on a (resp. hyper-

group)semihypergroup such that the quotient H/γ∗is

commutative (resp. group) semigroup.

Theorem 2.10 [4] If Hbe a hypergroup, then γ=

γ∗.

3 A new Relation α

In this section we introduce a new relation αon a

(resp. semi)hypergroup H, and reformulate the re-

lation γ∗based on the relation α.

Deﬁnition 3.1 Consider the relations αand δon a

group Gas follows:

g1αg2⇐⇒ ∃m∈N,∃(y1, y2, ..., ym)∈Gm,

∃σ∈Sm:g1∈Qm

i−1yi,and g2∈Qm

i=1yσ(i).

and

g1δg2⇐⇒ g1g−1

2∈G0,

where G0is derived group (or commutator sub-

group) of G.

As usual, we usually use, the congruence relation in-

stead, (strongly)regular relation on groups or semi-

groups.

Lemma 3.2 The relation αand δare congruence re-

lations on G.

Proof. By deﬁnition it is clear that δis a con-

gruence relation on G. Also, αis a symmetric

and reﬂexive relation. Let g1αg2and g2αg3,

then by deﬁnition α, there will be m, n ∈N,

σ∈Sn,τ∈Sm,(x1, x2, ..., xm)∈Gmand

(y1, y2, ..., yn)∈Gn, such that g1=Qm

i=1 xi,g2=

i=1 xσ(i)=Qn

j=1 yj,g3=Qn

j=1 yτ(j). So, g1g2=

x1x2...xmy1y2...ynαyj1yj2...yjnxi1xi2...xim=

g3gg2.

Now we can write

x1x2...xmy1y2...yn(y−1

ny−1

n−1...y−1

1)αyj1yj2...yjnxi1xi2...

xim(y−1

ny−1

n−1...y−1

1).Therefore, g1αg2. Sup-

pose g1αg2and g3∈G, so there are

n∈N,σ∈Snand (x1, x2, ..., xn)∈Gn, such

that g1=x1x2...xnαxi1xi2...xim=g2. So, one has

g3g1=g3x1x2...xnαg3xi1xi2...xing3g2.

Lemma 3.3 On a group α=δ.

Proof. We must show that δ⊆αand α⊆δ. Be-

cause, αis a congruence relation, then (Gα, •)is a

group, where [a1]α•[a2]α= [a1a2]αand eG/α =

[eG]α, and [a]−1

α= [a−1]α. Since, a1a2αa2a1, then

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[a1]α•[a2]α= [a1a2]α= [a2a1]α= [a2]α•[a1]α,

and hence (G/α, •)is abelian group. Thus δ⊆α,

because δis the smallest strongly regular relation on

Gsuch that G/α is an abelian group. Conversely,

suppose that aαb we must show that ab−1∈G0.

Since, aαb there will be m∈N,σ∈Smand

(x1, x2, a, xm)∈Gm, that is a=Qm

i=1 xiand

b=Qm

i=1 xσ(i). We know that xixj= [xi, xj]xjxi,

where [xi, xj] = xixjx−1

ix−1

j. So, there exists natu-

ral number kand elements aj,bj,(1 ≤j≤k)such

that

xi1xi2...xim=

[a1, b1][a2, b2]...[ak, bk]x1x2...xm,

where

g= [a1, b1][a2, b2]...[ak, bk]∈G0.

Therefore, ab−1=x1x2...xm(xi1xi2...xim)−1=

x1x2...xm(gx1x2...xm)−1=g−1∈G0.

Remark 3.4 Let ρbe an strongly regular relation on

H. Then it is easy to see that for each a, b ∈H;

a(α∗ρ)b⇐⇒ [a]ρα[b]ρ.

Lemma 3.5 Let ρis strongly regular relation on H.

Then α∗ρis also an strongly regular relation on H.

Proof. It is clear that α∗ρis an equivalence re-

lation on H. Let h1, h2, h ∈H, and h1(α∗ρ)h2.

Since h1(α∗ρ)h2, then [h1]ρα[h2]ρand [h]ρα[h]ρ.

