Polygroup-Theory-and-Related-System-by-Bijaan-Davvaz(1)

Introduction

Group theory is a fundamental concept in modern mathematics, focusing on the study of symmetry It defines a group as a collection of symmetries that preserve the structure of an object, indicating that all groups originate from this principle While permutations were examined prior to this, the formal development of group theory is credited to Évariste Galois.

Between 1811 and 1832, significant advancements in understanding polynomials emerged through the study of permutation groups of their roots This foundational work laid the groundwork for the development of group theory, which has since become integral to nearly every area of mathematics The historical roots of group theory can be traced back to three key influences.

(1) The theory of algebraic equations;

Early pioneers of group theory include notable mathematicians such as Euler, Gauss, Lagrange, Abel, and Galois Among them, Galois is particularly recognized for establishing the connection between group theory and field theory, leading to the development of what is now known as Galois theory.

Permutations were initially explored by Lagrange in the 1770s, focusing on the theory of algebraic equations His primary goal was to understand the algebraic solvability of cubic equations Through his examination of cubic equations, Lagrange aimed to uncover the underlying principles that govern their solutions.

Lagrange assumes the roots of a given cubic equation are x 0 , x 00 and x 000

Then, taking 1, w, w 2 as the cube roots of unity, he examines the expression

The expression R = x₀ + wx₀₀ + w²x₀₀₀ demonstrates that it yields only two distinct values across the six permutations of the roots x₀, x₀₀, and x₀₀₀ However, the author struggled to fully explore this concept, perceiving permutations merely as rearrangements rather than as composable bijections The significance of permutation composition is evident in other scholarly work.

Ruffini and Abbati about 1800; in 1815 Cauchy established the calculus of permutations.

Évariste Galois discovered that for the roots r1, r2, , rn of an equation, there exists a group of permutations of these roots such that any function of the roots that remains unchanged by the group's substitutions is rationally expressible Conversely, any rationally expressible function of the roots is invariant under these substitutions In addition to his work on group theory, Galois made significant contributions to modular equations and elliptic functions He published his first work on group theory at the age of eighteen in 1829, but it did not gain recognition until his collected papers were published in 1846.

The number-theoretic strand was started by Euler and taken up by

Carl Friedrich Gauss advanced the concepts of modular arithmetic and explored additive and multiplicative groups in relation to quadratic fields In 1761, mathematician Leonhard Euler investigated modular arithmetic, focusing on the remainders of powers of a number modulo n Although Euler did not frame his findings in terms of group theory, he illustrated the decomposition of an abelian group into cosets of a subgroup and established a key result that the order of a subgroup is a divisor of the order of the entire group.

In 1801, Gauss advanced Euler's work significantly by contributing extensively to modular arithmetic, laying the groundwork for the theory of abelian groups He investigated the orders of elements and demonstrated that for every divisor of a cyclic group's order, there exists a corresponding subgroup Additionally, Gauss explored various abelian groups, specifically analyzing binary quadratic forms of the type ax² + 2bxy + cy², where a, b, and c are integers.

Gauss studied how forms behave under transformations and substitutions, categorizing them into distinct classes and establishing a composition rule for these classes He demonstrated that the sequence in which three forms are composed does not affect the outcome, thereby confirming the associative law Gauss's findings indicate the existence of a finite abelian group, a concept later expanded upon by Schering in 1869 when he identified a basis for this group in Gauss's works.

In the early 19th century, geometry underwent significant transformations that paved the way for the emergence of the group concept The exploration of projective and non-Euclidean geometries marked a departure from traditional metric properties, while the investigation of geometry in multiple dimensions introduced greater abstraction Key contributions to this evolution came from influential figures such as Monge, his student Carnot, and notably Poncelet Additionally, the study of non-Euclidean geometry by scholars like Lambert, Gauss, Lobachevsky, and János Bolyai further enriched the discipline.

In 1827, Möbius initiated the classification of geometries based on properties invariant under specific groups, despite lacking awareness of the group concept By 1832, Steiner advanced the field by exploring synthetic geometry, laying the groundwork for the eventual study of transformation groups.

Arthur Cayley and Augustin Louis Cauchy were pioneers in recognizing the significance of mathematical theory, with Cauchy contributing several key theorems The subject gained widespread attention through the efforts of Serret.

Camille Jordan and Eugen Netto Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu.

It was Walther von Dyck who, in 1882, gave the modern definition of a group.

The systematic study of Lie groups and their discrete sub-groups as transformation groups began in 1884 with the work of Sophus Lie, and was subsequently advanced by notable mathematicians including Killing, Study, Schur, Maurer, and Cartan.

The discontinuous (discrete group) theory was built up by Felix Klein, Lie,

Poincar´e, and Charles Emile Picard, in connection in particular with modular forms.

Other important mathematicians in this subject area include Emil

Artin, Emmy Noether, Sylow, and many others.

The abstract definition of a group and some examples

The set of all integers, represented as {0, ±1, ±2, }, includes both positive and negative whole numbers When adding any two integers, the result is always an integer, and the following two rules of addition apply to any integers m, n, and p: the sum of m and n is an integer, and the properties of addition ensure consistent results across various combinations of integers.

Furthermore, for any two given integersmandn, the equation m+x=n has a unique solutionx=n−m, which is also an integer.

Similar scenarios arise across various branches of mathematics, extending beyond just integers For example, the focus can shift to the collection of all nonsingular 2×2 matrices, which includes every 2×2 matrix that is invertible.

Asuch that the determinant of Ais not zero) LetA andB be any 2×2 matrices,

Then, the product ofAandBis also a 2×2 matrix This product is defined as

In matrix multiplication, the order of multiplication matters, meaning that for two matrices A and B, AB does not equal BA However, the associative property holds true, as demonstrated by the equation (AB)C = A(BC) for any three 2×2 matrices A, B, and C When matrix A is nonsingular, the equations AX and Y A = B yield unique solutions, specifically X = A⁻¹B and Y = B A⁻¹, where both X and Y are 2×2 matrices.

A group is defined as a set of elements combined with an operation that adheres to the associative law In this context, a linear equation within the group has a unique solution that belongs to the set Therefore, the collection of nonsingular elements forms a cohesive group.

2×2 matrices together with multiplication is said to be group, as is the set of all the integers with addition.

We will now state the formal definition of a group.

A group is defined as a non-empty set G equipped with a binary operation, commonly referred to as multiplication, which maps each ordered pair of elements (a, b) in G to another element ab in G For G to qualify as a group under this operation, it must satisfy two specific properties.

(1) For any three elements a, band c of G, the associative law holds:

(2) For two arbitrary elementsaandb, there existsxandyofGwhich satisfy the equationsaxndya=b.

The following properties of a group are important.

(2 0 ) There is a unique element e in G such that for all g ∈ G, ge eg=g.

(2 00 ) For any elementa∈G, there is a unique elementa 0 ∈Gsuch that aa 0 =a 0 a=e, whereeis the element of Gdefined in (2 0 ).

(3) The solutions x and y of the equations ax = b and ya = b are unique and we have x=a 0 b andy 0 , where a 0 is the element associated with the elementa in (2 00 ).

To prove the properties of the set G, we start by selecting an element a from G, as G is non-empty According to property (2), we find solutions x and y such that ax = a and ya = a For any arbitrary element g in G, we can identify elements u and v in G that satisfy au = g = va, leading to the conclusion that g = (va)e = v(ae) = va This demonstrates that ge = g, confirming that the identity element e satisfies the equation Since g is arbitrary, we can specifically set g = e, resulting in e = e Furthermore, the identity element e maintains the property ge = g for any g in G, thus proving e = e Consequently, we establish that any solution to ax = a is equivalent to a solution for ya = a.

Thus, the uniqueness of the elementeis proved, and (2 0 ) holds.

The proof of (2 00) follows a similar approach as (2), demonstrating the existence of elements a0 and a00 in G such that aa0 = e = a00a By applying (1) and (2 0), we derive that a00 = a00e = a00(aa0) and subsequently show that (a00a)a0 is greater than 0, leading to the conclusion that a0 is unique Thus, the proof of the uniqueness of a0 parallels that of the uniqueness of e in (2 0).

To prove the statement, we start with Ifax = b By applying left multiplication of a⁻¹ (the inverse of a) to both sides, we obtain a⁻¹b = a⁻¹(ax) = (a⁻¹a)x = ex = x This demonstrates that the solution to the equation ax = b is x = a⁻¹b, and this solution is unique Similarly, the solution to the equation ya = b is uniquely determined as y = b⁻¹.

Corollary 1.2.3 A non-empty setG with an operation is a group if the conditions (1), (2 0 ) and (2 00 ) are satisfied.

The elementedefined in (2 0 ) is called theidentityofG, the elementa 0 defined in (2 00 ) is called the inverse of a The inverse of an element a is customary denoted bya −1

Theorem 1.2.4 We have (a −1 ) −1 = a and (ab) −1 = b −1 a −1 , for all a, b∈G.

Proof The first equality follows from (2 00 )(the uniqueness of the inverse).

The second one is proved by the equality

(ab)(b −1 a −1 ) = ((ab)b −1 )a −1 = (a(bb −1 ))a −1 = (ae)a −1 −1 =e and the uniqueness of the inverse Notice the change in the order of the factors fromabto b −1 a −1

In any group, the product of elements a1, a2, , an (where n ≥ 3) is defined inductively as a1 an = (a1 an−1)an The general associative law applies, ensuring that for any arbitrary elements x1, x2, , xn, the product remains uniquely determined regardless of the method of multiplication, as long as the order of the factors is preserved.

For example, (xy)((z(uv))w) =x(((y(zu))v)w).

If a1 = a2 = = an, then we use the power notation (for a =a1), a1a2 an = a n If n = −m is a negative integer, then we define a n = (a −1 ) m ; also, we define a 0 = e The formulas a m a n = a m+n and

(a m ) n =a mn hold for any elementa ofGand any pair of integersm and n.

We say that two elements a and b of a group G are commutative or commute ifab A group is said to beabelian or commutative, if any two elements commute.

The number of elements in a group Gis called the order of G and is denoted by|G| If|G|is finite, thenGis said to be a finite group; otherwise

Before delving into the properties of groups, it is essential to explore specific examples These examples will serve as a foundation for defining various important types of groups.

(1) The set of integersZ, the set of rational numbersQand the set of real numbersRare all groups under ordinary addition.

(2) The setZn={0,1, , n−1}forn≥1 is a group under addition modulon For any i in Zn, the inverse of i is n−i This group usually referred to as thegroup of integers modulon.

(3) For a positive integern, consider the set Cn ={a 0 , a 1 , , a n−1 }.

OnCn define a binary operation as follows: a l a m a l+m if l+m < n a (l+m)−n if l+m≥n.

For every positive integern,Cnis an abelian group The groupCn is called thecyclic groupof ordern.

(4) For all integers n≥1, the set of complex roots of unity cos2kπ n +isin2kπ n |k= 0,1,2, , n−1

(i.e., complex zeros ofx n −1) is a group under multiplication.

A dihedral group in mathematics represents the set of symmetries of a regular polygon, encompassing both rotations and reflections As one of the fundamental examples of finite groups, dihedral groups are significant in the fields of group theory, geometry, and chemistry.

(6) Thequaternion groupis a non-abelian group of order 8 It is often denoted by Qor Q8 and written in multiplicative form, with the following 8 elements

Here 1 is the identity element, (−1) 2 = 1 and (−1)a=a(−1) =−a for allainQ The remaining multiplication rules can be obtained from the following relation: i 2 =j 2 =k 2 =ijk=−1.

The general linear group, denoted as GL(n), is formed by all invertible n×n matrices over the real numbers, operating under matrix multiplication This group represents various transformations in n-dimensional Euclidean space, including rotations, reflections, dilations, and skew transformations that maintain the origin When we limit our focus to matrices with a determinant of 1, we derive the special linear group, SL(n), which preserves both orientation and volume in geometric solids Further narrowing our scope to orthogonal matrices leads us to the orthogonal group, O(n), comprising transformations that consist solely of rotations and reflections, preserving lengths and angles while fixing the origin.

Finally, if we impose both restrictions, then we obtain thespecial orthogonal group SO(n), which consists of rotations only These groups are first examples of infinite non-abelian groups.

Subgroups

The concept of subgroups is one of the most basic ideas in group theory.

Definition 1.3.1 A non-empty subset H of a group G is said to be a subgroupofGif the following conditions are satisfied:

Corollary 1.3.2 If H is a subgroup of G, thenH is a group in its own right.

Corollary 1.3.3 Let G be a group and H be a non-empty subset of G.

Then,H is a subgroup ofGifH is closed under division, i.e., ifab −1 is in

Corollary 1.3.4 LetH be a non-empty finite subset of a groupG Then,

H is a subgroup ofGif H is closed under the operation of G.

(1) Let Gbe the group of all real numbers under addition, and letH be the set of all integers Then,H is a subgroup ofG.

(2) Let Gbe the group of all nonzero real numbers under multiplication, and letH be the set of positive rational numbers Then,H is a subgroup ofG.

(3) LetGbe the group of all nonzero complex numbers under multiplication, and letH ={a+bi|a 2 +b 2 = 1} Then,H is a subgroup ofG.

(4) Let G be an abelian group Then, H = {x ∈ G | x 2 = e} is a subgroup ofG.

(5) LetGbe the multiplicative group of all nonsingular 2×2 matrices over complex numbers Let H be the set of the following eight matrices ±

(6) The centerZ(G) of a group Gis the subset of elements in Gthat commute with every element of G In symbols, Z(G) = {a ∈

G|ax=xa, for allx∈G} The center of a groupGis a subgroup ofG.

(7) If H is a subgroup of G, then by the centralizer C(H) of H we mean the set {x∈G| xh=hx for allh∈H} Then,C(H) is a subgroup ofG.

(8) Let a ∈ G, define N(a) = {x ∈ G| xa = ax} Then, N(a) is a subgroup ofG N(a) is usually called thenormalizerorcentralizer ofainG.

(9) IfGis a group anda∈G, then thecyclic subgroup generated bya, denoted by< a >, is the set of all the powers of a.

Among the subgroups of G, the subgroups G and {e} are said to be trivial A subgroup H is said to be a proper subgroupof G if H 6=G If

M is a proper subgroup ofG and if M ⊆H ⊆Gfor a subgroupH ofG implies that G=H orH =M, thenM is said to be amaximal subgroup ofG.

Proposition 1.3.6 Let H and K be two subgroups of a group G The intersectionH∩K of H andK is a subgroup of G In general if {Hi} i∈I is a family of subgroups ofG, then T i∈I

Definition 1.3.7 IfXis a subset of a groupG, then the smallest subgroup ofGcontainingX, denoted by < X >, is called thesubgroup generated by

X If X consists of a single element a, then < X >=< a >, the cyclic subgroup generated bya.

Theorem 1.3.8 If X is a non-empty subset of a group G, then the subgroup< X >is the set of all finite products of the form u1u2 un, where for each i, eitherui∈X oru −1 i ∈X.

Let H be the set of all finite products formed by elements ui or their inverses u −1 i from a set X, where n is any positive integer For elements x = a1a2 an and y = b1b2 bm in H, their product xy is expressed as a finite sequence of elements ai and bj, ensuring that each factor or its inverse belongs to X, thereby confirming that xy ∈ H Additionally, the inverse of x, denoted x −1, is represented as a sequence of inverses, where each ai or its inverse is also in X, leading to the conclusion that x −1 ∈ H This establishes that H is a subgroup of G, with X being a subset of H.

In any subgroup \( K \) of \( G \) that contains \( X \), each element \( u \) in \( X \) is also in \( K \), which implies that the inverse \( u^{-1} \) is also in \( K \) Consequently, if \( x \) is expressed as the product \( u_1 u_2 \ldots u_n \), where each \( u_i \) belongs to \( X \) or \( u_i^{-1} \) is in \( X \), then \( x \) must be in \( K \) because all \( u_i \) are elements of \( K \) Therefore, we can conclude that \( H \) is a subgroup of \( G \) generated by \( X \), leading to the result that \( H \subseteq K \).

Definition 1.3.9 Let G be a group and H be a subgroup of G For a, b ∈ G we say a is congruent to b mod H, written as a ≡ b mod H if ab −1 ∈H.

Lemma 1.3.10 The relation a≡b mod H is an equivalence relation.

A right coset of a subgroup H in a group G is defined as Ha = {ha | h ∈ H}, where a is an element of G Similarly, a left coset is represented as aH The index of H in G, denoted as [G:H], refers to the number of distinct right cosets of H within G.

Lemma 1.3.12 For all a∈G, we haveHa={x∈G|a≡x mod H}.

The following corollary contains the basic properties of right cosets and it is useful in many applications.

Corollary 1.3.13 Let H be a subgroup ofG.

(1) Every element a of G contained in exactly one coset of H This coset isHa.

(2) Two distinct cosets of H have no common element.

(3) The groupG is partitioned into a disjoint union of cosets ofH.

(4) There is a one to one correspondence between any two right cosets ofH in G.

(5) There is a one to one correspondence between the set of left cosets ofH in Gand the set of right cosets ofH inG.

Theorem 1.3.14 If G is a finite group and H is a subgroup of G, then

The above theorem of Lagrange is one of the basic results in finite group theory.

In group theory, the order of an element \( a \) in a group \( G \) is defined as the smallest positive integer \( n \) such that \( a^n = e \), where \( e \) is the identity element of the group If no such integer exists, the element \( a \) is said to have infinite order The notation \( o(a) \) is used to represent the order of the element \( a \).

There are many useful corollaries of Lagrange theorem.

Corollary 1.3.16 A finite cyclic group of prime order contains no non- trivial subgroup.

Corollary 1.3.17 The order of an element of a finite groupGdivides the order|G|.

Definition 1.3.18 LetA and B be two subsets of a group G The set

AB ={ab | a∈ A, b∈ B} consisting of the products of elements a∈ A andb∈B is said to be theproductofAandB.

The associative law of multiplication gives us (AB)C=A(BC) for any three subsetsA, B andC.

The product of two subgroups is not necessarily a subgroup We have the following theorem.

Theorem 1.3.19 Let A and B be subgroups of a group G Then, the following two conditions are equivalent:

(1) The productAB is a subgroup ofG;

Proof Suppose thatAB is a subgroup of G Then, for anya∈A,b∈B, we have a −1 b −1 ∈AB and so ba= (a −1 b −1 ) −1 ∈AB Thus, BA ⊆AB.

Now, ifxis any element ofAB, thenx −1 ∈ABand sox= (x −1 ) −1 (ab) −1 =b −1 a −1 ∈BA, soAB⊆BA Thus,AB.

