NEUTRAL ELEMENTS, FUNDAMENTAL

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International Journal of Algebra and Computation Vol. 19, No. 4 (2009) 567–583 c World Scientific Publishing Company

NEUTRAL ELEMENTS, FUNDAMENTAL RELATIONS AND n-ARY HYPERSEMIGROUPS

B. DAVVAZ∗,§ , W. A. DUDEK†,¶ and S. MIRVAKILI‡ ∗Department

of Mathematics, Yazd University, Yazd, Iran

†Institute

of Mathematics and Computer Science Wroclaw University of Technology, Wyb. Wyspia´ nskiego 27 50-370 Wroclaw, Poland ‡Department of Mathematics Payame Noor University, Yazd, Iran §[email protected] ¶[email protected]

Received 2 January 2009 Revised 26 April 2009 Communicated by J. Meakin The main tools in the theory of n-ary hyperstructures are the fundamental relations. The fundamental relation on an n-ary hypersemigroup is defined as the smallest equivalence relation so that the quotient would be the n-ary semigroup. In this paper we study neutral elements in n-ary hypersemigroups and introduce a new strongly compatible equivalence relation on an n-ary hypersemigroup so that the quotient is a commutative n-ary semigroup. Also we determine some necessary and sufficient conditions so that this relation is transitive. Finally, we prove that this relation is transitive on an n-ary hypergroup with neutral (identity) element. Keywords: n-ary operation; n-ary hyperoperation; n-ary hypersemigroup; fundamental equivalence relation. 2000 Mathematics Subject Classification: 08A60, 08A02, 20N20

1. Introduction Hyperstructure theory was first studied by Marty in 1934 [20]. This theory has been studied in the following decades and nowadays by many mathematicians, for example see [2, 3, 5, 7]. A recent book [4] contains a wealth of applications. There are applications to the following subjects: geometry, hypergraphs, binary relations, combinatorics, codes, cryptography, probability, etc. The fundamental relation β (resp. β ∗ ) was introduced on hypergroups by Koskas [18] and was studied mainly by Corsini [1] and Vougiouklis [23–25]. Also, the fundamental equivalence relation γ ∗ was studied on hypergroups by Freni [15, 16], Davvaz and Karimian [5, 7]. 567

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B. Davvaz, W. A. Dudek & S. Mirvakili

Hyperalgebras and power algebras are pairs (A; (fi )i∈I ) consisting of a set A and an indexed or non-indexed set of operations fi : A × · · · × A → P ∗ (A) in the first and fi : A×· · ·×A → P(A) in the second case. Here P(A) is the power set of A and P ∗ (A) = P(A)\∅. The general theory of hyperalgebras, poweralgebras, hypercoalgebras and power co-algebras can be studied as application of (F1 , F2 )-systems where F1 and F2 are appropriate set-valued functors [9]. There are applications in several branches of mathematics and in computer science. For instance, hyperalgebras are used to prove that any non-deterministic automaton is equivalent to a deterministic one. n-ary groups and n-ary semigroups are algebras with one nary operation which is associative and invertible (in the first case) in a generalized sense. The idea of investigations of n-ary algebras seems to be going back to Kasner’s lecture [17] at the 53rd annual meeting of the American Association of the Advancement of Science in 1904. But the first paper concerning the theory of nary groups was written (under inspiration of Emmy Noether) by D¨ ornte in 1928 (see [10]). Since then many papers concerning various n-ary algebras have appeared in the literature, for example see [13, 22]. The concept of an n-ary hypergroup is defined by Davvaz and Vougiouklis in [6], which is a generalization of the concept of hypergroup in the sense of Marty and a generalization of an n-ary group, too. In n-ary hypersemigroup theory, strongly compatible equivalence relations play a role analogous to congruences in n-ary semigroup theory. Indeed, it is known (see [6]) that if ρ is a strongly compatible equivalence on an n-ary hypersemigroup (H, f ), then we can define an n-ary operation f /ρ on the quotient set H/ρ such that (H/ρ, f /ρ) is an n-ary semigroup. If ρ is a relation on a set H, we denote ρ∗ as the transitive closure of ρ. Davvaz and Vougiouklis [6] introduced the relation β on an n-ary hypersemigroup H such that β ∗ is the smallest strongly compatible equivalence relation such that the quotient (H/β ∗ , f /β ∗ ) is an ordinary n-ary semigroup, see also [19]. Also, Pelea and Purdea in [21] defined a relation α on a multialgebra. In this paper we define a certain relation γ on an n-ary hypersemigroup, which will be defined in Sec. 4 and prove that γ ∗ is the smallest strongly compatible equivalence relation such that the quotient (H/γ ∗ , f /γ ∗ ) is a commutative n-ary semigroup. Also, we find some necessary and sufficient conditions so that the relation γ is transitive, that is, γ = γ ∗ . Finally, we prove that in a cancellative n-ary hypergroup containing at least one weak neutral element the relation γ is transitive.