Given that αis a strongly regular relation. Then

[h1]ρ•[h2]ρα[h2]ρ•[h1]ρ, and since ρis strongly reg-

ular, it concluded that

[h1]ρ•[hρ] = [h1oh]ρ= [z1]ρ.

and

[h2]ρ•[hρ] = [h2oh]ρ= [z2]ρ,

for all z1∈h1oh and for each z2∈h2oh. Therefore,

[z1]ρ= [h1oh]ρα[h2oh]ρ= [z2]ρ,

and for each z1∈h1oh, z2∈h2oh;z1(α∗ρ)z2.

Theorem 3.6 α∗β=γ.

Proof. By Lemma 3.5, (H/(α∗β), ?)is a

group. Let h1, h2∈H, by deﬁnition of α, one has

[h1]β•[h2]βα[h2]β•[h1]β. Since βis strongly reg-

ular , then [z1]β= [h1oh]βα[h2oh]β= [z2]β, for

each z1∈h1oh2, z2∈h2oh1. This means that

[h1oh2]α∗β= [h2oh1]α∗β,and since α∗βis strongly

regular, we have [h1]α∗β?[h2]α∗β= [h2]α∗β?[h1]α∗β.

Since (H, (α∗β), ?)is an abelian group and γis the

smallest relation such that H/γ is an abelian group,

it conclude that γ⊆α∗β. Suppose h1, h2∈

Hand h1(α∗β)h2. So, by deﬁnition, there are

m∈Nand ([x1]β,[x2]β, ..., [xm]β)∈(H/β)mand

σ∈Sm, such that [h1]β=Qm

i=1[xi]βand [h2]β=

i=1[xσ(i)]β. Since βis strongly regular relation,

then

h1]∈[h1]β= [x1]β•[x2]β•...•[xm]β= [x1◦x2◦...◦xm]β,

and

h2∈[h2]β= [xi1]β•[xi2]β•...•[xim]β= [xi1◦xi2◦...◦xim]β.

Let x∈x1◦x2◦...◦xmand y∈xi1◦xi2◦...◦xim.

Then we have xγy. Also, x∈[h1]βand y∈[h2]βim-

plies h1βx and yβh2. But, β⊆γ. Therefor, h1γx and

yγh2, this shows that h1γxγyγh2, and hence h1γh2,

as desired.

Theorem 3.7 Let ρbe an strongly regular relation on

a hypergroup (H, ◦). Then H/(α∗ρ)is an abelian

group and H/(α∗ρ)=(H/ρ)0.

Proof. By Theorem 2.7 we have β⊆ρ, and hence

α∗β⊆α∗ρ. Also, by Theorem 3.6 we have γ⊆α∗ρ.

Corollary 3.8 Let ϕ:H1→H2be a homomor-

phism of hypergroups. Let ρ1be a strongly regu-

lar relation on H1and ρ2be a strongly regular re-

lation on H2. Then ¯ϕ:H1/ρ1→H2/ρ2, where

¯ϕ([x]ρ1) = [ϕ(x)]ρ2is a homomorphism of groups.

Proof. It is obvious.

Corollary 3.9 Let H,Gand Abe the categories of

hypergroups, groups and abelian groups, respectively.

Let ρbe a strongly regular relation on H. Then the

mappings Fρ:H → G, and Fα:G → A, deﬁned

by Fρ(H) = H/ρ and Fα(G) = G/α are functors.

Moreover, Fα·Fρ=Fα∗ρ:H → A, and Fα∗ρ(H) =

H/(α∗ρ).

4 Conclusion

A new characterization for the fundamental relation

γ∗on a hypergroup, such that its quotient space be

abelian are given. In Precisely, it is shown that the

γ∗can be obtained as combination the fundamental

relation β∗and the commutator subgroup of the fun-

damental group derived from β∗.

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[7] M. Koskas, Groupoides, Demi-hypergroupes et

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155–192.

[8] F. Marty, Sur une generalization de la notion de

groupe, in: 8th Congress Math., Scandinaves,

Stockholm, Sweden, 1934, 45–49.

[9] S. Spartalis, T. Vougiouclis, The fundamental

relations on Hv−rings, Rivista. di Math. Pura

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resentations, Hadronic Press Monographs in

Mathematics, Hadronic Press, Florida, 1994.

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