To demonstrate that the set AB is a subgroup, we start by assuming that if \( a \in A \) and \( b \in B \), then \( ab = b_1a_1 \) for some \( a_1 \in A \) and \( b_1 \in B \) We need to verify two conditions: closure and the existence of inverses for every element in AB Let \( x \in AB \) and \( y = a_0 b_0 \in AB \) Then, \( xy = aba_0b_0 \) Since \( ba_0 \in BA \), we can express it as \( ba_0 = a_2b_2 \) with \( a_2 \in A \) and \( b_2 \in B \) Thus, \( xy = a(a_2b_2)b_0 = (aa_2)(b_2b_0) \in AB \) Moreover, the inverse of \( x \) can be expressed as \( x^{-1} = b^{-1}a^{-1} \in BA \), confirming that AB is indeed a subgroup.

Normal subgroups and quotient groups

There is one kind of subgroup that is especially interesting IfGis a group andH is a subgroup ofG, it is not always true thataH =Hafor alla∈G.

Certain situations in group theory highlight the critical importance of specific subgroups, a concept first recognized by Galois, who emphasized their significance in mathematical study.

Definition 1.4.1 A subgroupN of a groupGis called anormal subgroup ofGifaN =N afor alla∈G.

A groupGis said to besimpleifG6={e}andGcontains no non-trivial normal subgroup.

The only simple abelian groups are Zp withpprime.

There are several equivalent formulations of the definition of normality.

Lemma 1.4.2 Let Gbe a group andN be a subgroup ofG Then,

(1) N is normal in G if and only if a −1 na ∈ N for all a ∈ G and n∈N.

(2) N is normal in G if and only if the product of two right cosets of

N inGis again a right coset of N inG.

(1) The centerZ(G) of a group is always normal Indeed, any subgroup ofZ(G) is normal inG.

(2) IfH has only two left cosets inG, thenH is normal inG.

(3) Let Gbe the set of all real matrices a b

0d where ad 6= 0, under matrix multiplication Then, N 1b

(4) Let Qbe the set of all rational numbers and G={(a, b)| a, b ∈

Q, a6= 0} Define ∗ onG as follows: (a, b)∗(c, d) = (ac, ad+b).

Then, (G,∗) is a non-abelian group If we considerN ={(1, b)|b∈

Q}, thenN is a normal subgroup ofG.

(5) For everyn≥1,SL(n) is a normal subgroup ofGL(n).

Theorem 1.4.4 states that if N is a normal subgroup of a group G, then the collection of all cosets of N forms a set G For any two elements X and Y in G, their product XY is defined as the subset obtained by multiplying the subsets X and Y This product XY is also a coset of N, and under this multiplication, the set G satisfies the properties of a group.

Proof LetX andY be two elements ofG Then, there are elementsxand y ofG such thatX =N x and Y =N y By assumption, N is normal so thatN x=xN for anyx∈G Whence

The evidence demonstrates that XY is a coset of N For any element Z in G, the associative law for subset multiplication yields (XY)Z = X(YZ), confirming that the multiplication on G adheres to the associative property According to the definition, we find that (N x)(N y) = N xy, indicating that the coset containing the identity element e of G, specifically N, serves as the identity of G Furthermore, the inverse of the coset N x is represented by N x^(-1).

The group G which was defined in the above theorem is called the quotient groupofGbyNand is writtenG=G/N The mappingx−→N x fromGintoGis called thecanonical map The order of the quotient group

G/N is equal to the index of the normal subgroupN, i.e.,|G/N|= [G:N].

(2) Let G be a group such that (ab) p = a p b p for all a, b ∈ G, where p is a prime number Let N = {x ∈ G | x p m e for some m depending on x} Then, N is a normal subgroup ofG IfG=G/N and ifx∈Gis such thatx p =e, thenx=e.

We close this section with the following correspondence theorem.

Theorem 1.4.6 Let N be a normal subgroup of a group G, and let GG/N For any subgroupV ofG, there corresponds a subgroupV of Gsuch that

Subgroup V is defined as the collection of elements from group G that are included in certain elements of V, establishing a unique relationship with V Consequently, there is a one-to-one correspondence between the set of all subgroups of G and those subgroups of G that encompass N, represented as V ↔ V.

Group homomorphisms

Let G be a finite group with n elements a1, a2, , an A multiplication table forGis then×nmatrix withi, jentryai∗aj:

. an an∗a1 an∗a2 an∗an

Informally, we say that we “know” a finite group Gif we can write a multiplication table for it.

In this section, we consider one of the most fundamental notions of group theory-“homomorphism” The homomorphism term comes from the

Greek words “homo”, which means like and “morphe”, which means form.

In our discussion on groups, we explore how to uncover information about a specific group by analyzing its interactions with other groups through the lens of homomorphisms A group homomorphism is essential as it maintains the integrity of the group operation.

In examining two simple groups, we observe group G, which consists of the elements 1 and -1 under multiplication, and group ˆG, the additive group Z2 A comparison of the multiplication tables for these two groups reveals their structural similarities and differences.

Gand and Gare represent distinct groups; however, they share no significant differences This observation highlights the need for a more precise understanding of their similarities and distinctions.

A function \( f \) from a group \( G \) to a group \( \hat{G} \) is defined as a group homomorphism if it satisfies the condition \( f(xy) = f(x)f(y) \) for all elements \( x, y \) in \( G \) When \( f \) is surjective, meaning that \( f(G) = \hat{G} \), the group \( \hat{G} \) is considered homomorphic to \( G \) If \( f \) is both surjective and injective, it is classified as an isomorphism from \( G \) to \( \hat{G} \) In cases where an isomorphism exists between \( G \) and \( \hat{G} \), we denote that \( G \) and \( \hat{G} \) are isomorphic, represented as \( G \sim= \hat{G} \).

Letf be a homomorphism fromGinto ˆG The subset

H ={x∈G|f(x) is the identity of ˆG} is called thekerneloff and is denoted byKerf.

(1) Every canonical mapping is a homomorphism.

Let G represent the group of all positive real numbers under multiplication, while ˆG denotes the group of all real numbers under addition The function f: G → ˆG is defined by f(x) = log10(x) for all x in G This function satisfies the property log10(xy) = log10(x) + log10(y), indicating that f is a homomorphism Furthermore, f is both one-to-one and onto, establishing a bijective relationship between the two groups.

The multiplicative group of all nonsingular n×n matrices over the real numbers is denoted as GL(n), while R ∗ represents the multiplicative group of all nonzero real numbers We define a function f that maps GL(n) to R ∗ by f(A) = det(A) for any matrix A in GL(n) Given that the determinant of the product of two matrices equals the product of their determinants, we find that f(AB) = f(A)f(B) for any matrices A and B in GL(n) This establishes f as a homomorphism from GL(n) to R ∗, and it is also onto.

The dihedral group D2n consists of all formal symbols a^i b^j, where i ranges from 0 to 1 and j ranges from 0 to n-1, with the relations a^2 = e and b^n d ab = b^(-1) a Within this group, the subgroup N, which includes the elements {e, b, b^2, , b^(n-1)}, is normal in G Additionally, the quotient group D2n/N is isomorphic to H, where H is defined as {1, -1}, representing the group formed by the multiplication of real numbers.

Proposition 1.5.3 Let f be a homomorphism from a group G into a groupG The following propositions hold:ˆ

(3) The kernel off is a normal subgroup ofG.

(4) LetH be a subgroup of G The imagef(H) ={f(x)| x∈H} is a subgroup ofGˆ For a subgroup Hˆ of Gˆ, the inverse image f −1 ( ˆH) ={x∈G|f(x)∈H}ˆ is a subgroup ofG.

For any elements x and y in group G, the function f(x) equals f(y) if and only if x and y belong to the same coset of the kernel of f Notably, if f is surjective, then f acts as an isomorphism if and only if the kernel of f contains only the identity element {e}.

(6) If H is a normal subgroup of G, then f(H) is a normal subgroup off(G).

We are in a position to establish an important connection between homomorphisms and quotient groups Many authors prefer to call the next theorem the Fundamental theorem of group homomorphism.

Theorem 1.5.4 Let f be a homomorphism from a group Gonto a group

G Then, there exists an isomorphismˆ g from G/kerf onto Gˆ such that f =gϕ, whereϕis the canonical homomorphism fromGontoG/kerf In this case, we say the following diagram is commutative.

Gˆ Proof Suppose thatK=kerf Define a functiong fromG/K to ˆGby g(Kx) =f(x).

The function g is well defined and does not depends on the choice of a representative fromKx Sincef is a homomorphism, we have g(KxKy) =g(Kxy) =f(xy) =f(x)f(y) =g(Kx)g(Ky).

The function g is a homomorphism from the quotient group G/K onto the group ˆG When the coset Kx is in the kernel of g, it leads to the conclusion that e = g(Kx) = f(x), indicating that x belongs to K Therefore, g can be classified as an isomorphism Additionally, it is evident that the relationship f = gϕ is maintained, ensuring the commutativity of the diagram.

Corollary 1.5.5 states that for a normal subgroup N of a group G and the canonical homomorphism ϕ from G to G/N, a homomorphism f from G to another group G exists such that f can be expressed as the composition of a homomorphism g from G/N to G and ϕ if and only if N is a subset of the kernel of f In this scenario, the image of f, denoted f(G), is isomorphic to the quotient of G/N by the kernel of f over N.

Proof If there is a homomorphismgsatisfyingf =gϕ, we havef(N) =e; so N ⊆ kerf Conversely, suppose that N ⊆ kerf Set K = kerf and

G=G/N We define a functiong fromGinto ˆGby g(N x) =f(x).

The functiong is uniquely determined independent of the choice of a rep- resentativexfromN x, andg is a homomorphism fromGinto ˆG Clearly, f =gϕby definition, and we havekerg=K From Theorem 1.5.4, we get

Corollary 1.5.6 Letf be a homomorphism from a groupGonto a group

G Letˆ Nˆ be a normal subgroup of Gˆ and set N =f −1 ( ˆN) Then, N is a normal subgroup ofGandG/N ∼= ˆG/N.ˆ

Theorem 1.5.7 Let H be a normal subgroup of a group G, and let K be any subgroup of G Then, the following hold:

Proof Since H is a normal subgroup of G, HK =KH and so according to Theorem 1.3.19,HK is a subgroup of G.

Now, consider the canonical mapping from G onto the factor group

G/H Letf be the restriction of the canonical mapping onK Then,f is a homomorphism fromK into G/H By definition,f is a homomorphism fromK ontoHK/H Theorem 1.5.4 proves that

But, the kernel of the canonical homomorphism isK, whence we getkerfH∩K This completes the proof

(1) Any finite cyclic group of ordernis isomorphic toZn Any infinite cyclic group is isomorphic toZ.

(2) LetGbe the group of all real valued functions on the unit interval

[0,1], where for anyf, g∈G, we define additionf+g by

(3) The quotient groupR/Zis isomorphic to the groupS 1 of complex numbers of absolute value 1 (with multiplication) An isomorphism is given byf(x+Z) =e 2πxi for allx∈R.

(4) The quotient groups and subgroups of a cyclic groups are cyclic.

An automorphism of a group G is defined as an isomorphism that maps G onto itself For any element g in G, the function ig, defined by ig(x) = g⁻¹xg, represents an inner automorphism of G.

LetAut(G) denote the set of all automorphisms ofG For the product of elements ofAut(G) we can use the composition of mappings.

Lemma 1.5.10 If Gis a group, thenAut(G)is also a group.

Aut(G) refers to the group of automorphisms of G, while Inn(G), the subgroup of Aut(G), represents the group of inner automorphisms of G In the case where G is abelian, specific properties regarding these groups can be observed.

Theorem 1.5.11 The group of inner automorphisms of G is isomorphic to the quotient groupG/Z(G), whereZ(G)is the center ofG Furthermore,

Inn(G)is a normal subgroup of Aut(G).

Proof The functiong7→igis a homomorphism fromGontoInn(G) Thus,

G/K ∼= Inn(G), where K is the kernel of the above homomorphism An element g of Glies in the kernel K if and only if ig =e, i.e., g −1 xg=x, for allx∈G Therefore, we haveK=Z(G).

The last assertion follows immediately from the formula σ −1 igσ=iσ(g), which holds for anyg∈Gandσ∈Aut(G)

(2) Aut(G) = 1 if and only if|G| ≤2.

Permutation groups

In this section, we study certain group of functions called permutation group, from a setX to itself Permutation groups are an important class of groups.

Definition 1.6.1 LetX be a set A one to one function fromX ontoX is called apermutationon the setX We denote the set of all permutations ofX bySX.

A function pdefined on the set X is a permutation if and only if the following three conditions are satisfied:

(2) Givena∈X, there is an elementx∈X such thatp(x) =a;

In the special case where X = {x1, x2, , xn}, we denote the set of permutations as Sn, which has a total of n! elements Each function f within Sn represents a one-to-one mapping of the set X onto itself For example, we can express f by detailing its action on each element, such as f: x1 → x2, x2 → x4, x4 → x3, and x3 → x1.

But this is very cumbersome One short cut might be to writef out as x1 x2 x3 xn xi 1 xi 2 xi 3 xi n

, wherexi k is the image ofxk underf Return to our example just above,f might be represented by x1x2x3 x4 x2x4x1 x3

Although this notation is more convenient, it still contains unnecessary elements, particularly the symbol x, which serves no clear purpose We could just as effectively represent the permutation without it.

In the context of set theory, when p and q are permutations of a set X, their composition, denoted as p◦q, also constitutes a permutation of X This composition is commonly referred to as the product of p and q, represented by the juxtaposition pq For instance, consider the permutation p defined on the elements 1, 2, and 3.

2 3 1 are permutations of{1,2,3} The productpq is

1 Notice that some authors compute this product in the reverse order This authors will write functions on the right: instead of f(x), they write (x)f.

The set SX of permutations on X forms a group under the operation defined above We callSX thesymmetric group onX.

Definition 1.6.2 LetX be a set andS X be the symmetric group onX.

Any subgroup ofSX is called apermutation grouponX.

Ifx∈X andf ∈S X , thenf fixesiiff(i) =iandf movesiiff(i)6=i.

We sayσ1, σ2, , σk aredisjointif no element ofX is moved by more than one ofσ1, σ2, , σk It is easy to see that τ σ=στ wheneverσand τ are disjoint.

A cycle permutation, denoted briefly, is defined as a permutation \( f \) on the set {1, 2, , n}, where a subset of distinct elements \( i_1, i_2, , i_r \) (with \( 1 \leq r \leq n \)) is involved In this case, \( f \) fixes the remaining \( n - r \) integers, and the mapping follows the pattern \( f(i_1) = i_2, f(i_2) = i_3, , f(i_{r-1}) = i_r, f(i_r) = i_1 \).

(i1 i2 ir) ris called the lengthof the cycle.

We multiply cycles by multiplying the permutations they represent.

Lemma 1.6.4 Every permutation can be uniquely expressed as a disjoint cycles.

A cycle of length 2 is called a transposition.

Lemma 1.6.5 Every permutation is a product of transpositions.

A permutation f on the set X = {1, 2, , n} can be represented as a product of transpositions If f can be expressed as an even number of transpositions, then all representations of f will also require an even number of transpositions Conversely, if f is expressed as an odd number of transpositions, every representation will necessitate an odd number of transpositions.

(respectively odd) number of transpositions.

Definition 1.6.6 A permutationf ∈Snis said to be an even permutation if it can be represented as a product of an even number of transpositions.

We call a permutationoddif it is not an even permutation.

The rule for combining even and odd permutations is like that of combining even and odd numbers under addition.

Let An be the subset of Sn consisting of all even permutations Since the product of two even permutation is even, An must be a subgroup of

S n An is called thealternative groupof degreen.

Lemma 1.6.7 An is a normal subgroup of Sn of order n! 2

The alternative groups are among the most important examples of groups.

Lemma 1.6.8 For n > 3, each element of An is a product of cycles of length 3.

Proof Suppose thatσ∈ An Then,σ is an even permutation Let (x y) and (a b) be two transpositions If {a, b} ∩ {x, y}=∅, then

On the other hand, if{a, b} ∩ {x, y} 6=∅, then we letb=x So, (a b)(b y) (y a b) .

Lemma 1.6.9 LetN be a normal subgroup ofAn (n≥5) IfN contains a cycle of length 3, thenN =An.

Proof By Lemma 1.6.8, it is enough to show that each cycle of length 3 is belong toN Suppose that (a b c) is an arbitrary cycle of length 3 and

(x y z)∈N Considerσ∈Sn such thatσ(a) =x,σ(b) =y andσ(c) =z.

Then, σ −1 (x y z)σ = (a b c) If σ ∈ An, then (a b c) ∈ N If σ 6∈ An, then we can choose u, v distinct from x, y, z Since σ 6∈ An, σ is an odd permutation So,σ(u v)∈An Thus, ((u v)σ) −1 (x y z)((u v)σ)∈N But

Lemma 1.6.10 LetN be a normal subgroup ofAn(n≥5) IfN contains a product of two disjoint transpositions, thenN =An.

Proof Suppose that (x y)(a b)∈ N such that {x, y} ∩ {a, b} =∅ Since n≥5, we choosezdistinct fromx, y, a, b Now, we setσ= (a b z) Clearly, σ∈An So, σ −1 (x y)(a b)σ= (z b a)(x y)(a b)(a b z)∈N This implies that (a z)(x y) ∈ N Since N is a subgroup, (x y)(a b)(a z)(x y) ∈ N.

Thus, (a z b)∈N Hence,N contains a cycle of length 3, and so by Lemma

Theorem 1.6.11 An is simple for alln≥5.

Proof Suppose thatn≥5 andN is a non-trivial normal subgroup of An.

We show thatN =An by considering the possible cases.

(1) N contains a cycle of length 3; henceN =An by Lemma 1.6.9.

(2) N contains an element σ, the product of disjoint cycles, at least one of which has length r ≥4 Then,σ= (a1 a2 ar)τ (disjoint) Let δ= (a1 a2 a3)∈An Then, σ −1 (δσδ −1 ) =τ −1 (ar ar−1 a1)(a1 a2 a3)(a1a2 ar)τ(a3a2 a1)

(3)N contains an elementσ, the product of disjoint cycles, at least two of which have length 3, so thatσ= (a1a2a3)(a4a5a6)τ(disjoint) Letδ(a1a2a4)∈An Then as above,N containsσ −1 (δσδ −1 ) = (a1a4a2a6a3).

(4)N contains an elementσthat is the product of one cycle of length 3 and some transpositions, sayσ= (a1a2a3)τ(disjoint), withτa product of disjoint transpositions Then,σ 2 = (a1 a3 a2)∈N, soN =An by Lemma

(5) Each element of N is the product of an even number of disjoint transpositions Let σ ∈ N, with σ = (a1 a2)(a3 a4)τ (disjoint).

Let δ = (a1 a2 a3) ∈ An Then as above, N contains σ −1 (δσδ −1 ).

Now, σ −1 (δσδ −1 ) = (a1 a3)(a2 a4) Since n ≥ 5, there is an element b ∈ {1, , n} distinct from a1, a2, a3, a4 Since η = (a1 a3 b) ∈ An and ζ = (a1 a3)(a2 a4) ∈ N, we have ζ(ηζη −1 ) ∈ N But ζ(ηζη −1 ) (a1a3 b)∈N Hence, N=An by Lemma 1.6.9.