2. Basic Definitions and Facts In this section we shall first explain what is meant by an algebraic hypersystem and then give several examples of familiar algebraic hypersystems and discuss some of their properties. These examples show that different algebraic hypersystems may have several common properties. This observation provides a motivation for the study of abstract algebraic hypersystems.

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Let H be a non-empty set and f be a mapping f : H × · · · × H → P ∗ (H), where H appears n ≥ 2 times and P ∗ (H) is the collection of all non-empty subsets of H. Then f is called an n-ary hyperoperation and n is called the arity of this hyperoperation. An algebraic system (H, f ), where f is an n-ary hyperoperation defined on H, is called an n-ary hypergroupoid or an n-ary hypersystem. Since we can identify the set {x} with the element x, any n-ary groupoid is an n-ary hypergroupoid. We shall use the following abbreviated notation: the sequence xi , xi+1 , . . . , xj will be denoted by xji . For j < i, xji is the empty symbol. In this convention f (x1 , . . . , xi , yi+1 , . . . , yj , zj+1 , . . . , zn ) j n , zj+1 ). In the case when yi+1 = · · · = yj = y the last will be written as f (xi1 , yi+1 (j−i)

n expression will be written in the form f (xi1 , y , zj+1 ). Similarly, for subsets A1 , . . . , An of H we define f (An1 ) = f (A1 , . . . , An ) = {f (xn1 ) | xi ∈ Ai , i = 1, . . . n}.

Moreover, if for all xn1 ∈ H the set f (xn1 ) is singleton, then f is called an n-ary operation and (H, f ) is called an n-ary groupoid. If m = k(n − 1) + 1, then the m-ary hyperoperation h given by k(n−1)+1

h(x1

k(n−1)+1

) = f (f (· · · (f (f (xn1 ), x2n−1 n+1 ), . . .), x(k−1)(n−1)+2 ) k

will be denoted by f(k) . In certain situations, when the arity of m does not play a crucial role, or when it will differ depending on additional assumptions, we write f(·) to mean f(k) for some k = 1, 2, . . . . Obviously f (A1 , A2 , . . . , An ) = {f (an1 )|ai ∈ Ai }. In the case A1 = · · · = Ai = A, Ai+1 = · · · = An = B we will write f (Ai , B n−i ). An n-ary hyperoperation f is called weakly (i, j)-associative if j−1 n+i−1 ), x2n−1 , f (xn+j−1 ), x2n−1 f (xi−1 1 , f (xi j n+i ) ∩ f (x1 n+j ) = ∅,

and (i, j)-associative if j−1 n+j−1 n+i−1 ), x2n−1 ), x2n−1 f (xi−1 1 , f (xi n+i ) = f (x1 , f (xj n+j ),

holds for fixed 1 ≤ i < j ≤ n and all x1 , x2 , . . . , x2n−1 ∈ H. If the above condition is satisfied for all i, j ∈ {1, 2, . . . , n}, then we say that f is weakly associative (associative, respectively). An n-ary hypergroupoid with the (weakly) associative operation is called an n-ary hypersemigroup (Hv -semigroup, respectively). n An n-ary hypersemigroup (H, f ) in which for all ai−1 1 , ai+1 , b ∈ H and 1 ≤ i ≤ n the relation n b ∈ f (ai−1 1 xi , ai+1 )

has a solution xi ∈ H is called an n-ary hypergroup.

(1)

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n Observe that this condition can be reformulated as f (ai−1 1 , H, ai+1 ) = H. In the case when (H, f ) is an n-ary semigroup, Eq. (1) has the form: n b = f (ai−1 1 , xi , ai+1 ).

(2)

An n-ary semigroup in which the last equation has a solution for any 1 ≤ i ≤ n and n all ai−1 1 , ai+1 , b ∈ H is an n-ary group. One can prove (see [22]) that in this case the solution is unique. Moreover, as it is observed in [22], an n-ary semigroup is an n-ary group if (2) is solved at the place i = 1 and i = n or at the place 1 < i < n. In [11, 12, 14] one can find other interesting characterizations of n-ary groups. It is clear that for n = 2 we obtain a (binary) hypersemigroup, hypergroup, semigroup and group, respectively. Note that n-ary hypergroups can be characterized in a similar way as n-ary groups. Below we present two such characterizations inspired by results presented in [22] and [12] (see also [13]). Proposition 2.1. An n-ary hypersemigroup (H, f ) is an n-ary hypergroup if and only if for all an1 , b ∈ H, Eq. (1) is solved at the place i = 1 and i = n or at the place 1 < i < n. Proof. Indeed, if it is solved at the place i = 1 and i = n, then for all an1 , b ∈ H , xn ). Hence for there exists x1 , xn ∈ H such that b ∈ f (x1 , an2 ) and x1 ∈ f (an−1 1 arbitrary i = 2, . . . , n − 1 we have n−1 , xn ), an2 ) = f (ai−1 , xn , ai2 ), ani+1 ), b ∈ f (f (an−1 1 1 , f (ai