Since the cases listed cover all the possibilities,Anhas no proper normal subgroups and hence is simple

The English mathematician Cayley first noted that every group could be realized as a subgroup ofSX for someX.

Theorem 1.6.12 (Cayley’s theorem) Let G be a given group Then, there exists a set X such thatG is isomorphic to a permutation group on

Proof We choose the setX consisting of all the elements of G For each elementg∈G, letπg be a function onX defined by the formula πg(x) =xg (forx∈X).

It follows thatπg is a permutation onX Furthermore, the associative law proves πgh=πgπh (forg, h∈G).

Thus, the functionπis a homomorphism fromGinto the symmetric group

SX Clearly, πg is the identity function on X if and only if g = e This means that π is one to one Hence, G is isomorphic to the image π(G) which is a permutation group onX.

Direct product

Constructing new groups from existing ones can be achieved through various methods, with the direct product of groups being particularly significant This section explores the essential concepts of both direct and semidirect products of groups.

Definition 1.7.1 If H and K are groups, then theirdirect product, denoted byH×K, is the group with elements all ordered pairs (h, k), where h∈H andk∈K, and with the operation

It is easy to check that H ×K is a group; the identity is (e, e 0 ); the inverse (h, k) −1 is (h −1 , k −1 ) Notice that neither H norK is a subgroup of H ×K, but H×K does contain isomorphic replicas of each, namely,

(1) LetV ={e, a, b, c}under a binary operation defined by the following table:

Then, (G,∗) is a group This is known as Klein’s four group We haveV ∼=Z2×Z2.

Theorem 1.7.3 Let G be a group with normal subgroups H and K If

In the group G, any element a can be expressed as a product of elements h from subgroup H and k from subgroup K, denoted as a = hk It is important to note that the elements h and k are uniquely determined by a If we assume a can also be represented as a = h1k1 for some h1 in H and k1 in K, then equating the two expressions leads to the conclusion that h must equal h1 and k must equal k1, thereby confirming their uniqueness We define a function f: G → H × K by mapping each element a to the pair (h, k), where a = hk However, if we have another representation a' = h'k' for a different pair, the product aa' does not conform to the expected form for evaluating f, prompting us to analyze the expression h'kh'⁻¹k⁻¹.

(h 0 kh 0−1 )k −1 ∈K, sinceKis normal Similarly,h 0 (kh 0−1 k −1 )∈H, sinceH is normal Therefore, h 0 kh 0−1 k −1 ∈H∩K ={e} andh 0 and k commute.

Therefore, f(aa 0 ) = f(hh 0 kk 0 ) = (hh 0 , kk 0 ) = (h, k)(h 0 , k 0 ) = f(a)f(a 0 ).

Now, it is easy to see thatf is an isomorphism

Theorem 1.7.4 IfAis a normal subgroup ofH andB is normal subgroup of K, then A×B is a normal subgroup of H×K and

Proof The homomorphism f : H ×K −→ (H/A)×(K/B), defined by f(h, k) = (Ah, Bk), is surjective and kerf = A×B The fundamental homomorphism theorem now gives the result

It follows, in particular, that ifNis normal subgroup ofH, thenN×{e} is a normal subgroup ofH×K.

Definition 1.7.6 Let G and H be two groups If a homomorphism ϕ from G into AutH is given, we say that G acts onH via ϕand G is an operator grouponH; the homomorphismϕis called anactionofG.

The action ϕof Gis not necessarily an isomorphism In particular, ϕ can be the trivial action, i.e.,ϕ(g) =ifor allg∈G Via the trivial action, any group can act onH.

For every element g in group G, the mapping ϕ(g) acts as an automorphism of subgroup H We represent the image of an element h in H under this mapping as h g The action on H is established if the function ϕ(g), which transforms h into h g, is defined for each g in G and adheres to specific formulas involving elements u, v in H and x, y in G.

(uv) x =u x v x , u xy = (u x ) y , u e =u, whereeis the identity ofG.

Theorem 1.7.7 Let ϕ be an action of a group G on another group H.

LetL be the cartesian product ofGandH,i.e., the totally of pairs(g, h)of elementsg∈Gandh∈H We define the product of two elements ofLby the formula

(g, h)(g 0 , h 0 ) = (gg 0 , ϕ(g 0 )(h)h 0 ). Then, Lforms a group with respect to this operation.

Let eG denotes the identity of G; similarly, let eH denotes the identity of H We define forg∈Gandh∈H, γ(g) = (g, eH), η(h) = (eG, h).

In the context of group theory, let γ be an isomorphism from group G onto itself, and η be an isomorphism from group H onto itself We establish that H is a normal subgroup of the combined group L = G H, with the intersection G ∩ H containing only the identity element e = (eG, eH) of L Additionally, for any elements g in G and h in H, the relationship γ(g)⁻¹ η(h) γ(g) = η(h g) is maintained.

Proof First, we verify the associative law Letg, u, x∈Gandh, v, y∈H.

By definition, e= (eG, eH) is the identity; the inverse of (g, h) is given by

The set L, defined as (g −1 , ϕ(g −1 )(h) −1), forms a group under the specified operation This operation demonstrates that γ(gg 0 ) equals γ(g)γ(g 0 ) and η(hh 0 ) equals η(h)η(h 0 ), indicating that both γ and η are homomorphisms Consequently, G = γ(G) and H = η(H) are established as subgroups.

Hence,H is a normal subgroup ofL=G H Clearly, we haveG∩H ={e}.

The group constructed in the above theorem is called the semidirect productofGandH with respect to the actionϕ.

Solvable and nilpotent groups

The goal of this section is to introduce the concept of solvable and nilpotent groups.

{e}=G0⊆G1⊆ .⊆Gn=G of a groupGis called anormal seriesofG, ifGi is a normal subgroup of

Definition 1.8.2 A groupGis said to besolvableif it has a normal series

{e}=G0⊆G1⊆ .⊆Gn=Gsuch that each of its factor groupGi+1/Gi is abelian, for everyi= 0, , n−1.

The above series is referred to as asolvable seriesofG.

(3) The dihedral groups are solvable.

(4) Any group whose order has the formp m , wherepis prime, is solvable.

Theorem 1.8.4 Any subgroupH of a solvable groupGis solvable.

Proof Suppose that {e} =G0 ⊆G1 ⊆ ⊆Gn =Gis a solvable series forG We show that

The series {e}=H∩G0⊆H∩G1⊆ ⊆H∩Gn =H represents a solvable series for the group H Given that Gi is normal in Gi+1 for i ranging from 0 to n−1, it follows that Ni = H ∩ Gi is also normal in Ni+1 = H ∩ Gi+1 We then establish a mapping f: Ni+1 −→ Gi+1/Gi defined by f(x) = xGi for all x in Ni+1.

The function f is established as a homomorphism For any element x in Ni+1, it can be shown that x is in the kernel of f if and only if x belongs to Gi, which further implies that x is in the intersection of H and Gi This leads to the conclusion that the kernel of f is equal to the intersection of H and Gi, represented as Ni According to the fundamental theorem of group homomorphism, it follows that Ni+1/Ni is isomorphic to f(Ni+1) Since f(Ni+1) is a subgroup of the quotient group Gi+1/Gi, which is abelian, it can be concluded that f(Ni+1) is also abelian.

Ni+1/Ni is abelian This proves thatH is a solvable group

Theorem 1.8.5 If N is a normal subgroup of a solvable group G, then

Proof Suppose that {e} =G0 ⊆G1 ⊆ ⊆Gn =Gis a solvable series forG We consider the following series

Let 0≤i≤n−1 be an arbitrary element Suppose thatx∈Gi+1N Then, x=gyfor someg∈Gi+1 andy∈N Hence, xGiN =gyGiN =gyN Gi =gN Gi

=GigN =GigyN =GiN gy=GiN x.

This proves thatGiN is a normal subgroup ofGi+1N Thus,

Now, we define f : Gi+1 −→ Gi+1N/GiN by f(x) = GiN x, for all x ∈

In the context of group theory, we establish that if \( f \) is a homomorphism from \( Gi+1 \) to \( Gi+1N \), then for any element \( y \in Gi+1N \), it can be expressed as \( y = zg \) where \( z \in N \) and \( g \in Gi+1 \) This leads to the conclusion that \( GiN y \) is equivalent to \( GiN g = f(g) \), demonstrating that \( f \) is onto Given that \( Gi \subseteq \text{ker} f \), we define a new homomorphism \( f: Gi+1/Gi \rightarrow Gi+1N/GiN \) where \( f(Gix) = GiN x \) for all \( x \in Gi+1 \), which is also onto Therefore, \( Gi+1N/GiN \) serves as a homomorphic image of the abelian group \( Gi+1/Gi \), confirming that it is abelian This leads to the conclusion that each factor group is abelian, ultimately proving that \( G/N \) is solvable.

Theorem 1.8.6 LetN be a normal subgroup of a groupG If bothN and

G/N are solvable, then Gis solvable.

Proof Since N and G/N are solvable, there exist the following solvable series forN andG/N, respectively,

Here each Gi is a subgroup of Gcontaining N Since Gi/N is normal in

Gi+1/N, eachGi is normal inGi+1 Moreover,G0=N Now,

{e}=N0⊆N1⊆ .⊆Nm=N =G0⊆G1⊆ .⊆Gn=G is a solvable series forG Therefore, we conclude thatGis solvable

Corollary 1.8.7 If two groups H and K are solvable, then H ×K is solvable.

Proof LetG=H×K Alternatively, we can considerH ∼= (H× {e}) and

Since K is solvable, we have G/H is solvable Hence, G is solvable by

Lemma 1.8.8 LetN be a normal subgroup ofGand G/N be abelian If x, y∈G, thenxyx −1 y −1 ∈N.

Proposition 1.8.9 Sn is not solvable for n≥5.

Proof Suppose thatSk is solvable for somek≥5 and

{e}=G0⊆G1⊆ .⊆Gm−1⊆Gm=Sk is a solvable series forSk By induction, we show thatGm−icontains all of the cycles of length 3, where 0≤i≤m Indeed, sinceG0={e}, we obtain a contradiction and the proof completes.

Let (x y z) be a cycle of length 3 in Sk Since k ≥5, there exist u, v distinct fromx, y, z Now, suppose thatσ= (z u y) andδ= (y x v) Since

Gm−1is a normal subgroup ofSk andSk/Gm−1is abelian, by Lemma 1.8.8, we obtainσδσ −1 δ −1 ∈Gm−1 Therefore,

Now, let Gm−j contains all of the cycles of length 3 Since (x y z)∈

Gm−j and Gm−j−1 is a normal subgroup of Gm−j, similar to the above discussion, we obtain (x y z)∈Gm−j−1

The commutator subgroup \( G' \) of a group \( G \) is generated by the set of elements of the form \( x^{-1}y^{-1}xy \), where \( x, y \in G \) Each element of \( G' \) can be expressed as a product of commutators, denoted as \( a_{i_1}^{\epsilon_1} a_{i_2}^{\epsilon_2} \ldots a_{i_k}^{\epsilon_k} \), with \( a_j \) representing the commutator \( x^{-1}y^{-1}xy \), \( \epsilon_j \) being either \( +1 \) or \( -1 \), and \( k \) being any positive integer This subgroup is also referred to as the derived subgroup.

Lemma 1.8.10 Let G be a group and G 0 be its commutator subgroup.

(2) For any normal subgroup N of G,G/N is an abelian group if and ifN contains G 0

In the group G, if we assume elements a and b belong to G, it follows that the inverse of the commutator (a −1 b −1 ab) is also a commutator, specifically b −1 a −1 ba This indicates that every element of G 0 can be expressed as a product of a finite number of commutators When considering an element a in G and an element x in G 0, we can represent x as a product of commutators, denoted as x = g1g2 gk, where each gi is defined as a commutator gi = a −1 i b −1 i aibi for some elements ai, bi in G Consequently, we can manipulate this representation to express a −1 xa as the product of transformed commutators, leading to the expression a −1 gia = a −1 a −1 i b −1 i aibia, further illustrating the relationship between elements in G and their commutators.

Hence, a −1 gia is again a commutator Therefore, a −1 xa is a product of commutators which implies thata −1 xa∈G 0

(2) Suppose thatG/N is abelian anda, bare two elements ofG Then,

(aN)(bN) = (bN)(aN) or abN N So, a −1 b −1 ab ∈N We conclude that N contains every commutatora −1 b −1 ab SinceG 0 is generated by all the commutators,G 0 ⊆N.

Conversely, suppose thatG 0 ⊆N anda, bare two elements ofG Then, a −1 b −1 ab ∈ G 0 gives a −1 b −1 ab ∈ N which implies that abN = baN or

(aN)(bN) = (bN)(aN) This shows thatG/N is abelian

Corollary 1.8.12 A groupGis abelian if and only ifG 0 ={e}.

Note that by using Corollary 1.8.12 and Theorem 1.6.11, one can give a short proof for Proposition 1.8.9.

Definition 1.8.13 LetGbe a group We define the sequence of subgroups

Theorem 1.8.14 A groupGis solvable if and only ifG (n) ={e}for some n≥1.

Proof Suppose that Gis solvable and{e}=G0⊆G1⊆ ⊆Gn =Gis a solvable series forG We prove inductively that

G (k) ⊆Gn−k (∗) for all 0 ≤ k ≤ n If k = 0, then G (0) = G = G0 Let G (k) ⊆ Gn−k for some k This implies that G (k+1) = (G (k) ) 0 ⊆ (Gn−k) 0 Since

Gn−k/Gn−k−1 is abelian, by Lemma 1.8.10, (Gn−k) 0 ⊆ Gn−k−1 Conse- quently, G (k+1) ⊆Gn−k−1 Thus, by induction, (∗) holds In particular,

Conversely, assume thatG (n) ={e}for somen Then

{e}=G (n) ⊆G (n−1) ⊆ .⊆G (1) ⊆G (0) =G is a normal series for G such that G (j) /G (j+1) = G (j) /(G (j) ) 0 is abelian.

We now introduce the class of nilpotent groups.

Definition 1.8.15 A group Gis said to benilpotentif it has a series of subgroups

{e}=G0⊆G1⊆ .⊆Gn=G such that for every 1≤i≤n,

(1) Gi is a normal subgroup ofG;

Such series is called acentral seriesofG For a nilpotent group, the smallest nsuch thatGhas a central series of lengthnis called thenilpotency class ofG; andGis said to benilpotent of classn.

LetH andK be subgroups ofG We define

Lemma 1.8.16 A series of G, say{e}=G0 ⊆G1⊆ .⊆Gn=Gis a central series if and only if for eachi= 1, , n,

Proof If the given series is a central series, then for each i = 1, , n,

Gi−1 is a normal subgroup of G and Gi/Gi−1 ≤ Z(G/Gi−1) Then, for anyx∈Gi andy∈G,

(xGi−1)(yGi−1) = (yGi−1)(xGi−1), that is xyGi−1 = yxGi−1 Hence, x −1 y −1 xy ∈ Gi−1 This implies that

Conversely, suppose that for each i = 1, , n, [Gi, G] ≤ Gi−1 Let x∈Gi and y ∈G Then,x −1 y −1 xy∈Gi−1 In particular, since Gi−1≤

Gi, if x ∈ Gi−1 then y −1 xy ∈ Gi−1 Thus, Gi−1 is a normal subgroup of G Moreover, xyGi−1 = yxGi−1 for every x∈ Gi and y ∈ G, and so

Gi/Gi−1≤Z(G/Gi−1) Thus, the series is a central series

(2) The quaternion groupQ8 is nilpotent.

Theorem 1.8.18 If G is nilpotent, then all subgroups and all quotient groups of Gare nilpotent.

Proof Suppose thatG is nilpotent and {e} =G0 ⊆G1 ⊆ .⊆Gn =G is a central series for G Let H be a subgroup of G and N be a normal subgroup ofG Then, we have

{e}=H∩G0⊆H∩G1⊆ .⊆H∩Gn=H, (∗) {N}=G0N/N ⊆G1N/N ⊆ .⊆GnN/N =G/N (∗∗) For eachi= 1, , n, [Gi, G]≤Gi−1 Hence,

Therefore by Lemma 1.8.15, (∗) and (∗∗) are central series, so thatH and

In the following lemma, we present a non nilpotent group.

Proof Assume that S3 is nilpotent Then, there exist a central series

G1 ≤ Z(S3) = {e} which implies that G1 = {e} Similarly, we obtain

By the above lemma, we see that Theorem 1.8.6 is not true for nilpotent groups, i.e., ifN is a normal subgroup of a groupGand bothN andG/N are nilpotent, thenGis not nilpotent, in general.

Lemma 1.8.20 If two groups H and K are nilpotent, then H ×K is nilpotent.

Proof IfH andKare both nilpotent, then there are central series

By inserting repetition of terms if necessary, we may assume without loss generality thatm=n Then, we have

[Hi×Ki, G] = ([Hi, H]×[Ki, K])≤(Hi−1×Ki−1).

Now, by Lemma 1.8.16, we conclude thatGis nilpotent

Definition 1.8.21 We define subgroups Γn(G) and Zn(G) ofG respectively as follows Let Γ1(G) =GandZ0(G) ={e} Then, for each integer n >1, Γn(G) = [Γn−1(G), G] and for each integern >0,

G= Γ1(G)≥Γ2(G)≥Γ3(G)≥ , {e}=Z0(G)≤Z1(G)≤Z2(G)≤ .ã The first sequence is called the lower central series of G and the second sequence is called theupper central seriesofG.

It is not difficult to see that the terms of the lower and upper central series are normal subgroups ofG.

Theorem 1.8.22 The following three statements are equivalent:

Proof If Γn(G) ={e} for somen, then

G= Γ1(G)≥Γ2(G)≥ .≥Γn(G) ={e} is a central series ofG, and soGis nilpotent.

Similarly, if Zn(G) =Gfor somen, then

{e}=Z0(G)≤Z1(G)≤ .≤Zn(G) =G is a central series ofG, and soGis nilpotent.

Conversely, assume thatGis nilpotent and

In a central series of a group \( G \), denoted as \( G_0 \subseteq G_1 \subseteq \ldots \subseteq G_n = G \), we demonstrate through induction that \( G_i \) is a subgroup of the center \( Z_i(G) \) for all indices \( 0 \leq i \leq n \) This assertion holds trivially for \( i = 0 \) Assuming it is true for \( i - 1 \), we find that \( G_i/G_{i-1} \) is contained within the center \( Z(G/G_{i-1}) \), confirming the inductive step Thus, we establish that each \( G_i \) is indeed a subgroup of the corresponding center, reinforcing the structure of central series in group theory.

GiZi−1(G)/Zi−1(G)≤Z(G/Zi−1(G)) =Zi(G)/Zi−1(G).

Hence,Gi ≤Zi(G) Thus, the induction argument goes through In particular, sinceG=Gn≤Zn(G), we obtainZn(G) =G.

Now, we prove by induction on j that Γj+1(G)≤Gn−j, for all 0 ≤ j ≤ n This is trivial for j = 0 Assume that j > 0 and, inductively, that Γj(G)≤Gn−j+1 Then, by Lemma 1.8.16, [Gn−j+1, G]≤

Again, the induction argument goes through In particular, since Γn+1(G)≤G0={e}, we obtain Γn+1(G) ={e}

In a nilpotent group G, any central series {e}=G0⊆G1⊆ ⊆Gn=G satisfies the relationship Γn−i+1(G)≤Gi≤Zi(G) for all 0≤i≤n Furthermore, the smallest integer r for which Γr+1(G) = {e} is the same as the smallest integer r for which Zr(G) = G.

Proof Suppose thatr is the least integer such thatZr(G) =G We show thatris also the least integer such that Γr+1(G) ={e} Let Γk+1(G) ={e} for somek < r We setGi= Γk−i+1(G) Then,

Hence, {e} = G0 ⊆ G1 ⊆ ⊆ Gk = G is a central series Now, by

Theorem 1.8.22, it follows thatZk(G) =G; this is contrary to the definition ofr

In a nilpotent group G, the lower central series represents the fastest descending central series, while the upper central series serves as the fastest ascending central series.

Lemma 1.8.24 For each non-negative integer i,G (i) ≤Γi+1(G).

Proof We prove by induction on i If i = 0, then G (0) = G = Γ1(G).

Suppose thati >0 and, inductively, thatG (i−1) ≤Γi(G) Then,

Theorem 1.8.25 Each nilpotent group is solvable.

Proof Suppose that G is a nilpotent group By Corollary 1.8.23, there exist a non-negative integerrsuch that Γr+1={e} By Lemma 1.8.24, we getG (i) ≤Γi+1(G) ={e} Now, by Theorem 1.8.14, we conclude that Gis solvable

Lemma 1.8.19 shows that the converse of Theorem 1.8.25 is not true in general.

Hypergroups 33

Introduction and historical development of hypergroups 33

The hypergroup notion was introduced in 1934 by F Marty [103], at the

The 8th Congress of Scandinavian Mathematicians featured discussions on hypergroups, which serve as a suitable generalization of traditional groups In a standard group, the composition of two elements results in a single element; however, in a hypergroup, this composition yields a set of elements The published notes explore the application of hypergroups in various contexts, including algebraic functions, rational fractions, and non-commutative groups.

The motivating example was the following: Let Gbe a group andK be a subgroup ofG Then,G/K={xK |x∈G}becomes a hypergroup, where the composition is defined in a usual manner.

Prenowitz explored various geometries, including projective, descriptive, and spherical, as hypergroups Collaborating with Jantosciak, they established geometries based on join spaces, a specific type of hypergroup In recent decades, these join spaces have proven to be valuable tools in the analysis of graphs, hypergraphs, and binary relations.

Hypergroup theory encompasses a variety of extensively studied types, including regular hypergroups, reversible regular hypergroups, canonical hypergroups, cogroups, and cyclic hypergroups The complexities and diversities encountered in hypergroup theory often surpass those found in traditional group theory Notably, hypergroups feature multiple types of homomorphisms and several classifications of subhypergroups, including closed, invertible, ultraclosed, and conjugable subhypergroups.

Around the 1940’s, the general aspects of the theory, the connections with groups and various applications in geometry were studied in France by

F Marty, M Krasner, M Kuntzmann, R Croisot, in U.S.A by M Dresher,

O Ore, W Prenowitz, H.S Wall, J.E Eaton, H Campaigne, L Griffiths, in Russia by A Dietzman, A Vikhrov, in Italy by G Zappa, in Japan by

In the past two decades, significant advancements in the study of hyperstructures have emerged, with notable contributions from various researchers A Orsatti in Italy explored semiregular hypergroups, while K Drbohlav in Czechoslovakia focused on hypergroups of two-sided classes Additionally, M Benado in Romania investigated hyperlattices, further enriching the field.

Starting in the 1970s, significant advancements were made in the theory as its research scope expanded In France, researchers M Krasner, M Koskas, and Y Sureau explored the theory of subhypergroups and the relationships established within hyperstructures Meanwhile, in Greece, J Mittas and his students, including M Konstantinidou, contributed to this evolving field of study.

K Serafimidis, S Ioulidis and C.N Yatras studied the canonical hypergroups, the hyperrings, the hyperlattices, Ch Massouros obtained important results about hyperfields and other hyperstructures G Massouros, together with J Mittas studied applications of hyperstructures to Automata.

D Stratigopoulos continued some of Krasner ideas, studying in depth noncommutative hyperrings and hypermodules T Vougiouklis, L Konguetsof and later S Spartalis, A Dramalidis analyzed especially the cyclic hypergroups, theP-hyperstructures.

The study of regular hypergroups, complete hypergroups, and hypergroup homomorphisms has seen significant contributions, particularly in their applications to Combinatorics and Geometry.

Italian mathematician P Corsini and his group of research, among whom we mention M.de Salvo, R Migliorato, F de Maria, G Romeo, P Bo- nansinga.

Also around 1970’s, some connections between hyperstructures and ordered systems, particularly lattices, were established by T Nakano and

J.C Varlet Around the 1980’s and 90’s, associativity semyhypergroups were analyzed in the context of semigroup theory by T Kepka and then by

J Jezec, P Nemec and K Drbohlav, and in Finland by M Niemenmaa.

In the United States, R Roth applied canonical hypergroups to address challenges in character theory related to finite groups, while S Comer explored the interplay between hypergroups, combinatorics, and relation theory Additionally, J Jantosciak advanced the study of join spaces, originally introduced by W Prenowitz, by generalizing them for noncommutative contexts and examining the connections between homomorphisms and their corresponding relations.

In America, hyperstructures have been studied both in U.S.A (at Char- leston, South Carolina – The Citadel, New York-Brooklyn College, CUNY,

Cleveland, Ohio – John Carroll University) and in Canada (at Universit´e de Montr´eal).

A big role in spreading this theory is played by the Congresses on Al- gebraic Hyperstructures and their Applications.

P Corsini played a pivotal role in the development of Hyperstructure Theory, organizing the first three Congresses in Italy His significant contributions include delivering numerous lectures on hyperstructures and their applications across various countries, including multiple presentations in Romania.

Thailand, Iran, China, Montenegro, making known this theory After his visits in these countries, hyperstructures have had a substantial development.

Coming back to the Congresses on Algebraic Hyperstructures, the first two were organized in Taormina, Sicily, in 1978 and 1983, with the names:

“Sistemi Binari e loro Applicazioni” and “Ipergruppi, Strutture Multivoche e Algebrizzazione di Strutture d’Incidenza” The third Congress, called

“Ipergruppi, altre Strutture Multivoche e loro Applicazioni” was organized in Udine in 1985.

The fourth congress, organized by T Vougiouklis in Xanthi in 1990, used already the name of Algebraic Congress on Hyperstructures and their

The AHA Congress, held every three years since 1990, has been a significant event for Romanian mathematicians The fifth AHA Congress marked a pivotal moment in the ongoing exploration of Hyperstructure Theory, which has remained a central focus in the field since the 1990s.

Since 1993, the University "Al.I.Cuza" of Iasi, under the guidance of M Stefanescu, has been a hub for research in modern algebra, particularly in the study of hyperstructures This area has garnered significant interest among Romanian researchers, leading to numerous publications in both national and international journals, presentations at various conferences and congresses, and the development of Ph.D theses focused on this field.

The sixth AHA Congress was organized in 1996 at the Agriculture Uni- versity of Prague by T Kepka and P Nemec, the seventh was organized in

1999 by R Migliorato in Taormina, Sicily, then the eighth was organized in

2002 by T Vougiouklis in Samothraki, Greece All these congresses were organized in Europe Nowadays, one works successfully on Hyperstructures in the following countries of Europe:

• in Greece, at Thessaloniki (Aristotle University), at Alexan- dropoulis (Democritus University of Thrace), at Patras (Patras University), Orestiada (Democritus University of Thrace), at Athens;

• in Italy at Udine University, at Messina University, at Rome (Uni- versita’ “La Sapienza”), at Pescara (D’Annunzio University), at Teramo (Universita’ di Teramo), Palermo University;

• in Romania, at Iasi (“Al.I Cuza” University), Cluj (“Babes- Bolyai” University), Constanta (“Ovidius” University);

• in Czech Republic, at Praha (Charles University, Agriculture University), at Brno (Brno University of Technology, Military Academy of Brno, Masaryk University), Olomouc (Palacky Uni- versity);

• in Montenegro, at Podgorica University.

Let us continue with the following AHA Congresses.

The ninth congress on hyperstructures, organized in 2005 by R Ameri in Babolsar, Iran, was the first of this kind in Asia In the past millenniums,

Iran gave fundamental contributions to Mathematics and in particular, to

Algebra has been significantly advanced by notable figures such as Khwarizmi, Kashi, Khayyam, and Zadeh, who have recognized the critical role of hyperstructures Their contributions highlight the theoretical importance of these mathematical concepts and their practical applications across diverse scientific fields.

Nowadays, hyperstructures are cultivated in many universities and research centers in Iran, among which we mention Yazd University, Shahid

Bahonar University of Kerman, Mazandaran University, Kashan Univer- sity, Ferdowsi University of Mashhad, Tehran University, Tarbiat Modarres

University, Zahedan (Sistan and Baluchestan University), Semnan Univer- sity, Islasmic Azad University of Kerman, Shahid Beheshti University of

Tehran, Center for Theoretical Physics and Mathematics of Tehran, Zan- jan (Institute for Advanced Studies in Basic Sciences).

Another Asian country where hyperstructures have had success is Thai- land In Chulalornkorn University of Bangkok, important results have been obtained by Y Kemprasit and her students Y Punkla, S Chaoprakhoi,

N Triphop, C Namnak on the connections among hyperstructures, semigroups and rings.

Asia hosts several prominent research centers focused on hyperstructures, including the University of Calcutta and Aditanar College of Arts and Sciences in Tiruchendur, Tamil Nadu, India, as well as Chiungju National University in Korea.

Chiungju National University of Education, Gyeongsang National Univer- sity, Jinju), Japan (Hitotsubashi University of Tokyo), Sultanate of Oman

(Education College for Teachers), China (Northwest University of Xian,

Hyperstructures have been developed in various countries, including Germany, the Netherlands, Belgium, Macedonia, Serbia, Slovakia, Spain, Uzbekistan, and Australia The tenth AHA Congress took place in Brno, Czech Republic, in autumn 2008, organized by Šárka Hošková at the Military Academy of Brno Subsequently, the eleventh AHA Congress was held in Pescara, Italy, in autumn 2011, organized by A Maturo at the Università degli Studi.

Over 800 papers and several books have been published on hyperstructures, with many focusing on their applications in various fields This article highlights key areas related to hyperstructures and names notable mathematicians who have contributed to each topic.

• Geometry(W Prenowitz, J Jantosciak, and later G Tallini),

• Cryptography and Probability(L Berardi, F Eugeni, S Innamorati,

• Median Algebras, Relation Algebras, C-algebras (S Comer),

• Categories (M Scafati, M.M Zahedi, C Pelea, R Bayon, N.

Ligeros, S.N Hosseini, B Davvaz, M.R Khosharadi-Zadeh),

• Topology(J Mittas, M Konstantinidou, M.M Zahedi, R Ameri,

• Binary Relations (J Chvalina, I.G Rosenberg, P Corsini, V Leo- reanu, B Davvaz, S Spartalis, I Chajda, S Hoˇskov´a, I Cristea,

• Graphs and Hypergraphs(P Corsini, I.G Rosenberg, V Leoreanu,

• Lattices and Hyperlattices( J.C Varlet, T Nakano, J Mittas, A.

Rosenberg, B Davvaz, G Calugareanu, G Radu, A.R Ashrafi),

• Fuzzy Sets and Rough Sets (P Corsini, M.M Zahedi, B Davvaz,

R Ameri, R.A Borzooei, V Leoreanu, I Cristea, A Kehagias,

A Hasankhani, I Tofan, C Volf, G.A Moghani, H Hedayati),

• Intuitionistic Fuzzy Hyperalgebras (B Davvaz, R.A Borzooei, Y.B Jun, W.A Dudek, L Torkzadeh),

• Generalized Dynamical Systems(M.R Molaei) and so on.

Another topic which has aroused the interest of several mathematicians, is that one of Hv-structures, introduced by T Vougiouklis and studied then also by B Davvaz, M.R Darafsheh, M Ghadiri, R Migliorato, S.

Spartalis, A Dramalidis, A Iranmanesh, M.N Iradmusa, A Madanshekaf.

Hv-structures are a special kind of hyperstructures, for which the weak associativity holds Recently,n-ary hyperstructures, introduced by B Davvaz and T Vougiouklis, represent an intensively studied field of research.

Therefore, there are good reasons to hope that Hyperstructure Theory will be one of the more successful fields of research in algebra.

Definition and examples of hypergroups

A hypergroupoid is defined as a pair (H, ◦), where H is a non-empty set and ◦: H × H → P*(H) represents a hyperoperation, with P*(H) denoting the collection of non-empty subsets of H For any two non-empty subsets A and B of H, along with an element x in H, specific operations can be performed to explore the relationships within the hypergroupoid structure.

Definition 2.2.1 A hypergroupoid (H,◦) is called asemihypergroupif for alla, b, cofH we have (a◦b)◦c=a◦(b◦c), which means that

A hypergroupoid (H,◦) is called aquasihypergroupif for allaofH we have a◦H =H◦a=H This condition is also called thereproduction axiom.

Definition 2.2.2 A hypergroupoid (H,◦) which is both a semihypergroup and a quasihypergroup is called ahypergroup.

A hypergroup where the hyperproduct of any two elements results in exactly one element qualifies as a group Specifically, if (H,◦) is a hypergroup such that for all elements x and y in H, the cardinality |x◦y| equals 1, then (H,◦) functions as a semigroup Consequently, for any elements a and b in H, there exist elements x and y such that a equals b◦x and a equals y◦b, leading to the conclusion that (H,◦) is indeed a group.

Now, we look at some examples of hypergroups.

(1) IfH is a non-empty set and for allx, yofH, we definex◦y=H, then (H,◦) is a hypergroup, called thetotal hypergroup.

(2) Let (S,ã) be a semigroup andPbe a non-empty subset ofS For all x, yofS, we definex◦y=xP y Then, (S,◦) is a semihypergroup If(S,ã) is a group, then (S,◦) is a hypergroup, called aP-hypergroup.

(3) IfGis a group and for allx, yofG,< x, y >denotes the subgroup generated byxand y, then we define x◦y =< x, y > We obtain that (G,◦) is a hypergroup.

(4) If (G,ã) is a group,H is a normal subgroup ofGand for allx, yof

G, we definex◦y=xyH, then (G,◦) is a hypergroup.

(5) Let (G,ã) be a group and letH be a non-normal subgroup of it If we denoteG/H ={xH |x∈G}, then (G/H,◦) is a hypergroup, where for allxH, yHofG/H, we havexH◦yH ={zH|z∈xHy}.

(6) If (G,+) is an abelian group, ρ is an equivalence relation in G, which has classes x={x,−x}, then for allx, y of G/ρ, we define x◦y ={x+y, x−y} We obtain that (G/ρ,◦) is a hypergroup.

In an integral domain D with its field of fractions F, we define a hyperoperation on the quotient F/U, where U represents the group of invertible elements of D For any elements x and y in F/U, the operation is given by x◦y = {z | ∃(u, v) ∈ U² such that z = ux + vy} This definition establishes that (F/U,◦) forms a hypergroup, showcasing the algebraic structure derived from the interactions of elements within the field and its invertible components.

(8) Let (L,∧,∨) be a lattice with a minimum element 0 If for alla∈L,

F(a) denotes the principal filter generated froma, then we obtain a hypergroup (L,◦), where for alla, bofL, we havea◦b=F(a∧b).

(9) Let (L,∧,∨) be a modular lattice If for all x, y of L, we define x◦y={z∈L|z∨x=x∨y=y∨z}, then (L,◦) is a hypergroup.

(10) Let (L,∧,∨) be a distributive lattice If for allx, yofL, we define x◦y ={z∈L|x∧y≤z≤x∨y}, then (L,◦) is a hypergroup.

(11) Let H be a non-empty set and à : H −→ [0,1] be a function If for all x, yof H we define x◦y ={z ∈L |à(x)∧à(y)≤à(z)≤ à(x)∨à(y)}, then (H,◦) is a hypergroup.

Let H be a non-empty set with an equivalence relation R defined on it For every element x in H, the equivalence class R(x) contains a minimum of three elements Additionally, for any subset A of H, the notation R(A) represents the corresponding set S.

R(x) The couple (R(A), R(A)) is called a rough set If for all x, y ofH, we definex◦y=R({x, y})\R({x, y}), then (H,◦) is a hypergroup.

(13) Let H be a non-empty set and à, λ be two functions from H to

[0,1] For allx, yofH we define x◦y={u∈H |à(x)∧λ(x)∧à(y)∧λ(y)≤à(u)∧λ(u) andà(u)∨λ(u)≤à(x)∨λ(x)∨à(y)∨λ(y)}.

Then, the hyperstructure (H,◦) is a commutative hypergroup.

(14) Define the following hyperoperation on the real setR: for allx∈R, x◦x=xand for all different real elementsx, y,x◦y is the open interval betweenxandy Then, (R,◦) is a hypergroup.

Some kinds of subhypergroups

A non-empty subset K of a semihypergroup (H,◦) is called a subsemihy- pergroupif it is a semihypergroup In other words, a non-empty subsetK of a semihypergroup (H,◦) is a subsemihypergroup ifK◦K⊆K.

Definition 2.3.1 A non-empty subsetK of a hypergroup (H,◦) is called asubhypergroupif it is a hypergroup.

Hence, a non-empty subsetKof a hypergroup (H,◦) is a subhypergroup if for allaofK we havea◦K=K◦a=K.

There are several kinds of subhypergroups In what follows, we introduce closed, invertible, ultraclosed and conjugable subhypergroups and some connections among them.

Among the mathematicians who studied this topic, we mention F.

Marty, M Dresher, O Ore, and M Krasner conducted an analysis of closed and invertible subhypergroups, while M Koskas explored the concept of complete parts within subhypergroups Additionally, Y Sureau investigated ultraclosed, invertible, and conjugable subhypergroups Corsini achieved significant findings regarding ultraclosed and complete parts, and Leoreanu contributed further insights into the study of subhypergroups.

Let us present now the definition of these types of subhypergroups Let

(H,◦) be a hypergroup and (K,◦) be a subhypergroup of it.