which means that for 1 < i < n, Eq. (1) is solved by xi ∈ f (an−1 , xn , ai2 ). i Conversely, if (1) is solved at some place 1 < i < n, then for all an1 , b ∈ H there exists xi ∈ H such that i+1 i−1 n n n n b ∈ f (ai−1 1 , xi , f (ai+1 , a2 ), ai+2 ) = f (f (a1 , xi , ai+1 ), a2 ). n So, (1) for i = 1 is solved by x1 ∈ f (ai−1 1 , xi , ai+1 ). Similarly n−1 i−1 n−1 n n , f (ai−1 b ∈ f (ai−2 1 , f (ai−1 , a1 ), xi , ai+1 ) = f (a1 1 , xi , ai+1 )) n proves that (1) for i = n is solved by xn ∈ f (ai−1 1 , xi , ai+1 ).

Proposition 2.2. An n-ary hypersemigroup (H, f ) is an n-ary hypergroup if and only if for some a ∈ H and all b, c ∈ H there exists x, y ∈ H such that (n−2)

(n−2)

b ∈ f (c, a , x) ∩ f (y, a , c).

(3)

Proof. If for some a ∈ H and all b, c ∈ H there are x, y ∈ H such that (3) holds, then also for some a ∈ H and all b, an1 ∈ H there are x , y ∈ H such that (n−2)

(n−2)

b ∈ f (f (an1 ), a , x ) ∩ f (y , a , f (an1 )).

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(n−2)

Thus b ∈ f (an−1 , f (an , a , x )) ∩ f (f (y , a , a1 ), an2 ). This means that Eq. (1) 1 (n−2)

at the place i = 1 is solved by x1 ∈ f (y , a , a1 ), at the place i = n by xn ∈ (n−2)

f (an , a , x ). So, by Proposition 2.1, (H, f ) is an n-ary hypergroup. The converse statement is obvious. Proposition 2.3. An n-ary hypersemigroup (H, f ) is an n-ary hypergroup if and only if for all a, b ∈ H there exists x, y ∈ H such that (n−1)

(n−1)

b ∈ f ( a , x) ∩ f (y, a ). The proof is analogous to the proof of the previous proposition. An n-ary hypergroupoid (H, f ) is (i, j)-commutative if n , aj , aj−1 f (an1 ) = f (ai−1 i i+1 , ai , aj+1 )

holds for all an1 ∈ H and fixed 1 ≤ i < j ≤ n. If this equation holds for all pairs (i, j) and all an1 ∈ H, then an n-ary hypergroupoid (H, f ) is called commutative. In this case f (a1 , . . . , an ) = f (aσ(1) , . . . , aσ(n) ) for all σ ∈ Sn and all an1 ∈ H. In the sequel, the expression f (aσ(1) , . . . , aσ(n) ) will be written in the abreviated σ(n) form as f (aσ(1) ). An n-ary hypergroupoid (H, f ) is said weak commutative, if σ(n) f (aσ(1) ) = ∅ σ∈Sn

an1

for all ∈ H. It is not difficult to verify that the following proposition is valid. Proposition 2.4. Any (1, n)-commutative n-ary hypersemigroup satisfies the identity 2n nn n1 n2 nn f (f (x1n 11 ), f (x21 ), . . . , f (xn1 )) = f (f (x11 ), f (x12 ), . . . , f (x1n )).

The last identity means that we obtain the same result if the hyperoperation f will be applied to the matrix [xij ]n×n (from left) to rows and (from right) to columns. An n-ary hypersemigroup (H, f ) is weakly i-cancellative, if there exist elements a2 , . . . , an ∈ H such that f (ai2 , x, ani+1 ) = f (ai2 , y, ani+1 ) implies

x=y

for all x, y ∈ H. If this implication is valid for all i = 1, 2, . . . , n, then we say that (H, f ) is weakly cancellative and elements a2 , . . . , an are called cancellable. An n-ary hypergroupoid in which this implication holds for all a2 , . . . , an ∈ H

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is called i-cancellative. An n-ary hypergroupoid which is i-cancellative for every i = 1, 2, . . . , n is called cancellative. The cancellativity in hypersemigroups is characterized by the following proposition proved in [8]. Proposition 2.5. An n-ary hypersemigroup is cancellative if and only if it is icancellative for i = 1 and i = n, or equivalently, if and only if it is i-cancellative for some 1 < i < n. Any n-ary group is cancellative but there are n-ary hypergroups which are not cancellative. Example 2.6. Let H = {a, b, c} be a ation f defined as follows:  x     b f (x, y, z) =  z     {a, c}

set with a commutative ternary hyperoperfor x = y = z, for x = y = z, for x = y, x = z, x = b, for x = y = b, z = b.