Definition 2.3.2 We say thatK is:

• closed on the left (on the right) if for allk1, k2 of K and xof H, fromk1∈x◦k2 (k1∈k2◦x, respectively), it follows thatx∈K;

• invertible on the left(on the right) if for allx, yofH, fromx∈K◦y (x∈y◦K), it follows that y∈K◦x(y∈x◦K, respectively);

• ultraclosed on the left (on the right) if for all x of H, we have

• conjugable on the rightif it is closed on the right and for allx∈H,there existsx 0 ∈H such thatx 0 ◦x⊆K Similarly, we can define the notion ofconjugable on the left.

We say that K is closed (invertible, ultraclosed, conjugable) if it is closed (invertible, ultraclosed, conjugable, respectively) on the left and on the right.

(1) Let (A,◦) be a hypergroup, H =A∪T, where T is a set with at least three elements andA∩T =∅ We define the hyperoperation

⊗onH, as follows: if (x, y)∈A 2 , thenx⊗y=x◦y; if (x, t)∈A×T, thenx⊗t=t⊗x=t; if (t1, t2)∈T×T, then t1⊗t2=t2⊗t1=A∪(T\ {t1, t2}).

Then, (H,⊗) is a hypergroup and (A,⊗) is an ultraclosed, non- conjugable subhypergroup ofH.

In a total hypergroup (A,◦) with at least two elements, we introduce a set T = {ti} i∈N, ensuring A and T are disjoint and that all ti are distinct We define a hyperoperation ⊗ on H = A∪T with the following rules: for any elements x and y in A, the operation x⊗y results in A; when x is in A and t is in T, the operation x⊗t and t⊗x yields the set (A\{x})∪T; and for any distinct elements ti and tj in T, the operation ti⊗tj and tj⊗ti produces the set A∪{ti+j}.

Then, (H,⊗) is a hypergroup and (A,⊗) is a non-closed subhypergroup ofH.

In the context of the group (Z,+) and its subgroups Si = 2^i Z, where i is a non-negative integer, each non-zero integer x has a unique integer n(x) such that x belongs to Sn(x) but not to Sn(x)+1 We define a commutative hyperoperation on Z\{0} as follows: if n(x) is less than n(y), then x◦y equals x plus Sn(y); if n(x) equals n(y), then x◦y equals Sn(x) excluding zero; and if n(x) is greater than n(y), then x◦y equals y plus Sn(x) It is important to note that when n(x) is less than n(y), the relationship n(x+y) equals n(x) holds true Consequently, (Z\{0},◦) forms a hypergroup, and for every i in the natural numbers, (Si\{0},◦) constitutes an invertible subhypergroup of Z\{0}.

Other examples can be found in [22].

Lemma 2.3.4 A subhypergroup K is invertible on the right if and only if

Proof IfKis invertible on the right andz∈x◦K∩y◦K, thenx, y∈z◦K, whencex◦K⊆z◦Kandy◦K⊆z◦K It follows thatx◦K=z◦K=y◦K.

Conversely, if{x◦K}x∈H is a partition ofH andx∈y◦K, thenx◦K⊆ y◦K, whencex◦K=y◦K and so we havex∈y◦K=x◦K Hence, for allxofH we havex∈x◦K From here, we obtain thaty∈y◦K=x◦K.

Similar to Lemma 2.3.4, we can give a necessary and sufficient condition for invertible subhypergroups on the left.

The following theorems present some connections among the above types of subhypergroups.

IfAandBare subsets ofHsuch that we haveH=A∪BandA∩B =∅, then we denoteH =A⊕B.

Theorem 2.3.5 If a subhypergroupKof a hypergroup(H,◦)is ultraclosed, then it is closed and invertible.

Proof First we check thatKis closed Forx∈K, we haveK∩x◦(H\K)=∅ and fromH=x◦K∪x◦(H\K), we obtainx◦(H\K) =H\K, which means that K◦(H \K) =H \K Similarly, we obtain (H\K)◦K =H \K, hence K is closed Now,we show that {x◦K}x∈H is a partition of H.

Let \( y \in x^\circ K \cap z^\circ K \), which implies that \( y^\circ K \subseteq x^\circ K \) and \( y^\circ (H \setminus K) \subseteq x^\circ K^\circ (H \setminus K) = x^\circ (H \setminus K) \) From the equation \( H = x^\circ K \oplus x^\circ (H \setminus K) = y^\circ K \oplus y^\circ (H \setminus K) \), we conclude that \( x^\circ K = y^\circ K \) and similarly \( z^\circ K = y^\circ K \) Therefore, the set \( \{ x^\circ K \}_{x \in H} \) forms a partition of \( H \), indicating that \( K \) is invertible on the right A parallel argument demonstrates that \( K \) is also invertible on the left.

Theorem 2.3.6 If a subhypergroupKof a hypergroup(H,◦)is invertible, then it is closed.

We denote the set {e∈H | ∃x∈H, such thatx∈x◦e∪e◦x} byIp and we call it theset of partial identitiesofH.

Theorem 2.3.7 A subhypergroup K of a hypergroup(H,◦)is ultraclosed if and only if K is closed andIp⊆K.

To prove that K is left-invertible, we start by noting that K is closed and that Ip is a subset of K Assume there exist elements x and y in H such that x belongs to the composition of K and y, while y does not belong to the composition of K and x This implies that y is in the composition of the complement of K and x, leading to the conclusion that x is in the composition of K and the complement of K, which is a subset of the composition of the complement of K and x Since K is closed, this results in the intersection of Ip and the complement of K being non-empty, which contradicts our initial assumption.

Hence,K is invertible on the left Now, we check thatKis ultraclosed on the left Suppose that there areaandxinHsuch thata∈K◦x∩(H\K)◦x.

Since K is invertible on the left, it follows that x belongs to K◦a Given that K is closed, we find that a is included in (H\K)◦x, which is a subset of (H\K)◦K◦a and ultimately within (H\K)◦a This implies that the intersection of Ip and (H\K) is not empty, leading to a contradiction Consequently, we conclude that K is ultraclosed on both the left and right sides.

Conversely, suppose thatKis ultraclosed According to Theorem 2.3.5,

Kis closed Now, suppose thatIp∩(H\K)6=∅, which means that there is e∈H\K and there isx∈H, such thatx∈e◦x, for instance We obtain x∈(H\K)◦x, whenceK◦x⊆(H\K)◦x, which contradicts thatK is ultraclosed Hence,Ip⊆K

Theorem 2.3.8 If a subypergroupKof a hypergroup(H,◦)is conjugable, then it is ultraclosed.

Proof Letx∈H DenoteB=x◦K∩x◦(H\K) SinceK is conjugable it follows that K is closed and there existsx 0 ∈H, such thatx 0 ◦x⊆K.

Hence,B =∅, which means that K is ultraclosed on the right Similarly, we check thatKis ultraclosed on the left.

Homomorphisms of hypergroups

Homomorphisms of hypergroups are studied by Dresher, Ore, Krasner,

Kuntzmann, Koskas, Jantosciak, Corsini, Freni, Davvaz and many others In this section, we study several kinds of homomorphisms The main references are [22; 86].

Definition 2.4.1 Let (H1,◦) and (H2,∗) be two hypergroups A map f :H1−→H2, is called

(1) a homomorphismor inclusion homomorphismif for allx, yof H1, we havef(x◦y)⊆f(x)∗f(y);

(2) a good homomorphism if for all x, y of H1, we have f(x◦y) f(x)∗f(y);

(3) anisomorphismif it is a one to one and onto good homomorphism.

Iff is an isomorphism, thenH1andH2are said to beisomorphic,which is denoted byH1∼=H2.

Example 2.4.2 LetH1={a, b, c}andH2={0,1,2}be two hypergroups with the following hyperoperations:

2H2 1 {1,2} and let f : H1 −→ H2 is defined by f(a) = 0, f(b) = 1 and f(c) = 2.

Clearly,f is an inclusion homomorphism.

Proposition 2.4.3 Let (H1,◦) and (H2,∗) be two hypergroups and f :

H1 −→ H2 be a good homomorphism Then, Imf is a subhypergroup of

Proof For everya, b∈H1, we havef(a)∗f(b) =f(a◦b)⊆Imf Moreover, there existx, y ∈ H1 such that a∈ x◦b and a∈ b◦y and consequently f(a)∈f(x)∗f(b) andf(a)∈f(b)∗f(y)

We employ for simplicity of notation xf =f −1 (f(x)) and for a subset

Notice that the defining condition for an inclusion homomorphism is equivalent to x◦y⊆f −1 (f(x)∗f(y)).

It is also clear for an inclusion homomorphism that

The defining condition for an inclusion homomorphism is also valid for sets.

That is, ifA, B are non-empty subsets of H1, then it follows that f(A◦B)⊆f(A)∗f(B).

Applying the above relation forA=xf andB=yf, we obtain xf◦yf ⊆f −1 (f(x)∗f(y)) and

In the literature, various types of homomorphisms are defined and analyzed, each characterized by specific properties that serve as conditions on the inverse image of the product of the function values, f −1 (f(x)∗f(y)) This article focuses on four distinct types of homomorphisms, providing a comprehensive overview of their definitions and properties.

Definition 2.4.4 Let (H1,◦) and (H2,∗) be two hypergroups and f : H1 −→ H2 be a mapping Then, given x, y ∈ H1, f is called a homomorphism of type 1, iff −1 (f(x)∗f(y)) = (xf◦yf) f ; type 2, iff −1 (f(x)∗f(y)) = (x◦y)f; type 3, iff −1 (f(x)∗f(y)) =xf◦yf; type 4, iff −1 (f(x)∗f(y)) = (x◦y)f =xf◦yf.

Note thatx◦y⊆(x◦y)f,x◦y⊆xf◦yfand that (x◦y)f ⊆(xf◦yf) f ,xf◦ yf ⊆(xf◦yf) f Hence, a homomorphism of any type 1 through 4 is indeed an inclusion homomorphism Observe that a one to one homomorphism of

H1 ontoH2 of any type 1 through 4 is an isomorphism.

Proposition 2.4.5 Let (H1,◦) and(H2,∗)be two hypergroups, A, B are non-empty subsets of H1 and f : H1 −→H2 be a mapping Then, f is a homomorphism of

Proof Each part is established by a straightforward set theoretic argument.

Proposition 2.4.6 Let (H1,◦) and (H2,∗) be two hypergroups and f :

H1−→H2 be a mapping Then,f is a homomorphism of

(1) type 4 if and only if f is a homomorphism of type 2 and type 3;

(2) type 1 if f is a homomorphism of type 2 or type 3.

(2) Suppose that x, y∈H1 andf is a homomorphism of type 2 Then,

(x◦y)f ⊆(xf◦yf)f⊆f −1 (f(x)∗f(y)) = (x◦y)f. Similarly, iff is a homomorphism of type 3, then xf◦yf ⊆(xf◦yf)f⊆f −1 (f(x)∗f(y)) =xf◦yf. Hence, in either case,f is a homomorphism of type 1 Thus, (2) holds

The defining condition for a homomorphism of type 1 or type 2 can easily be simplified if the homomorphism is onto.

Proposition 2.4.7 Let (H1,◦) and (H2,∗) be two hypergroups and f : H1 −→ H2 be an onto mapping Then, given x, y ∈ H1, f is a homomorphism of

(1) type 1 if and only if f(xf◦yf) =f(x)∗f(y);

(2) type 2 if and only if f(x◦y) =f(x)∗f(y).

Corollary 2.4.8 Let(H1,◦)and(H2,∗)be two hypergroups,A, B be non- empty subsets ofH1 andf :H1−→H2 be an onto mapping Then,f is a homomorphism of

Example 2.4.9 Consider H1 ={0,1,2} and H2 ={a, b} together with the following hyperoperations:

∗ a b a a {a, b} b {a, b} b and suppose thatf :H1−→H2is defined byf(0) =f(1) ndf(2) =b.

Then,f is a good homomorphism of type 4.

In the context of a hypergroup H, we examine equivalence relations that allow the family of equivalence classes to form a hypergroup through the hyperoperation derived from H For any equivalence relation ρ on H, we denote the equivalence class of an element x ∈ H as xρ or ρ(x) Additionally, for a non-empty subset A of H, the equivalence class can be represented as Aρ = ∪{xρ | x ∈ A}.

In the context of hyperoperations, we denote the family of classes of ρ as We let H/ρ (read H modulo ρ) = {xρ | x ∈ H} The hyperoperation on H gives rise to a new operation ⊗ on H/ρ, defined by xρ ⊗ yρ = {zρ | z ∈ xρ ◦ yρ}, where x and y belong to H This structure, (H/ρ, ⊗), is referred to as a factor or quotient structure Additionally, in the definition of ⊗, the condition z ∈ xρ ◦ yρ can be substituted with z ∈ (xρ ◦ yρ)ρ or zρ ⊆ (xρ ◦ yρ)ρ It is clear that the union of (xρ ⊗ yρ) is contained within (xρ ◦ yρ)ρ.

Proposition 2.4.10 Let (H,◦) be a hypergroup Then, (H/ρ,⊗) is a hypergroup if and only if for allx, y, z∈H,

Similarly, xρ⊗(yρ⊗zρ) = {tρ | t ∈ xρ◦ (yρ◦ zρ)ρ} Therefore, ⊗ is associative Reproducibility in (H/ρ,⊗) is a consequence of reproducibility in H Suppose thatxρ, yρ∈H/ρ Letu, v∈H such thaty∈x◦u,v◦x.

Then, obviousely,yρ∈xρ⊗uρ,vρ⊗xρ Hence, the proposition holds

Definition 2.4.11 Let ρ be an equivalence relation on a hypergroup

(H,◦) Then, givenx, y∈H,ρis said to be of type 1, ifH/ρis a hypergroup; type 2, ifxρ◦yρ⊆(x◦y)ρ; type 3, if (x◦y)ρ⊆xρ◦yρ; type 4, ifxρ◦yρ= (x◦y)ρ.

Observe that being of type 2 is equivalent to (xρ◦yρ)ρ = (x◦y)ρ, and of type 3 to (xρ◦yρ)ρ =xρ◦yρ Note that for an equivalence of type 2, xρ⊗yρ={zρ |z∈x◦y}.

Proposition 2.4.12 Let ρ be an equivalence relation on a hypergroup

(1) type 4 if and only if ρis of type 2 and type 3;

(2) type 1 if ρis of type 2 or type 3.

(2) Letx, y, z∈H Suppose thatρis of type 2 Then, we obtain

Suppose thatρis of type 3 Then, we have

Hence, in either case, Proposition 2.4.10 applies and yields that H/ρ is a hypergroup Therefore,ρis of type 1 and (2) holds

Homomorphisms between hypergroups and equivalence relations on hypergroups are closely related The following results are fundamental.

Theorem 2.4.13 Let (H1,◦) and (H2,∗) be two hypergroups and f :

In this article, we discuss the concept of onto mapping in relation to the equivalence relation denoted by f on the set H1 The equivalence classes formed by this relation consist of the family {xf | x∈H1} For n values of 1, 2, 3, or 4, f is classified as a non-type equivalence relation on H1.

H1/f is canonically isomorphic toH2 if and only iff is a homomorphism of typen.

Proof Letn= 1 Suppose thatf is a homomorphism of type 1 In order to show f is an equivalence relation of type 1 onH1, Proposition 2.4.10 is employed Letx, y, z∈H1 By Corollary 2.4.8 (1),

((xf◦yf)f◦zf) f =f −1 (f((xf◦yf)f ◦zf))

Similarly, we obtain (xf◦(yf◦zf)f)f=f −1 (f(x)∗f(y)∗f(z)) Thus,f is an equivalence relation of type 1 The canonical mappingθ ofH1/f onto

H2 is given for x∈H1 byθ(xf) =f(x) It is clearly well defined and one to one Moreover, forx, y∈H1, θ(xf⊗yf) =θ({zf | z∈xf◦yf}) ={f(z)|z∈xf◦yf}=f(xf◦yf) and θ(xf)∗θ(yf) =f(x)∗f(y).

Therefore, Proposition 2.4.7 (1) yields thatH1/f is canonically isomorphic to H2 if and only if f is a homomorphism of type 1 The theorem is then established forn= 1.

Now, let n > 1 If f is a homomorphism of type n, then Proposition

2.4.6 and the theorem forn= 1 imply thatH1/f is canonically isomorphic to H2 On the other hand, ifH1/f is canonically isomorphic to H2, then the above relations yields forx, y∈H1thatf −1 (f(x)∗f(y)) = (xf◦yf)f.

Therefore, for n= 2,3 or 4, f is an equivalence relation of type nif and only iff is a homomorphism of typen

Corollary 2.4.14 Let (H,◦) be a hypergroup Let θ be an equivalence relation onH We denote also byθthe canonical mapping ofH ontoH/θ.

Then, for n = 1,2,3 or 4, θ is an equivalence relation of type n on H if and only if H/θ is a hypergroup and θ is a homomorphism of typen ofH ontoH/θ.

Proof By Proposition 2.4.13, ifθis an equivalence relation of type 1,2,3 or

4, thenH/θ is a hypergroup Note that for x∈H, θ(x) =xθ, an element ofH/θ, and thatθ −1 (θ(x)) =xθ, a subset ofH This compatibility allows the theorem to be applied and yields the corollary

Three key equivalence relations on a hypergroup, known as the fundamental equivalences, play a crucial role in distinguishing between pairs of elements through hyperoperation.

Definition 2.4.15 Let (H,◦) be a hypergroup Let x, y∈ H Then,x andy are said to beoperationally equivalentor0-equivalentifx◦a=y◦a and a◦x= a◦y for every a∈ H The elements xand y are said to be inseparable or i-equivalent if x∈ a◦b when and only when y ∈ a◦b for everya, b∈H Also,xandy are said to beessentially indistinguishableor e-equivalentif they are both operationally equivalent and inseparable.

Obviously, the relations o-, i- and e-equivalence, denoted respectively by o, i, e, are equivalence relations on a hypergroup H For x∈ H, the o-, i- and e-equivalence classes of x are hence denoted by xo, xi and xe, respectively.

Proposition 2.4.16 Let (H,◦)be a hypergroup and x, y∈H.

(1) xo◦yo=x◦y;o-equivalence is of type 2.

(3) xe=xo∩xi;xe◦ye= (x◦y)e=x◦y;e-equivalence is of type4.

Proof It is an immediate consequence of definition

Corollary 2.4.17 Given thatH is a hypergroup, H/o,H/i andH/e are hypergroups The canonical mappings of H onto H/o, H/i and H/e are homomorphisms of type 2,3 and4, respectively.

Definition 2.4.18 A hypergroup H in which xo = xfor each x ∈ H, xi=xfor eachx∈H or xe=xfor eachx∈H is said respectively to be o-reduced, i-reducedor e-reduced Ane-reduced hypergroup is simply said to bereduced.