It is easy to see that (H, f ) is a commutative ternary hypergroup. It is not cancellative because f (a, c, b) = f (c, c, b) = b. Similarly, f (b, b, a) = f (b, b, c) = {a, c}, but a = c. 3. Neutral Elements An n-ary hypergroupoid (H, f ) has a weak neutral element (weak identity) if there exists an element e ∈ H such that (i−1)

(n−i)

x ∈ f ( e , x, e ) holds for all x ∈ H and all 1 ≤ i ≤ n. If for all x ∈ H and all 1 ≤ i ≤ n, we have (i−1)

(n−i)

x = f ( e , x, e ) then e is called a neutral element. Obviously, each neutral element is weak neutral but not conversely. Example 3.1. It is not difficult to see that the set H = {a, b, e} with the hyperoperation “◦” defined by the table ◦ e a b

e e a b

a b a b {e, b} {e, a} {e, a} {e, a}

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is a commutative hypersemigroup with the identity e. So, (H, f ), where f (x, y, z) = x ◦ y ◦ z, is a commutative ternary hypersemigroup and e is its neutral element. Elements a and b are weak neutral but they are not neutral. In the ternary hypergroup defined in Example 2.6 we have two neutral elements: a and c. The third element is weak neutral. Example 3.2. Let Z4 be the additive group of order 4 and let H = Z4 ∪ {θ} and xn1 ∈ Z4 . Define the commutative n-ary hyperoperation f as follows: • if (x1 + x2 + · · · + xn + 2) = 0 (mod 4) then f (xn1 ) = {0, θ}, • if (x1 + x2 + · · · + xn + 2) = 0 (mod 4) then (i)

(i)

f (xn1 ) = (x1 + x2 + · · · + xn + 2) (mod 4) and f ( θ , xni+1 ) = f ( 0 , xni+1 ). Then (H, f ) is a commutative n-ary hypergroup. For n = 3 it has two weak neutral elements 1 and 3, for n = 4 it has only one weak neutral element 2, for n = 5 it has no weak neutral elements. There are n for which the commutative n-ary hypergroup has no neutral element. Fixing in an n-ary hyperoperation f elements a2 , . . . , an−1 we obtain a binary , y). Choosing different elements a2 , . . . , an−1 we hyperoperation x ◦ y = f (x, an−1 2 obtain different binary operations. Such obtained hypergroupoids are called retracts of (H, f ). Obviously retracts of an n-ary hypersemigroup are hypersemigroups, retracts of an n-ary hypergroup are hypergroups. As a consequence of Proposition 2.2 we obtain the following characterization of n-ary hypergroups. Corollary 3.3. An n-ary hypersemigroup (H, f ) is an n-ary hypergroup if and only (n−2)

if there exists a ∈ H such that (H, ◦), where x ◦ y = f (x, a , y), is a hypergroup. Example 3.4. The ternary hypersemigroup (H, f ) defined in Example 3.1 gives three hypersemigroups: (H, ◦), (H, ), (H, ), where x ◦ y = f (x, e, y), x y = f (x, a, y) and x y = f (x, b, y). The first hypersemigroups has a neutral element and coincides with the hypersemigroup (H, ◦) defined in Example 3.1. The multiplication tables of hypersemigroups (H, ), (H, ) are as follows:

e

e a

e

a {e, b}

b {e, a}

e b

a b {e, a} {e, a}

a {e, b}

{e, a}

H

a {e, a}

H

H

b {e, a}

H

H

b {e, a}

H

H

It is clear that these three hypersemigroups are not isomorphic. Hypersemigroups (H, ) and (H, ) have no neutral elements but have two weak neutral elements. This example shows that retracts induced by weak neutral elements may not be isomorphic.

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Theorem 3.5. For a given n-ary hypersemigroup retracts induced by neutral elements are isomorphic. Proof. Let e1 , . . . , en−1 be neutral elements of an n-ary hypersemigroup (H, f ). (n−2)

Consider two retracts (H, ◦) and (H, ), where x ◦ y = f (x, e1 , y) and x y = , y). Then ϕ(x) = f (e1 , x, en−1 ) is an isomorphism of (H, ) onto (H, ◦). f (x, en−1 2 2 Indeed, (n−2)

), e1 , f (e1 , y, en−1 )) ϕ(x) ◦ ϕ(y) = f (f (e1 , x, en−1 2 2 (n−2)

= f (f (e1 , x, en−1 ), f ( e1 , e1 , y), en−1 ) 2 2 = f (e1 , f (x, en−1 , y), en−1 ) = ϕ(x y). 2 2 Hence ϕ is a homomorphism. It is also an endomorphism since for every z ∈ H there (n−2)

(n−2)

(n−2)

exists x = f ( e1 , z, en−1 , . . . , e2 ) ∈ H such that z = ϕ(x). Moreover, ϕ(x) = ) = f (e1 , y, en−1 ), whence multiplying this equality by ϕ(y) implies f (e1 , x, en−1 2 2 (n−2) e1

(n−2) (n−2)

(n−2)

(on left) and by en−1 , en−2 , . . . , e2 (on right), we obtain x = y. This means that ϕ is an isomorphism.

Corollary 3.6. If an n-ary hypersemigroup derived from (H, ◦) has a neutral element, then all its retracts are isomorphic with (H, ◦). Theorem 3.7. If an n-ary hypersemigroup (H, f ) has a weak neutral element, then there exists a binary hypersemigroup (H, ◦) and its (n − 2) weak endomorphisms ϕ2 , . . . , ϕn−1 such that f (xn1 ) ⊆ x1 ◦ ϕ2 (x2 ) ◦ ϕ3 (x3 ) ◦ · · · ◦ ϕn−1 (xn−1 ) ◦ xn

(4)

for all xn1 ∈ H. Proof. Let e be a weak neutral element of (H, f ). Then (H, ◦), where x ◦ y = (n−2)

(k−1)

(n−k)

f (x, e , y), is a binary hypersemigroup and ϕk (x) = f ( e , x, e ) is its weak endomorphism, i.e., ϕk (x ◦ y) ⊆ ϕk (x) ◦ ϕk (y) for every k = 1, 2, . . . , n. (n−1)

(n−1)

(n−1)

(n−1)

Moreover x ∈ f ( e , x). Hence f ( e , x) ⊆ f ( e , f ( e , x)). Consequently (n−1)

(n−1)

x ∈ f ( e , f ( e , x)). Whence in view of the associativity of f we obtain (n−1)

(n−1)

(n−1)

(n−1)

(n−1)

f (xn1 ) ⊆ f (x1 , f ( e , x2 ), f ( e , f ( e , x3 )), f ( e , f ( e , x4 )), (n−1)

(n−1)

(n−1)

. . . , f ( e , f ( e , xn−1 )), f ( e , xn )) (n−2)

(n−2)

= f(·) (x1 , e , f (e, x2 , e ), (n−2)

(n−2)

(n−3)

(n−2)

(2)

(n−3)

e , f ( e , x3 , e ), (2)

e , . . . , e , f ( e , xn−2 , e ),

(n−2)

(n−2)

(3)

e , f ( e , x4 ,

e , f ( e , xn−1 , e),

= x1 ◦ ϕ2 (x2 ) ◦ ϕ3 (x3 ) ◦ · · · ◦ ϕn−1 (xn−1 ) ◦ xn , which completes the proof.

(n−2)

(n−2)

(n−4)

e ),

e , xn )

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Corollary 3.8. If an n-ary hypersemigroup (H, f ) has a weak neutral element, then there exists a binary hypersemigroup (H, ◦) and its weak endomorphism ϕ such that f (xn1 ) ⊆ x1 ◦ ϕ(x2 ) ◦ ϕ(x3 ) ◦ · · · ◦ ϕ(xn−1 ) ◦ xn

(5)

for all xn1 ∈ H. Proof. Let (H, ◦) and ϕk be as in the previous proof. Then ϕk (x) ⊆ ϕn−1 (x) ◦ e = ϕ(x) for every x ∈ H and k = 2, . . . , n − 1. Since ϕ is a weak endomorphism of (H, ◦), from (4) we obtain (5). For n-ary hypersemigroups having neutral element the following stronger results is proved in [8]. Theorem 3.9. An n-ary hypersemigroup (H, f ) has a neutral element if and only if there exists a binary hypersemigroup (H, ◦) such that f (xn1 ) = x1 ◦ x2 ◦ x3 ◦ · · · ◦ xn

(6)

for all xn1 ∈ H. An n-ary hypersemigroup (H, f ) having the above form is called derived from a hypersemigroup (H, ◦). (n−2)

The hyperoperation “◦” used in (6) has the form x ◦ y = f (x, e , y), where e is a neutral element of (H, f ). This means that the following corollary is true. Corollary 3.10. Any (1, n)-commutative n-ary hypersemigroup containing a neutral element is a commutative n-ary hypersemigroup derived from a commutative binary hypersemigroup. Proposition 3.11. If e1 and e2 are neutral elements of a ternary hypersemigroup (H, f ) then ({e1 , e2 }, f ) is a ternary group contained in (H, f ). Proof. Indeed, by the assumption, f (ei , ei , ej ) = f (ei , ej , ei ) = f (ej , ei , ei ) = ej for ei , ej ∈ {e1 , e2 }. So, ({e1 , e2 }, f ) is a ternary semigroup. Moreover, as it is not difficult to see, the solutions of the equations f (x, ei , ej ) = f (ej , y, ei ) = f (ei , ej , z) = ek , where ei , ej , ek ∈ {e1 , e2 }, have the form x = y = z = f (ej , ek , ei ) ∈ {e1 , e2 }. This means that ({e1 , e2 }, f ) is a ternary group contained in (H, f ). Theorem 3.12. The set of all neutral elements of a (1, n)-commutative n-ary hypersemigroup (H, f ) forms an n-ary group.