Example 2.4.19 LetH ={a, b, c, d} Let the hyperoperation◦onH be given by the following table:

One easily checks that (H,◦) is a hypergroup The o-equivalence classes are seen to be{a, b},candd, whereas thei-equivalence classes area,band

{c, d} Therefore, thee-equivalence classes are all singletons, that is to say,

Example 2.4.20 LetH =Z×Z ∗ , whereZ ∗ is the set of non-zero integers.

Letρbe the equivalence relation that puts equivalent “fraction” into classes, that is, for (x, y)∈H,

The hyperoperation◦onH is given for (w, x),(y, z)∈H by

(w, x)◦(y, z) = (wz+xy, xz)ρ. Then, (H,◦) is a hypergroup in which

Proposition 2.4.21 Let (H1,◦),(H2,∗)and(H3,•)be hypergroups For n= 1,2,3or4, letf be a homomorphism of typenofH1ontoH2andgbe a homomorphism of type nof H2 ontoH3 Then,gf is a homomorphism of typenof H1 ontoH3.

Proof Suppose thatx, y∈H1 We first observe that forz∈H1, zgf =f −1 g −1 gf(z) =f −1 (f(z)g).

Letn= 1 By the above relation, we obtain gf(xgf◦ygf) =gf f −1 (f(x)g)◦f −1 (f(y)g)

. Sincef is onto, Corollary 2.4.8 (1) applies and yields gf f −1 (f(x)g)◦f −1 (f(y)g)

Then, Proposition 2.4.7 (1) gives g(f(x)g∗f(y)g) =gf(x)•gf(y).

Hence,gf is a homomorphism of type 1 by the above relations and Propo- sition 2.4.7 (1).

Letn= 2 Similar to the previous case, but simpler.

Letn= 3 Sinceg is of type 3, f −1 g −1 (gf(x)•gf(y)) =f −1 (f(x)g∗f(y)g).

Sincef is onto, Proposition 2.4.5 (3) applies and gives f −1 (f(x)g∗f(y)g) =f −1 (f(x)g)◦f −1 (f(y)g. Now, we obtain f −1 (f(x)g)◦f −1 (f(y)g) =xgf◦ygf.

So, by the above relations, we conclude thatgf is a homomorphism of type

Letn= 4 Then,gf is a homomorphism of type 4 by Proposition 2.4.6

In the following definition, we introduce more types of homomorphisms of hypergroups that appeared in the literature under various names.

Definition 2.4.22 Let (H1,◦) and (H2,∗) be two hypergroups and f :

H1−→H2 be a mapping We say thatf is a homomorphism of type 5, if for allx, y∈H1,f is a good homomorphism and furthermore

(2)f(y\x) =f(f −1 (f(y)\x)); type 6, if for allx, y∈H1,f is a good homomorphism and furthermore

(4)f(f −1 (f(y))\x) =f(y)\f(x); type 7, if for allx, y∈H1,f is a good homomorphism and furthermore

Theorem 2.4.23 If f :H1−→H2 is a homomorphism of type 7, thenf is a homomorphism of type 4.

Proof In general, iff is an inclusion homomorphism, then for everyx, y∈

Now, let f be a homomorphism of type 7 Suppose that z ∈ f −1 (f(x)∗ f(y)) Then,f(z)∈f(x)∗f(y), which implies thatf(y)∈f(x)\f(z) and sof(y)∈f(x\z) Thus, there existsy 0 ∈x\z such thatf(y) =f(y 0 ), con- sequentlyz∈x◦y 0 ⊆f −1 (f(x))◦f −1 (f(y)) Therefore,f −1 (f(x)∗f(y))⊆ f −1 (f(x))◦f −1 (f(y))

Theorem 2.4.24 If f :H1 −→H2 is an onto homomorphism of type 4, thenf is a homomorphism of type 6.

An onto homomorphism of type 4 is classified as a good homomorphism Given \( u \in f(z)/f(x) \), there exists a \( y \) such that \( f(y) = u \), leading to \( f(z) \in f(y) * f(x) \) This implies that \( z \) belongs to the intersection of the preimages \( f^{-1}(f(y) * f(x)) \) and \( f^{-1}(f(y)) \circ f^{-1}(f(x)) \) Consequently, we can find elements \( a \) and \( b \) such that \( z \) is in the composition \( a \circ b \), where \( a \) is in \( f^{-1}(f(y)) \) and \( b \) is in \( f^{-1}(f(x)) \) Thus, we have \( f(y) = f(a) \), \( f(x) = f(b) \), and therefore \( u = f(a) \) is an element of \( f(z/b) \), which is a subset of \( f(z/f^{-1}(f(x))) \) It is important to note that the reverse inclusion always holds Additionally, we can demonstrate that \( f(f^{-1}(f(y)) \setminus x) = f(y) \setminus f(x) \).

Notice that, in general, a homomorphism of type 4 is not good.

Example 2.4.25 LetH(Z) be the hypergroup (N,◦), where for allx, y, x◦y={x+y, |x−y|}.

Letf :H(Z)−→ H(Z) be the function defined by f(x) 0 if x∈2N

Then,f is a homomorphism of type 4 but is not a good homomorphism.

Regular and strongly regular relations

Strong regular relations enable the connection between semihy-pergroups and semigroups, as well as hypergroups and groups These specific equivalence relations play a crucial role in understanding the structural relationships among these algebraic systems.

By utilizing a strong regular relation on a (semi)hypergroup, we can establish a (semi)group structure on the corresponding quotient set This leads to the inquiry of whether regular relations also exist, to which the answer is affirmative Regular relations indeed offer new (semi)hypergroup structures on these quotient sets.

Let us define these notions First, we do some notations.

Let (H,◦) be a semihypergroup andRbe an equivalence relation onH.

IfAandB are non-empty subsets of H, then

ARB means that∀a∈A,∃b∈B such thataRband

∀b 0 ∈B,∃a 0 ∈Asuch thata 0 Rb 0 ;ARB means that∀a∈A,∀b∈B, we have aRb.

Definition 2.5.1 The equivalence relationR 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 allxofH, fromaRb, it follows that (a◦x)R(b◦x) ((x◦a)R(x◦b) respectively);

(3) Ris calledregular(strongly regular) if it is regular (strongly regular) on the right and on the left.

Theorem 2.5.2 Let (H,◦) be a semihypergroup andR be an equivalence relation onH.

(1) If R is regular, then H/Ris a semihypergroup, with respect to the following hyperoperation: x⊗y={z|z∈x◦y};

(2) If the above hyperoperation is well defined onH/R, thenR is regular.

Proof (1) First, we check that the hyperoperation ⊗ is well defined on

H/R Consider x=x1 and y =y1 We check thatx⊗y =x1⊗y1 We have xRx1 and yRy1 SinceR is regular, it follows that (x◦y)R(x1◦y),

In the context of the relation R, if (x1◦y)R(x1◦y1), then it follows that (x◦y)R(x1◦y1) Consequently, for every element z in x◦y, there is a corresponding element z1 in x1◦y1 such that zRz1, indicating that z equals z1 This leads to the conclusion that x⊗y is a subset of x1⊗y1, and we can similarly establish the reverse inclusion Additionally, we examine the associativity of the operation ⊗ For arbitrary elements x, y, and z in H/R, if u belongs to (x⊗y)⊗z, it implies the existence of an element v in x⊗y such that u is contained in v⊗z.

In this analysis, we establish the existence of elements v1 within the composition of sets x and y, and u1 within the composition of sets v and z, where the relation R holds for v and v1, as well as for u and u1 Given that R is regular, we can identify an element u2 within the composition of v1 and z, which is also a subset of the composition of x and (y and z), ensuring that u1 relates to u2 Consequently, we find an element u3 within the composition of y and z, leading to the conclusion that u2 is part of the composition of x and u3 This implies that the composition of (x and y) with z is a subset of the composition of x with (y and z) A similar argument allows us to derive the converse inclusion.

(2) Let aRb and x be an arbitrary element ofH If u ∈ a◦x, then u∈a⊗x=b⊗x={v |v∈b◦x} Hence, there existsv∈b◦xsuch that uRv, whence (a◦x)R(b◦x) Similarly we obtain thatR is regular on the left

Corollary 2.5.3 If(H,◦)is a hypergroup andRis an equivalence relation onH, thenR is regular if and only if(H/R,⊗)is a hypergroup.

Proof IfH is a hypergroup, then for allxofH we haveH◦x=x◦H=H, whence we obtain H/R⊗x=x⊗H/R=H/R According to the above theorem, it follows that (H/R,⊗) is a hypergroup

Notice that ifRis regular on a (semi)hypergroupH, then the canonical projection π: H −→ H/Ris a good epimorphism Indeed, for all x, y of

H and z ∈ π(x◦y), there exists z 0 ∈ x◦y such that z = z 0 We have z=z 0 ∈x⊗y=π(x)⊗π(y) Conversely, ifz∈π(x)⊗π(y) =x⊗y, then there existsz1∈x◦y such thatz=z1∈π(x◦y).

Theorem 2.5.4 If(H,◦)and(K,∗)are semihypergroups andf :H −→K is a good homomorphism, then the equivalence ρ f associated with f, that is xρ f y ⇔ f(x) = f(y), is regular and ϕ : f(H) −→ H/ρ f , defined by ϕ(f(x)) =x, is an isomorphism.

Proof Leth1ρ f h2 andabe an arbitrary element ofH Ifu∈h1◦a, then f(u)∈f(h1◦a) =f(h1)∗f(a) =f(h2)∗f(a) =f(h2◦a).

In the context of the function f, there exists an element v in h2◦a such that f(u) equals f(v), indicating that u is related to v through the relation ρ f This demonstrates that the relation ρ f is regular on the right side A similar argument can be made to show that ρ f is also regular on the left side Furthermore, for any functions f(x) and f(y) in the set H, the relationship ϕ(f(x)∗f(y)) equals ϕ(f(x◦y)), which can be expressed as the set of elements z that belong to the operation x◦y, thereby resulting in x⊗y being equivalent to ϕ(f(x))⊗ϕ(f(y)).

Moreover, ifϕ(f(x)) =ϕ(f(y)), thenxρ f y, soϕis injective and clearly, it is also surjective Finally, for allx, yofH/ρ f we have ϕ −1 (x⊗y) =ϕ −1 ({z|z∈x◦y}) ={f(z)|z∈x◦y}

Theorem 2.5.5 Let (H,◦) be a semihypergroup andR be an equivalence relation onH.

(1) If R is strongly regular, then H/R is a semigroup, with respect to the following operation: x⊗y=z, for allz∈x◦y;

(2) If the above operation is well defined on H/R, then R is strongly regular.

Proof (1) For allx, yofH, we have (x◦y)R(x◦y) Hence,x⊗y={z|z∈ x◦y}={z}, which means that x⊗y has exactly one element Therefore,

If \( R \) is an arbitrary element of \( H \), we verify that \( (a \circ x) R (b \circ x) \) For any \( u \in a \circ x \) and \( v \in b \circ x \), it follows that \( u = a \otimes x = b \otimes x = v \), indicating that \( uRv \) This demonstrates that \( R \) is strongly regular on the right, and a similar argument shows that it is also strongly regular on the left.

Corollary 2.5.6 If(H,◦)is a hypergroup andRis an equivalence relation onH, thenR is strongly regular if and only if(H/R,⊗)is a group.

Theorem 2.5.7 If (H,◦)is a semihypergroup, (S,∗) is a semigroup and f :H −→S is a homomorphism, then the equivalenceρ f associated withf is strongly regular.

Proof Letaρ f b,x∈H andu∈a◦x It follows that f(u) =f(a)∗f(x) =f(b)∗f(x) =f(b◦x).

Hence, for allv∈b◦x, we havef(u) =f(v), which means thatuρ f v Hence, ρ f is strongly regular on the right and similarly, it is strongly regular on the left

The fundamental relation has an important role in the study of semihypergroups and especially of hypergroups.

Definition 2.5.8 For alln >1, we define the relationβn on a semihyper- groupH, as follows: a βnb⇔ ∃(x1, , xn)∈H n : {a, b} ⊆ n

Q i=1 xi, and β = S n≥1 βn, where β1 ={(x, x) | x∈H} is the diagonal relation on

H Clearly, the relationβ is reflexive and symmetric Denote byβ ∗ the transitive closure ofβ.

Theorem 2.5.9 β ∗ is the smallest strongly regular relation on H.

(1) β ∗ is a strongly regular relation onH;

(2) IfRis a strongly regular relation onH, then β ∗ ⊆R.

(1) Let a β ∗ b and x be an arbitrary element of H It follows that there existx0=a, x1, , xn =b such that for alli∈ {0,1, , n−1} we have xiβ xi+1 Let u1 ∈ a◦x and u2 ∈ b◦x We check that u1β ∗ u2.

From xiβ xi+1 it follows that there exists a hyperproduct Pi, such that

In the given context, we establish that if {xi, xi+1} is a subset of Pi and both xi◦x and xi+1◦x are included in Pi◦x, it follows that xi◦xβxi+1◦x Consequently, for every i in the range {0, 1, , n−1} and for all elements si in xi◦x, the relationship siβsi+1 holds true By defining s0 as u1 and sn as u2, we derive the conclusion that u1β∗u2 This indicates that β∗ is strongly regular on both the right and left sides.

(2) We have β1 ={(x, x)|x∈H} ⊆ R, sinceR is reflexive Suppose thatβn−1⊆R and show thatβn⊆R Ifaβnb, then there existx1, , xn inH, such that{a, b} ⊆ n

Q i=1 xi Hence, there existsu, v in n−1

In the given context, we analyze the relationship between elements a and b within the sets u and v, respectively, ensuring that a belongs to the closure of xn while b is within the closure of xn Given that uβn−1v and following the initial hypothesis, we conclude that uRv Due to the strong regularity of relation R, it follows that a is related to b (aRb) Consequently, we establish that βn is a subset of R By employing induction, we can assert that β is also a subset of R, leading to the conclusion that β* is contained within R as well.

Hence, the relation β ∗ is the smallest equivalence relation on H, such that the quotientH/β ∗ is a group.

Definition 2.5.10 β ∗ is called thefundamental equivalence relationonH andH/β ∗ is called the fundamental group.

If H is a hypergroup, then the relationship β equals β ∗ The canonical projection ϕH maps H to the quotient H/β ∗ The heart of H is defined as the set ωH, which includes all elements x in H such that ϕH(x) equals 1, where 1 represents the identity element of the group H/β ∗ This concept was introduced by Koskas and has been primarily explored by researchers including Corsini, Davvaz, Freni, and Leoreanu.

Freni in [73] introduced the relationγas a generalization of the relation β.

Definition 2.5.11 LetH be a semihypergroup Then, we set γ1={(x, x)|x∈H} and for every integern >1,γn is the relation defined as follows: x γny⇐⇒ ∃(z1, , zn)∈H n , ∃σ∈Sn : x∈ n

Obviously, for n ≥ 1, the relations γn are symmetric, and the relation γ= S n≥1 γn is reflexive and symmetric.

Letγ ∗ be the transitive closure ofγ IfH is a hypergroup, thenγ=γ ∗

Theorem 2.5.12 The relation γ ∗ is a strongly regular relation.

The relation γ ∗ is established as an equivalence relation To demonstrate that it is strongly regular, we need to prove that if xγy, then both (x◦a)γ(y◦a) and (a◦x)γ(a◦y) hold true for any element a in H Given that xγy, there exists an integer n in N such that xγny, indicating the presence of a sequence (z1, , zn) in H n along with a permutation σ in Sn, confirming the relationship between x and y.

For every a ∈ H, set a =zn+1 and letτ be a permutation of Sn+1 such that τ(i) =σ(i),∀i∈ {1,2, , n}; τ(n+ 1) =n+ 1.

For allv∈x◦aand for allw∈y◦a, we havev∈x◦a⊆ n

Q i=1 zτ(i) So,v γn+1wand hencev γ w Thus, (x◦a) γ (y◦a) In the same way, we can show that

Moreover, ifx γ ∗ y, then there existm∈Nand

(w0=x, w1, , wm−1, wm=y)∈H m+1 such thatx=w0γ w1γ γ wm−1γ wm=y Now, we obtain x◦a=w0◦a γ w1◦a γ w2◦a γ γ wm−1◦a γ wm◦a=y◦a.

Finally, for all v ∈ x◦ a = w0 ◦a and for all w ∈ wm◦ a = y◦ a, taking z1 ∈ w1 ◦ a, z2 ∈ w2 ◦ a, , zm−1 ∈ wm−1 ◦ a, we have v γ z1γ z2γ γ zm−1γ w, and sov γ ∗ w Therefore,x◦aγ ∗ y◦a Simi- larly, we obtaina◦xγ ∗ a◦y Hence,γ ∗ is strongly regular

Corollary 2.5.13 The quotient H/γ ∗ is a commutative semigroup Fur- thermore, if H is a hypergroup, thenH/γ ∗ is a commutative group.

Proof Sinceγ ∗ is a strongly regular relation, the quotientH/γ ∗ is a semigroup under the following operation: γ ∗ (x1)⊗γ ∗ (x2) =γ ∗ (z), for allz∈x1◦x2.

Moreover, ifH is a hypergroup, thenH/γ ∗ is a group Finally, if σis the cycle ofS2 such thatσ(1) = 2, for allz∈x1◦x2 andw∈xσ(1)◦xσ(2), we havezγ2w, sozγ ∗ wandγ ∗ (x1)⊗γ ∗ (x2) =γ ∗ (z) =γ ∗ (x2)⊗γ ∗ (x1)

Theorem 2.5.14 The relationγ ∗ is the smallest strongly regular relation on a semihypergroup H such that the quotientH/γ ∗ is commutative semigroup.

Proof Suppose that ρ is a strongly regular relation such that H/ρ is a commutative semigroup and ϕ : H −→ H/ρ is the canonical projection.

Then, ϕ is a good homomorphism Moreover, if x γny, then there exist

(z1, , zn)∈H n andσ∈Sn such thatx∈ n

Q i=1 zσ(i), whence ϕ(x) =ϕ(z1)⊗ .⊗ϕ(zn) andϕ(y) =ϕ(zσ(1))⊗ .⊗ϕ(zσ(n)) By the commutativity ofH/ρ, it follows thatϕ(x) =ϕ(y) andx ρ y Thus,x γny impliesx ρ y, and obviously, x γ yimplies thatx ρ y.

Finally, if x γ ∗ y, then there existm∈Nand

(w0=x, w1, , wm−1, wm=y)∈H m+1 such thatx=w0γ w1γ γ wm−1γ wm=y Therefore, x=w0ρ w1ρ ρ wm−1ρ wm=y, and transitivity ofρimplies thatx ρ y Therefore,γ ∗ ⊆ρ

Complete parts were introduced and studied for the first time by

M Koskas [94] Later, this topic was analyzed by P Corsini [29], Y Sureau

B Davvaz et al have extensively contributed to the general theory of hypergroups, while M De Salvo has analyzed complete parts from a combinatorial perspective Additionally, a generalization known as n-complete parts has been introduced, building upon these foundational studies.

R Migliorato Other authors gave a contribution to the study of complete parts and of the heart of a hypergroup Among them, V Leoreanu analyzed the structure of the heart of a hypergroup in her Ph.D Thesis.

We present now the definitions.

Definition 2.5.15 Let (H,◦) be a semihypergroup andAbe a non-empty subset ofH We say thatAis acomplete partofH if for any non-zero natural numbernand for alla1, , an ofH, the following implication holds:

Theorem 2.5.16 If(H,◦)is a semihypergroup andRis a strongly regular relation onH, then for allz of H, the equivalence class of z is a complete part of H.

Proof Leta1, , an be elements ofH, such that z∩ n

Q i=1 ai, such that y R z The homomorphism π :

H −→ H/R is good and H/R is a semigroup It follows that π(z) π(y) =π n Q i=1 ai n

Now, we want to determine some necessary and sufficient conditions so that the relationγ is transitive.

Definition 2.5.17 LetM be a non-empty subset ofH We say thatM is aγ-partofHif for any non-zero natural numbern, for all (z1, , zn)∈H n and for allσ∈Sn, we have

Lemma 2.5.18 Let M be a non-empty subsets ofH Then, the following conditions are equivalent:

Proof (1=⇒2) If (x, y)∈ H 2 is a pair such that x∈M and x γ y, then there existn∈N, (z1, , zn)∈H n andσ∈Sn such thatx∈M ∩ n

Q i=1 zσ(i) Since M is a γ-part ofH, we have n

(2=⇒3) Assume that (x, y) ∈ H 2 such that x ∈ M and x γ ∗ y Ob- viously, there exist m ∈N and (w0 = x, w1, , wm−1, wm =y)∈ H m+1 such thatx=w0γ w1γ γ wm−1γ wm =y Since x∈M, applying (2) mtimes, we obtainy∈M.

Q i=1 xi For every σ∈Snand for everyy∈ n

Finally, by (3), we obtainy∈M, whence n

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

LetH be a semihypergroup For allx∈H, we set

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

Proof For allx, y∈H, we have xγy ⇐⇒ ∃n∈N,∃(x1, , xn)∈H n ,∃σ∈Sn: x∈ n

Theorem 2.5.20 Let H be a semihypergroup Then, the following conditions are equivalent:

(3) Pσ(x)is aγ-part ofH, for allx∈H.

Proof (1=⇒2) By Lemma 2.5.19, for allx, y∈H, we have y∈γ ∗ (x)⇐⇒xγ ∗ y⇐⇒xγy⇐⇒y∈Pσ(x).

According to Lemma 2.5.18, a non-empty subset M of H qualifies as a γ-part of H if it represents a union of equivalence classes modulo γ∗ Notably, each equivalence class modulo γ∗ is itself considered a γ-part of H.

(3=⇒1) If x γ y and y γ z, then there exist m, n ∈ N, (x1, , xn) ∈

Tn(x), (y1, , ym) ∈ Tm(y), σ ∈ Sn and τ ∈Sm such that y ∈ n

Q i=1 yτ(i) SincePσ(x) is aγ-part ofH, we have x∈ n

Complete hypergroups

In this section, we study the concept of complete hypergroups.

Definition 2.6.1 Let A be a non-empty subset of H The intersection of the parts ofH which are complete and containA is called thecomplete closure of A in H; it will be denoted by C(A) A semihypergroup H is complete, if it satisfies one of the following conditions:

A hypergroup is complete if it is a complete semihypergroup If (H,◦) is a complete semihypergroup, then either there exist a, b∈ H such that β ∗ (x) =a◦b orβ ∗ (x) ={x}.

An elemente∈H is called anidentityif a∈e◦a∩a◦e for all a∈H.

An elementx 0 is called an inverseofxif an identityeexists such that e∈x◦x 0 ∩x 0 ◦x.

Definition 2.6.2 A regular hypergroup H is a hypergroup which it has at least one identity and every element has at least one inverse A regular hypergroupHis said to bereversible, if it satisfies the following conditions:

IfH is regular, for everyx∈H, we denotei(x) the set of the inverses ofx.

Theorem 2.6.3 If(H,◦)is a complete hypergroup, then

(1) ωH ={e∈H : ∀x∈H, x∈x◦e∩e◦x},which means thatωH is the set of two-sided identities ofH.

(2) H is regular(i.e H has at least one identity and any element has an inverse)and reversible.

Proof (1) Ifu∈ωH, then for alla∈H,we havea∈C(a) =a◦ωH=a◦u.

Similarly we havea∈u◦a,which means that uis a two-sided identity of

H Conversely, any two-sided identity u of H is an element of ωH, since ϕ(u) = 1.

(2) Let a, a 0 , a 00 be elements of H and e be a two-sided identity, such thate∈a 0 ◦a∩a◦a 00 Then,a 0 ◦a=ωH=a◦a 00 anda◦a 0 ⊆a◦a 0 ◦a◦a 00 ⊆ a◦ωH◦a 00 =ωH◦a◦a 00 =ωH.Hence,a◦a 0 =ωH,soa 0 is an inverse ofa.

Moreover, if a∈b◦c, thenωH=a 0 ◦a⊆a 0 ◦b◦c, so for any inversec 0 ofc, we have c 0 ∈ωH◦c 0 ⊆a 0 ◦b◦c◦c 0 =a 0 ◦b◦ωH =a 0 ◦b.

Similarly, from here we obtainb 0 ∈c◦a 0 ,and sob 0 ◦a⊆c◦a 0 ◦a=C(c), whencec∈C(c) =b 0 ◦a.In a similar way, we obtainb∈a◦c 0

Definition 2.6.4 A hypergroup (H,◦) is called flat if for all subhyper- groupK ofH, we haveωK=ωH∩K.

Theorem 2.6.5 Any complete hypergroup is flat.

Proof LetH be a complete hypergroup and letK be a subhypergroupH.

Moreover, y ∈ CK(x) ⇒ yβKx ⇒ yβHx⇒ y ∈ CH(x), which means that CK(x) ⊆ CH(x) Clearly, ωH∩K 6= ∅ If x∈ ωH∩K ⊆ ωK, then

CK(x) =ωK, CH(x) =ωH.Hence,ωK ⊆ωH whenceωK ⊆ωH∩K.Hence, ωK=ωH∩K.

Corollary 2.6.6 IfKis a subhypergroup of a complete hypergroup(H,◦), thenωK=ωH.

Proof Setx∈ωH∩K.We haveωH =C(x◦x) =x◦x⊆ωH∩K, whence ωH ⊆ωH∩K,then we apply the above theorem Hence,ωK=ωH.

Theorem 2.6.7 Let H, H 0 be complete hypergroups and f :H →H 0 be a good homomorphism Then, we have f(ωH) =ω H 0

Proof Let x∈ ωH Then, x◦x=ωH, whence f(x)◦f(x) =f(ωH) On the other hand,f(x) is an identity ofH 0 , sincexis an identity ofH,which means thatf(x)∈ωH 0 Hence,ω H 0 =f(x)◦f(x) =f(ωH).

Definition 2.6.8 A hypergroupH is said to be ann-complete hypergroup if for all (z1, , zn)∈H n , n

A hypergroup H is said to ben ∗ -complete if there exists n ∈N such thatβ n ∗ =β andβ ∗ n−1 6=β n ∗

Definition 2.6.9 A hypergroup H is said to be γn-complete if for all

(z1, z2, , zn)∈H n and for allσ∈Sn : γ n Q i=1 zi n

Corollary 2.6.10 If H is a commutative hypergroup, then H is a γn- complete hypergroup if and only ifH is an n-complete hypergroup.

We begin with some properties which are valid in every hypergroup.

They concern the relation γn and will be prove only for n > 1 The case n= 1 is always trivially true We suppose thatH= (H,◦) is a hypergroup.

Proposition 2.6.11 For any positive integer n, we have

Proof (1) Ifx γny, then there exist (z1, z2, , zn)∈H n andσ∈Sn such that x ∈ n

Q i=1 zσ(i) Since H is a hypergroup, so there exists

Now let z 0 i = zi for 1 ≤ i ≤ n−1 and z n 0 = t1, z n+1 0 = t2, so x ∈ z 1 0 ◦z 0 2 ◦z 0 n◦z 0 n+1 Now, letσ(j) =nand we defineσ 0 ∈Sn+1as follows: σ 0 (i) 

Proposition 2.6.12 A hypergroupH isγn-complete if and only if for all σ∈Sn and for allx∈ n

Q i=1 zσ(i). Proof Suppose thatH is γn-complete, for σ∈Sn If x∈ n

Conversely, for every σ∈Sn, (z1, z2, , zn)∈H n andx∈Qn i=1zi we have γ(x) n

Proposition 2.6.13 If H is aγn-complete hypergroup, thenγ ∗ =γn.

Proof We know that in every hypergroup γ ∗ = γ, so it is suffices to prove that: γ ⊆ γn Suppose that x γ y Then, there exists m ∈ N such that x γmy If m ≤ n, then γm ⊆ γn If m > n, then there exist

Q i=1 zσ(i) There existst1∈Hsuch thatt1∈zn◦zn+1◦ zm, x∈ n−1

Example 2.6.14 IfHis a semihypergroup, then Proposition 2.6.13 maybe not correct: letH be the following semihypergroup:

4 1,3 1,3 1,2,3 1,2,3 Then,H isγ2-complete, 4γ ∗ 4 but not 4γ24.

Proof If a γnb, then there exist (z1, z2, , zn) ∈ H n and σ ∈ Sn such that a ∈ n

Q i=1 zσ(i))◦x Now letui :=zi, for all 1≤i≤n, un+1 :=xand σ 0 ∈Sn+1 withσ 0 (i) =σ(i), σ 0 (n+ 1) =n+ 1 Then, we have a◦x⊆ n+1

Q i=1 uσ 0 (i), which implies that (a◦x)γn+1 (b◦x) The rest can be proved in an analogous way

Proof Ifa γ n ∗ b, then there exists (z1, z2, , zm)∈H m such that a=z1γnz2γn zm−1γn zm=b.

Now, we have a◦x=z1◦x γn+1 z2◦x zm−1◦x γn+1 zm◦x=b◦x.

It means that (a◦x)γ n+1 ∗ (b◦x) The rest can be proven similarly

Letγ 0 ∗ =∅ We define the γ n ∗ -complete hypergroups.

Definition 2.6.17 A hypergroup H is said to beγ n ∗ -completeif nis the smallest natural number such that (γ n ∗ )H=γH and (γ n ∗ )H6= (γ ∗ n−1 )H.

Example 2.6.18 LetH be the following hypergroup:

Corollary 2.6.19 Let H be a commutative hypergroup Then, H is a γ n ∗ -complete hypergroup if and only ifH is an n ∗ -complete hypergroup.

Proposition 2.6.20 A hypergroup H is γ 1 ∗ -complete if and only if H is an abelian group.

Proof Suppose thatHis aγ 1 ∗ -complete hypergroup, soγ 1 ∗ =γandγ2⊆γ1.

Now, for eachx∈z1◦z2andy∈z2◦z1, we havex γ2y, sox=y Therefore, z1◦z2=z2◦z1 is singleton, and soH is an abelian group.

Conversely, ifHis an abelian group, then∀(x, y)∈H 2 ; n

Q i=1 zi|= 1 By definition,x γnyif and only ifx n

Corollary 2.6.21 H is a γ n ∗ -complete hypergroup if and only ifn is the minimum integer such that H/γ ∗ n is an abelian group.

Proposition 2.6.22 In every hypergroups(H,◦)the following conditions are equivalent:

Proposition 2.6.23 Every finite hypergroup is γ ∗ n-complete.

Proof Since H is finite, the succession γ 1 ∗ ⊆γ 2 ∗ ⊆ is stationary Thus,

Proposition 2.6.24 If H is a γn-complete hypergroup, then there exists m≤nsuch that H isγ m ∗ -complete.

Proof One proves that, ifH isγn-complete, then γn =γ, soγ n ∗ =γ and there exists am≤nsuch thatγm ∗ =γ andγ m−1 ∗ 6=γm ∗

Letϕbe the canonical projection Then, we defineDH =ϕ −1 (1H/γ ∗ ).

Proof (1) Ifx γ y, then there exists (v, w)∈D 2 H such thaty∈x◦vandy∈ x◦w, by hypothesisv γnw Now, using Lemma 2.6.15, (x◦v)γn+1 (x◦w), whencex γn+1y, so γ⊆γn+1.

(2) It follows from (a) and Lemma 2.6.16

Theorem 2.6.26 If (v, w)∈D H 2 ,v γ n ∗ w and there exists (u 0 , w 0 )∈D 2 H such that u 0 6∈γ n−1 ∗ w 0 , thenH isγn ∗ -complete orγ n+1 ∗ -complete.

Example 2.6.27 Both of the two possibilities of Theorem 2.6.26 are verifiable, as the following examples shows:

Proposition 2.6.28 IfH is aγn-complete hypergroup, then for all(z1, z2,

, zn)∈H n and for allσ∈Sn, n

Proof By using Proposition 2.6.18, the proof is straightforward

Corollary 2.6.29 If H isγ2-complete, then H is a complete hypergroup.

Corollary 2.6.30 If H is a γ2-complete hypergroup, then

(1) DH is the set of identity elements ofH.

Definition 2.6.31 AKH hypergroupis a hypergroup constructed from a hypergroup H = (H,◦) and a family{A(x)}x∈H of non-empty subsets of

A(x) and defining the following hyperoperation∗:

(H,◦) is a hypergroup if and only if (KH,∗) is a hypergroup In this case,

KH is said to be aKH hypergroup generated by H For allP∈P ∗ (H), let

Proposition 2.6.32 (H,◦) is a hypergroup if and only if (KH,∗) is a hypergroup.

Theorem 2.6.33 If P is a γ-part of (H,◦), then K(P) is γ-part of

Proof 1 Suppose that (z1, z2, , zm)∈K H m , and∗ m

Q i=1 zi∩K(P), so there exists (x1, x2, , xm)∈H m such that for all

Q i=1 x i denotes the hyperproduct of the elements x i by hyperoperation ( ◦ ).

Since u ∈ K(P), there exists y2 ∈ P such that u ∈ A(y2) Therefore,

Q i=1 xi∩P SinceP is a γ-part of (H,◦), for allσ∈Sm,◦ m

Q i=1 xσ(i)⊆P Now, forσ∈Sm, assume that v∈ ∗ m

Theorem 2.6.34 For every (x, y) ∈ H 2 and (u, v) ∈ A(x)×A(y), the following conditions are pairwise equivalent:

Proof (1=⇒2) Letu(γn)K H v Then, there exist (z1, z2, , zn)∈K H n and σ∈Sn such that u∈ ∗ n

Q i=1 zσ(i) Now, for every 1≤i≤n there existsyi withzi∈A(yi) such that u∈ S w∈◦ Q n i=1 y i

Q i=1 yσ(i) such that u∈ A(w1), v∈A(w2).So,w1=x, w2=y and we obtainx(γn)Hy.

(2=⇒3) Suppose thatx(γn)Hy Then, there exist (z1, z2, , zn)∈H n and σ∈Sn such thatx∈ ◦ n

Theorem 2.6.35 For every (x, y) ∈ H 2 and (u, v) ∈ A(x)×A(y), the following conditions are pairwise equivalent.

Proof (1=⇒2) Letu(γn ∗ )K H v Then, there existm∈N, (x0, x1, , xm)∈

K H m+1 such thatu=x0(γn)K H x1 xm−1(γn)K H xm=v.For all 0≤j < m−1, there exist (z j 1 , z 2 j , , z n j )∈K H n andσ j ∈Snsuch thatxj∈ ∗ n

Q i=1 z j σ j (i) Now, for every 1≤ i≤n, 0 ≤j < m−1, there existsy j i such that z i j ∈A(y i j ), xj∈ S w∈◦ Q n i=1 y i j

Q i=1 y σ j j (i), where xj ∈A(wj) and xj+1 ∈ A(wj+1) such that w0(γn)H w1 wm−1(γn)Hwm So, w0=x, wm=y Therefore, we have x(γ n ∗ )Hy.

(2=⇒3) Suppose that x(γ n ∗ )H y Then, there exist m ∈ N and

(x0, x1, , xm)∈H m+1 such that x=x0(γn)H x1 xm−1(γn)Hxm y.For all 0≤j < m−1, there exist (z 1 j , z j 2 , , z j n )∈H n andσ j ∈Snsuch thatxj∈ ◦ n

Theorem 2.6.36 If H is hypergroup andn≥2, then

Proof (1) Suppose that (γn)H=γH It is enough to showγK H ⊆(γn)K H

If u γK H v, then there exists (x, y) ∈ H 2 such that u ∈ A(x) and v ∈

A(y) So,x γHy andx(γn)H y Thus, we haveA(x) (γn)K H A(y), and so u(γn)K H v.

Conversely, if (γn)K H =γK H , then it is enough to showγH⊆(γn)H We haveA(x) (γ)K H A(y) Hence,A(x) (γn)K H A(y), so we havex(γn)Hy.

(2) The proof is similar to the proof of part (1)

Corollary 2.6.37 Forn≥2,H isγ n ∗ -complete if and only if KH isγ ∗ n - complete.

Definition 2.6.38 Let Abe a non-empty subset of H The intersection of the γ-parts of H which containA is calledγ-closureof Ain H It will be denotedCγ(A).

Theorem 2.6.39 Let A be a non-empty subset ofH We pose

Proof It is necessary to prove

(2) IfA⊆B andB is a γ-part of H, thenG(A)⊆B.

Q i=1 xi∩G(A)6=∅ Then, there existsn∈Nsuch that p

Gn(A) 6= ∅ For every σ ∈ Sp and y ∈ p

Q i=1 xσ(i)⊆G(A), and so G(A) is aγ-part of H.

(2) We have A=G1(A)⊆B Suppose that B is a γ-part of H and

Gn(A)⊆B We prove that this impliesGn+1(A)⊆B For every z ∈ Gn+1(A) there exist p ∈ N, (x1, , xp) ∈ H p and σ ∈ Sp such that z ∈ p

Now, we proceed by induction Suppose thatGn−1(G2(x)) =Gn(x) Then,

(2) We prove by induction It is clear that x∈ G2(y) ⇔ y ∈ G2(x).

Suppose thatx∈Gn−1(y)⇔y∈Gn−1(x) If x∈Gn(y), then there exist q∈N, (a1, , aq)∈H q andσ∈S q such that x∈ q

Q i=1 ai∩Gn−1(y)6=∅, by this it follows that there existsv∈ q

Q i=1 ai∩Gn−1(y) Therefore,v∈G2(x) is obtained Fromv∈Gn−1(y) we have y∈Gn−1(G2(x)) =Gn(x)

Theorem 2.6.41 The relation x G y ⇔ x ∈ G({y}) is an equivalence relation.