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Proof. Assume that the set E of neutral elements of (H, f ) is non-empty. Then, in view of Proposition 2.4 or Theorem 3.9, for an arbitrary element x ∈ H, e1 , . . . , en ∈ H and 1 ≤ k ≤ n we have f (f (en1 ), . . . , f (en1 ), x, f (en1 ), . . . , f (en1 )) (k−1)

(n−k)

= f (f (en1 ), . . . , f (en1 ), f ( ek , x, ek ), f (en1 ), . . . , f (en1 )) (k−1)

(n−k)

(k−1)

(n−k)

(k−1)

(n−k)

= f (f ( e1 , ek , e1 ), . . . , f ( ek−1 , ek , ek−1 ), f ( ek , x, ek ), (k−1)

(n−k)

(k−1)

(n−k)

f ( ek+1 , ek , ek+1 ), . . . , f ( en , ek , en )) (k−1)

(n−k)

= f ( ek , x, en ) = x, which proves that the set E is closed under the hyperoperation f . Thus (E, f ) is an n-ary semigroup contained in (H, f ). Since for every k = 1, 2, . . . , n, the equation , y, enk+1 ) = ek f (ek−1 1 is solved by (n−2)

(n−2)

(n−2)

(n−2)

y = f(n−1) ( ek−1 , . . . , e1 , ek , en , . . . , ek+1 ) ∈ E, (E, f ) is an n-ary group. Corollary 3.13. The set of all neutral elements of a commutative n-ary hypersemigroup (H, f ) forms an n-ary group. 4. Fundamental Relation Definition 4.1. If (H, f ) is an n-ary hypersemigroup and ρ an equivalence relation on H, we set AρB ⇐⇒ aρb for all a ∈ A, b ∈ B. Definition 4.2. Let (H, f ) be an n-ary hypersemigroup. An equivalence relation ρ on H is strongly compatible if a1 ρb1 , . . . , an ρbn =⇒ f (a1 , . . . , an )ρf (b1 , . . . , bn ). If ρ is a strongly compatible relation on an n-ary hypersemigroup (H, f ), then the quotient (H/ρ, f /ρ) is an n-ary semigroup such that f/ρ(ρ(a1 ), . . . , ρ(an )) = ρ(x) for all x ∈ f (a1 , . . . , an ), where a1 , . . . an ∈ H and ρ(ai ) is an equivalence class of ai .

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Example 4.3. Let (H, f ) be the commutative ternary hypergroup defined in Example 2.6. The relation ρ = {(a, a), (b, b), (c, c), (a, c), (c, a)} is strongly compatible on (H, f ) and H/ρ = {ρ(a), ρ(b)} with a ternary operation f/ρ defined by ρ(x) for ρ(x) = ρ(y) = ρ(z), f/ρ(ρ(x), ρ(y), ρ(z)) = ρ(z) for ρ(x) = ρ(y), and ρ(x) = ρ(z) is a commutative ternary group isomorphic to the ternary group (Z2 , g), where g(x, y, z) = (x + y + z)(mod 2). This ternary group has two neutral elements.

Definition 4.4. Let γ be the transitive closure of the relation γ = k≥0 γk , where γ0 is the diagonal relation on H, i.e., γ0 = {(x, x) | x ∈ H} and γk for k > 0 is defined as follows: σ(m) x ∈ f(k) (am 1 ) and y ∈ f(k) (aσ(1) ) xγk y ⇐⇒ for some am 1 ∈ H, m = k(n − 1) + 1 and σ ∈ Sm . The following example proves that the relation γ is not transitive in general. Example 4.5. Assume that H = {a, b, c, d, . . .} has at least four elements and define on H an n-ary (n ≥ 3) hyperoperation f by putting H \ {a, b} for x1 = · · · = xn = a, n f (x1 ) = H \ {a, c} otherwise . n+i−1 2n−1 ), xn+i ) = H \ {a, c} for all x2n−1 ∈H Since f (xn1 ) = a, we have f (xi−1 1 , f (xi 1 and i = 1, 2, . . . , n. So, (H, f ) is an n-ary hypersemigroup. Moreover, f(k) (xm 1 ) = ∈ H, m = k(n − 1) + 1. Thus bγd and dγc, but not bγc. H \ {a, c} for all xm 1

Definition 4.6. By a commutative fundamental equivalence relation on an n-ary hypersemigroup (H, f ) we mean the smallest equivalence relation γ ∗ on H such that the quotient (H/γ ∗ , f /γ ∗ ) is a commutative n-ary semigroup. Theorem 4.7. The fundamental relation γ ∗ is the transitive closure of the relation . γ, i.e., γ ∗ = γ Proof. First we show that the quotient H/ γ is an n-ary semigroup. The n-ary operation f / γ on H/ γ is defined in the usual manner: (xn )) = { γ (y) | y ∈ f ( γ (x1 ), . . . , γ (xn ))} f / γ ( γ (x1 ), . . . , γ for all x1 , . . . , xn ∈ H. (x1 ), . . . , bn ∈ γ (xn ). Then Suppose b1 ∈ γ   (∃x11 , . . . , x1m1 +1 ) such that x11 = b1 , x1m1 +1 = x1 , x1 ⇐⇒ x1i1 ∈ f(k1 ) (at11 ) and x1i1 +1 ∈ f(k1 ) (aσ(t1 ) ) for all 1 ≤ i1 ≤ m1 b1 γ σ(1)   and some at11 ∈ H, t1 = k1 (n − 1) + 1 and some σ ∈ St1 .