Proof We write Cγ(x) instead of Cγ({x}) Clearly, G is reflexive Now, suppose thatxGyandyGz IfP is aγ-part ofH andz∈P, thenCγ(z)⊆

P, y∈P and consequentlyx∈Cγ(y)⊆P For this reasonx∈Cγ(z) that isxGz The symmetrically ofGfollows in a direct way from the preceding lemma

Theorem 2.6.42 For all x, y∈H, one getsx G y⇔x γ ∗ y.

Proof Letxγ y Then, there existsn∈Nsuch thatx γny So, there exist

(z1, , zn) ∈H n and σ ∈ Sn such that x∈ n

Conversely, ifxGy, then there existsn∈Nsuch thatx∈Gn+1(y), from this it follows that∃m∈N, ∃(z 1 1 , , z m 1 )∈H m , ∃σ 1 ∈Sm: x∈ m

Q i=1 z i 1 ∩Gn(y) Therefore,x γ x1 andx1∈Gn(y) and so there existr∈N,(z 1 2 , , z 2 r)∈H r , σ 2 ∈Sr such that x1∈ r

So, as a consequence one obtains

Q i=1 z n i ∩G n−(n−1) (y) =⇒xn∈G1(y) ={y}=⇒xn=y, and sox γ x1 γ xn=y Therefore,G⊆γ ∗

Theorem 2.6.43 If B is a non-empty subset ofH, then

Proof It is clear for every b∈ B, Cγ(b)⊆Cγ(B), because every γ-part containing B contains {b} Therefore, S b∈B

Cγ(b) ⊆ Cγ(B) In order to prove the converse remember thatCγ(B) = S

We demonstrate the theorem by induction Suppose that it is true for n, that is,Gn(B)⊆ S b∈B

Gn(b) and we prove thatGn+1(B)⊆ S b∈B

Gn+1(b) If z ∈ Gn+1(B), then there exist q∈ N,(x1, , xq)∈ H q and σ∈ Sq such that z∈ q

Q i=1 xi∩Gn(B)6=∅, by the hypothesis induction q

6= ∅ Hence, there exists b 0 ∈ B such that q

Q i=1 xσ(i), one gets z ∈ Gn+1(b 0 ) and so one has prove Gn+1(B) ⊆ S b∈B

Theorem 2.6.44 If A∈ P ∗ (H), one hasDH◦A=A◦DH=Cγ(A).

Corollary 2.6.45 Let A∈ P ∗ (H) Then,A is aγ-part ofH if and only if A◦DH =A.

Corollary 2.6.46 IfAis aγ-part ofH, then for everyB∈ P ∗ (H),A◦B andB◦A areγ-parts ofH.

Proof We have: Cγ(A◦B) =A◦B◦DH =A◦DH◦B=Cγ(A)◦B=A◦B.

Join spaces

W Prenowitz introduced join spaces to create a unified algebraic framework for axiomatizing and studying classical geometries, utilizing a hypergroup as the underlying algebraic structure This concept was further developed by Prenowitz and J Jantosciak.

Non-Euclidean geometry has led to the development of various branches, including descriptive, projective, and spherical geometry This framework has facilitated the construction of significant examples of join spaces related to binary relations, graphs, lattices, fuzzy sets, and rough sets In this section, we will explore the concept of join space, drawing on key references for a comprehensive understanding.

In order to define a join space, we recall the following notation: If a, b are elements of a hypergroupoid (H,◦), then we denote a/b = {x ∈ H | a∈x◦b} Moreover, byA/B we intend the set S a∈A,b∈B a/b.

Definition 2.7.1 A commutative hypergroup (H,◦) is called ajoin space if the following condition holds for all elementsa, b, c, dofH: a/b∩c/d6=∅ =⇒ a◦d∩b◦c6=∅ (transposition axiom).

Elements of H are calledpointsand are denoted by a, b, c, Sets of points are denoted by A, B, C,ã ã ã The elementary algebra of join space theory for sets of points include the result

A set of points M is said to be linear if it is closed under join and extension (a, b ∈M imply a◦b ⊆M and a/b⊆M) IfM is linear, then

M = M ◦M = M/M The linear sets are the closed subhypergroups of

H For a set of points A, the intersection of all linear sets containing A

(the least linear set containing A) is denoted by < A > and is called the linear spacespannedor generatedbyA We use < A, B >for< A∪B >.

Important results concerning linear sets are a formula for the linear span of two intersecting linear sets and a weak modularity property

Both of these results depend on the transposition axiom.

A point e is said to be a (scalar) identity if e◦a = a for every point a IfH has an identity, it is unique In a join spaceH with identityethe following hold:

(1) For eachathere exists a uniquea −1 such thate∈a◦a −1 ;

(4) e∈M for any non-empty linear setM.

The transposition axiom is essential for demonstrating the uniqueness of the inverse point, as well as for establishing that an extension can be reduced to a join Additionally, join spaces with identity have been explored in [112] under the term canonical hypergroups.

Some important examples of join spaces were presented in Section 2.2

(see Examples 2.2.3 (9), (10), (11), (12), (14)) We give here some other examples.

In a distributive lattice (L,∧,∨), we can define a hyperoperation a◦b as the set of elements x in L such that x equals (a∧b)∨(a∧x)∨(b∧x) This structure (L,◦) forms a non-geometrical join space where every element acts as an identity Furthermore, this hyperoperation can be extended to the broader framework of a median semilattice, which is characterized as a meet semilattice (S,∧) that satisfies specific conditions.

• Every principal ideal is a distributive lattice;

•Any three elements ofShave an upper bound whenever each pair of them has an upper bound.

(2) LetV be a vector space over an ordered fieldF If for alla, bofV we define a◦b={λa+àb|λ >0, à >0, λ+à= 1}, then (V,◦) is a join space, called anaffine join spaceoverF.

(3) LetG= (V, E) be a connected simple graph We say that a subset

AofV is convex if for all different elementsa, bofA, we have that

Acontains all points on all geodetics fromatob Denote by (a, b) the least convex set containing {a, b} A convex set P is called primeifV\P is convex Finally,Gis called astrong prime convex intersection graphif:

•For any convex setAand any pointx, which does not belong toA, there exists a prime convex setP, such thatA⊆P,x∈V\P;

• For any (a, b), (c, d) such that (a, b)∩(c, d) =∅, there exists a convex prime setP such that (a, b)⊆P and (c, d)⊆V \P.

IfGsatisfies the above two conditions and for all different elements a, bofV we definea◦b= (a, b) anda◦a=a, then (V,◦) is a join space.

In this article, we introduce a hyperoperation defined on the Cartesian plane R², specifically for an open real interval denoted as )a, b( For any distinct elements (x1, x2) and (y1, y2) in R², the operation is defined as (x1, x2)◦(y1, y2) = {(z1, z2) | z1 ∈ )x1, x2( and z2 ∈ )x2, y2(} Additionally, for any element (x1, x2) in R², it holds that (x1, x2)◦(x1, x2) = (x1, x2) Consequently, the structure (R², ◦) forms a geometric join space, although it lacks identity elements.

In a connected simple graph \( G = (V, E) \), we define a hyperoperation on the vertex set \( V \) such that for any distinct elements \( x \) and \( y \) in \( V \), the operation \( x \circ x \) equals \( x \), while \( x \circ y \) represents the set of all vertices \( z \) in \( V \) that are part of some paths \( \gamma: x - y \) Consequently, \( (V, \circ) \) forms a non-geometric join space where each element acts as an identity.

Three classical types of geometry are readily formulated as join spaces.

Definition 2.7.3 (1) A descriptive join spaceor ordered join geometryis a join space that satisfies the axioms a◦a=a/a=a; a, b, cdistinct andc∈ {a, b} implyc∈a◦b,b∈a◦cora∈b◦c.

(2) A spherical join spaceis a join space with identity e that satisfies the axioms a◦a=a; a/a={e, a, a −1 }.

(3) A projective join spaceis a join space with identity ethat satisfies the axiom a◦a=a/a⊆ {e, a}.

Proposition 2.7.4 IfH is one of the classical join spaces, then the “line” spanned by pointsaandb, i.e.,< a, b >is given by

LetM be a non-empty linear subset of the join spaceH For any point a the coset of M in H containing a is denoted by (a)M and is given by

(a)M = (a◦M)/M The family (H :M) ={(a)M | a∈H} of all cosets of

M in H is a partition ofH For the set of pointsAlet

Then, the join cosets (a)M and (b)M satisfies

(a)M◦(b)M ⊆(a◦b)M, so that the equivalence relation on H corresponding to the partition ofH into cosets ofM is a regular equivalence relation.

Proposition 2.7.5 The family(H :M)of cosets M in H under the join operation

(a)M ?(b)M ={(x)M | x∈(a)M ◦(b)M} is a join space with identity M = (m)M, wherem∈M.

Proof Note that (a) −1 M = M/a and that a◦ b∩M 6= ∅ if and only if

(H :M) under ?is known as the factor join spaceH moduloM.

The three classical isomorphism theorems of group theory have analogous principles in join spaces In this context, let (H, ◦) and (K, ã) represent join spaces A mapping ϕ: H → K is defined as a good homomorphism if it satisfies the condition ϕ(a ◦ b) = ϕ(a) ã ϕ(b) When ϕ is both one-to-one and onto, it is classified as an isomorphism, denoted by H ∼= K Additionally, if ϕ is a good homomorphism and K possesses an identity element e, the set {x ∈ H | ϕ(x) = e} is referred to as the kernel of ϕ, symbolized as ker ϕ.

Theorem 2.7.6 (First Isomorphism Theorem) Let H and K be join spaces Let K has identity Let ϕ be a good homomorphism of H onto

K Then,kerϕis a linear subset ofH and(H :kerϕ)∼=K.

Theorem 2.7.7 (Second Isomorphism Theorem) Let H be a join space.

Let M andN be linear subsets such that M∩N6=∅ Then,

Theorem 2.7.8 (Third Isomorphism Theorem) Let H be a join space.

Let M andN be linear subsets such that ∅ 6=N ⊆M Then,(M :N)is a linear subset of(H :N)and(H:M)∼= ((H :N) : (M :N)).

A version of Jordan-H¨older theorem also holds for join spaces.

Theorem 2.7.9 (Jordan-H¨older Theorem)Let H be a join space LetM andN be linear subsets such that ∅ 6=N ⊆M Suppose that

N =A0⊆ ã ã ã ⊆Am=M and N =B0⊆ ã ã ã ⊆Bn=M where each Ai and each Bj are linear and for i = 1,ã ã ã, m and for j 1,ã ã ã , nthat Ai−1 andBj−1 are maximal proper linear subsets ofAi and

Bj respectively Then, m=n and there exists a one to one corresponding between the families of factor spaces

{(Ai:Ai−1)|i= 1,ã ã ã , m}and {(Bj :Bj−1)|j = 1,ã ã ã , n} such that the correspondents are isomorphic.

Proof The chainsA0 ⊆ ã ã ã ⊆Am and B0 ⊆ ã ã ã ⊆Bn are refined respectively by

Ai,j =< Ai−1, Ai∩Bj > fori= 1,ã ã ã, mandj= 0,ã ã ã, n and

Bj,i=< Bj−1, Bj∩Ai> fori= 0,ã ã ã, mandj= 1,ã ã ã , n.

Next, by the second isomorphism theorem followed by the weak modularity property

(Ai,j :Ai,j−1) = (< Ai−1, Ai∩Bj >:< Ai−1, Ai∩Bj−1>)

= (, Ai∩Bj >:< Ai−1, Ai∩Bj−1>)

∼= (Ai∩Bj:< Ai−1, Ai∩Bj−1>∩(Ai∩Bj))

= (Ai∩Bj:< Ai−1∩(Ai∩Bj), Ai∩Bj−1>)

= (Ai∩Bj:< Ai−1∩Bj, Ai∩Bj−1>).

(Bj,i:Bj,i−1)∼= (Bj∩Ai:< Bj−1∩Ai, Bj∩Ai−1>).

(Ai,j :Ai,j−1)∼= (Bj,i:Bj,i−1) fori= 1,ã ã ã , mandj = 1,ã ã ã , n.

Therefore, given the maximality condition onAi−1 andBj−1 inAi andBj respectively, the conclusion of the theorem readily follows

IfN is a closed subhypergroup of a join space H and{x, y} ⊆H, then we define the following binary relation: xJNy ifx◦N∩y◦N 6=∅.

Theorem 2.7.10 JN is an equivalence relation onH and the equivalence class of an elementaisJN(a) = (a◦N)/N In particular,JN(a) =N for alla∈N.

The relation JN is proven to be reflexive, symmetric, and transitive, confirming that it is an equivalence relation on the set H For any element a in H, JN(a) is defined as (a◦N)/N If an element d belongs to JN(a), it implies that d◦N intersects with a◦N, indicating the existence of an element v in a◦N and m in N such that v is in d◦m, leading to the conclusion that d is included in (a◦N)/N Conversely, if an element y is in (a◦N)/N, there exists u in a◦N and m in N such that u is in y◦m, which confirms that y belongs to JN(a) This establishes the inclusion of (a◦N)/N within JN(a) Additionally, if a is an element of N, the relation holds true.

Canonical hypergroups are a particular case of join spaces The structure of canonical hypergroups was individualized for the first time by M.

Krasner as the additive structure of hyperfields In 1970, J Mittas was the first who studied them independently from the other operations In

In 1973, P Corsini conducted an analysis of sd-hypergroups, a specific category of canonical hypergroups By 1975, Roth applied canonical hypergroups to the character theory of finite groups Additionally, W Prenowitz and J Jantosciak highlighted the significance of canonical hypergroups in the field of geometry.

Mullen and J.F Price emphasized the significance of generalized canonical hypergroups in both harmonic analysis and particle physics P Corsini, P Bonansinga, K Serafimidis, M Kostantinidou, J Mittas, and De Salvo introduced and examined various connected hyperstructures related to canonical hypergroups Notable examples include strongly canonical hypergroups, i.p.s hypergroups, quasi-canonical hypergroups (also known as polygroups), and feebly canonical hypergroups.

Let us see now what a canonical hypergroup is.

Definition 2.7.11 We say that a hypergroup (H,◦) iscanonicalif

(2) it has a scalar identity (also called scalar unit), which means that

(3) every element has a unique inverse, which means that for allx∈H, there exists a uniquex −1 ∈H, such thate∈x◦x −1 ∩x −1 ◦x,

In a canonical hypergroup, the property of reversibility indicates that if an element x belongs to the composition of y and z (denoted as y◦z), there exist unique inverses y −1 and z −1 such that z is in the composition of y −1 and x, and y is in the composition of x and z −1 Additionally, the identity element of a canonical hypergroup is singular; for instance, if e represents a scalar identity and e 0 is the identity of the hypergroup (H,◦), it follows that e is contained within the composition of e and e 0, which results in {e 0 }.

Some interesting examples of a canonical hypergroup is the following ones (see [9]).

(1) Let C(n) = {e0, e1,ã ã ã , ek(n)}, wherek(n) = n/2 if n is an even natural number andk(n) = (n−1)/2 ifnis an odd natural number.

For alles, et of C(n), definees◦et={ep, ev},wherep= min{s+ t, n−(s+t)},v=|s−t| Then (C(n),◦) is a canonical hypergroup.

(2) Let (S, T) be a projective geometry, i.e., a system involving a setS of elements calledpointsand a setT of sets of points called lines, which satisfies the following postulates:

• Any lines contains at least three points;

• Two distinct points a, bare contained in a unique line, that we shall denote byL(a, b);

• Ifa, b, c, d are distinct points and L(a, b)∩L(c, d)6=∅, then L(a, c)∩L(b, d)6=∅.

Letebe an element which does not belong toSand letS 0 =S∪{e}.

We define the following hyperoperation onS 0 :

•For all different pointsa, bofS, we considera◦b=L(a, b)\{a, b};

•Ifa∈Sand any line contains exactly three points, leta◦a={e}, otherwisea◦a={a, e};

In what follows, we present some basic results of canonical hypergroups.

Theorem 2.7.13 If (H,◦) is a canonical hypergroup, then the following implication holds for allx, y, z, t ofH: x◦y∩z◦t6=∅ =⇒ x◦z −1 ∩t◦y −1 6=∅.

Proof Letu∈x◦y∩z◦t SinceH is reversible, we obtainu −1 ∈z −1 ◦t −1 , whence u◦u −1 ⊆ x◦y ◦z −1 ◦t −1 If e is an identity of H, we obtain e∈(x◦z −1 )◦(t◦y −1 ) −1 Hence there exists an elementv∈x◦z −1 ∩t◦y −1

Theorem 2.7.14 A commutative hypergroup is canonical if and only if it is a join space with a scalar identity.

Proof Suppose that (H,◦) is a canonical hypergroup For alla, bofH we havea/b=a◦b −1 Then the implication =⇒follows by the above theorem.

The uniqueness of an element's inverse can be demonstrated by considering the scalar identity, e If e is in the intersection of a◦b and a◦c, it follows that a belongs to the intersection of e/b and e/c This leads to the conclusion that e◦c and e◦b are not empty, thus establishing that b and c are equal to a's inverse Additionally, the reversibility of H is confirmed by noting that a is in the relation b◦c if and only if b is in a/c From the condition e being in b◦b −1, we derive that b is in e/b −1, which implies that the intersection of a◦b −1 and e◦c is not empty, indicating that c is in a◦b −1.

Theorem 2.7.15 If(H,◦)is a join space andN is a closed subhypergroup ofH, then the quotient(H/JN,⊗)is a canonical hypergroup, where for all a, bof H/JN, we havea⊗b={c|c∈a◦b}.

To establish the validity of the hyperoperation ⊗, we must demonstrate that for any elements a1 and a2 in N, and any x in H, if z belongs to a1◦x, then there exists a w in a2◦x such that z is related to w under the operation JN Given a1JN a2, we can find m, n in N and v in H, ensuring that v is an element of both a1◦m and a2◦n If z is in a1◦x, it follows that a1 is in the intersection of z/x and v/m, leading to the conclusion that the intersection of z◦N and v◦x is non-empty Consequently, this implies that there exists a w in a2◦x such that zJNw, confirming that the hyperoperation is well-defined.

The hyperoperation ⊗ is clearly defined, establishing that (H, ◦) functions as a join space Consequently, (H/JN, ⊗) also qualifies as a join space Additionally, N serves as a scalar identity for (H/JN, ⊗) According to the aforementioned theorem, this leads us to conclude that (H/JN, ⊗) is recognized as a canonical hypergroup.

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Polygroups 83

Weak Polygroups 139

Combinatorial Aspects of Polygroups 163

Tiêu đề	Polygroup Theory And Related Systems
Trường học	World Scientific Publishing Co. Pte. Ltd.
Thể loại	book
Năm xuất bản	2013
Thành phố	Singapore

Định dạng
Số trang	209
Dung lượng	2,45 MB