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.....................................................................  (∃xn1 , . . . , xnmn +1 ) such that xn1 = bn , xnmn +1 = xn ,   n) bn γ xn ⇐⇒ xnin ∈ f(kn ) (at1n ) and xnin +1 ∈ f(k1 ) (aσ(t σ(1) ) for all 1 ≤ i1 ≤ mn   and some at1n ∈ H, tn = kn (n − 1) + 1 and some σ ∈ Stn . Therefore f (x1i1 , x21 , . . . , xn1 ) ⊆ f(k1 ) (at11 ) 1 ≤ i 1 ≤ m1 , σ(t ) 1 ≤ i 1 ≤ m1 , f (x1i1 +1 , x21 , . . . , xn1 ) ⊆ f(k1 ) (aσ(1)1 ) f (x1m1 +1 , x2i2 , . . . , xn1 ) ⊆ f(k2 ) (at12 ) 1 ≤ i 2 ≤ m2 , σ(t2 ) f (x1m1 +1 , x2i2 +1 , . . . , xn1 ) ⊆ f(k2 ) (aσ(1) ) 1 ≤ i 2 ≤ m2 , ....................................... ............ 1 ≤ i n ≤ mn , f (x1m1 +1 , x2m2 +1 , . . . , xnin ) ⊆ f(kn ) (at1n ) σ(tn ) f (x1m1 +1 , x2m2 +1 , . . . , xnin +1 ) ⊆ f(kn ) (aσ(1) ) 1 ≤ in ≤ mn . So, every element y ∈ f (x11 , x21 , . . . , xn1 ) = f (b1 , b2 , . . . , bn ) is equivalent to every element z ∈ f (x1m1 +1 , x2m2 +1 , . . . , xnmn +1 ) = f (x1 , x2 , . . . , xn ). Therefore γ (xn )) is singleton. So, we can write f / γ ( γ (x1 ), . . . , (xn )) = γ (y) f / γ ( γ (x1 ), . . . , γ for all y ∈ f ( γ (x1 ), . . . , γ (xn )). Moreover, since f is associative, also f / γ is associative and consequently, (H/ γ , f / γ ) is an n-ary semigroup. σ( n) ) implies aγb, (H/ γ , f / γ ) is commutative because a ∈ f (xn1 ) and b ∈ f (xσ(1) (xn )) = f / γ ( γ (xσ(1) ), . . . , γ (xσ(n) )) whence γ (a) = γ (b). Therefore f / γ ( γ (x1 ), . . . , γ (xn ) ∈ H/ γ. for all γ (x1 ), . . . , γ Now, let θ be an equivalence relation in H such that H/θ is a commutative n-ary semigroup. Denote by θ(a) the equivalence class of a. Then f/θ(θ(x1 ), . . . , θ(xn )) = θ(y) for all x1 , . . . , xn ∈ H and y ∈ f (θ(x1 ), . . . , θ(xn )). But also, for every σ ∈ Sn , x1 , . . . , xn ∈ H and Ai ⊆ θ(xi ), i = 1, . . . , n, we have f/θ(θ(x1 ), . . . , θ(xn )) = θ(f (xσ(1) , . . . , xσ(n) )) = θ(f (Aσ(1) , . . . , Aσ(n) )). σ(m)

Therefore θ(x) = θ(f(k) (aσ(1) )) for all k ≥ 0 and x ∈ f(k) (am 1 ). So, for every a ∈ H, x ∈ γ(a) implies x ∈ θ(a). But θ is transitively closed, so x∈γ (a) implies x ∈ θ(a). Hence, the relation γ is the smallest equivalence relation on H such that H/ γ is a commutative n-semigroup, i.e., γ = γ∗.

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5. Transitivity Conditions for γ In this section, we determine some necessary and sufficient conditions so that the relation γ is transitive. We consider the following definition: Definition 5.1. Let M be a non-empty subset of an n-ary hypersemigroup (H, f ). We say that M is a γ-part of (H, f ) if for every k ∈ N, m = k(n − 1) + 1 and z 1 , z2 , . . . , zm ∈ H σ(m) f(k) (z1m ) M = ∅ =⇒ f(k) (zσ(1) ) ⊆ M for every σ ∈ Sm . In particular f(k) (z1m ) M = ∅ implies f(k) (z1m ) ⊆ M. Proposition 5.2. For a non-empty subset M of an n-ary hypersemigroup (H, f ) the following conditions are equivalent: (1) M is a γ-part of (H, f ), (2) x ∈ M, xγy =⇒ y ∈ M, (3) x ∈ M, xγ ∗ y =⇒ y ∈ M. Proof. (1) ⇒ (2) If (x, y) ∈ H 2 is a pair such that x ∈ M and xγy, then x ∈ σ(m) f(k) (z1m ) M and y ∈ f(k) (zσ(1) ) for some z1 , . . . , zm ∈ H, m = k(n − 1) + 1 and σ(m)

some σ ∈ Sm . Since M is a γ-part of H, we have f(k) (zσ(1) ) ⊆ M. So, y ∈ M. (2) ⇒ (3) Let (x, y) ∈ H 2 be such that x ∈ M and xγ ∗ y. Then for some x = w0 , w1 , . . . , wp−1 , wp = y ∈ H we have x = w0 γw1 γ . . . γwp−1 γwp = y. Since x ∈ M, applying (2) p times, we obtain y ∈ M. (3) ⇒ (1) Let f(k) (z1m ) M = ∅. If x ∈ f(k) (z1m ) M, then obviously x ∈ M. σ(m)

Since for every σ ∈ Sm and every y ∈ f(k) (zσ(1) ) we have xγy, thus xγ ∗ y. This, by σ(m)

(3), implies y ∈ M. Therefore f(k) (zσ(1) ) ⊆ M . So, M is a γ-part of (H, f ). Corollary 5.3. γ ∗ (x) is a γ-part for each x ∈ H. For every element x of an n-ary hypersemigroup (H, f ) we define the following three sets: Tk (x) = {(x1 , . . . , xm ) ∈ H m | m = k(n − 1) + 1, x ∈ f(k) (xm 1 )}, σ(m) Pk (x) = {f(k) (xσ(1) ) | σ ∈ Sm , (x1 , . . . , xm ) ∈ Tk (x), m = k(n − 1) + 1}, Pσ (x) = Pk (x). k≥1

Lemma 5.4. Pσ (x) = {y ∈ H | xγy}. σ(m)

Proof. Indeed, xγy if and only if x ∈ f(k) (xm 1 ) and y ∈ f(k) (xσ(1) ) for some (x1 , . . . , xm ) ∈ H m and some σ ∈ Sm , or equivalently, if and only if y ∈ Pk (x) for some k ∈ N, i.e., if and only if y ∈ Pσ (x).

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Proposition 5.5. For an n-ary hypersemigroup (H, f ) the following conditions are equivalent: (1) γ is transitive, (2) γ ∗ (x) = Pσ (x) for every x ∈ H, (3) every Pσ (x) is a γ-part of H. Proof. (1) ⇒ (2) By Lemma 5.4, for every pair (x, y) of elements of H we have: y ∈ γ ∗ (x) ⇐⇒ xγ ∗ y ⇐⇒ xγy ⇐⇒ y ∈ Pσ (x). (2) ⇒ (3) According to Proposition 5.2, a non-empty subset M of H is a γ-part of (H, f ) if and only if it is a union of equivalence classes modulo γ ∗ . In particular, every equivalence class γ ∗ (x) = Pσ (x) is a γ-part of (H, f ). σ(m) (3) ⇒ (1) If xγy and yγz, then x ∈ f(k) (xm 1 ) and y ∈ f(k) (xσ(1) ) for some

σ (m )

x1 , . . . , xm ∈ H and σ ∈ Sm . Similarly y ∈ f(k ) (y1m ) and z ∈ f(k ) (yσ (1) ) for some Pσ (x) y1 , . . . , ym ∈ H and σ ∈ Sm . Since Pσ (x) is a γ-part of H, x ∈ f(k) (xm 1 ) σ(m) m implies f(k) (xσ(1) ) ⊆ Pσ (x). So, y ∈ Pσ (x). Thus y ∈ f(k ) (y1 ) Pσ (x), whence σ (m )

we obtain f(k ) (yσ (1) ) ⊆ Pσ (x). Therefore z ∈ Pσ (x), i.e., xγz. Let ϕ : H → H/γ ∗ be the canonical projection, i.e., ϕ(x) = γ ∗ (x) for every x ∈ H. Proposition 5.6. ϕ−1 (ϕ(M )) = M for any γ-part M of an n-ary hypersemigroup (H, f ). Proof. Obviously M ⊆ ϕ−1 (ϕ(M )) for every M ⊆ H. Moreover, if M is a γ-part of (H, f ) and x ∈ ϕ−1 (ϕ(M )), then there exists an element b ∈ M such that ϕ(x) = ϕ(b). Hence x ∈ γ ∗ (x) = γ ∗ (b) ⊆ M by Proposition 5.2. Thus ϕ−1 (ϕ(M )) ⊆ M which completes the proof. Proposition 5.7. If an n-ary hypersemigroup (H, f ) has a weak neutral element e, then f (Di−1 , M, Dn−i ) ⊆ ϕ−1 (ϕ(M )) for every i = 1, . . . , n, D = ϕ−1 (γ ∗ (e)) and an arbitrary non-empty subset M ⊆ H. Proof. If e is a weak neutral element of (H, f ), then, as it is not difficult to verify, ε = ϕ(e) = γ ∗ (e) is a neutral element of (H/γ ∗ , f /γ ∗ ). Moreover, for every x ∈ f (Di−1 , M, Dn−i ), there exist d1 , . . . , dn ∈ D and b ∈ M such that x ∈ (i−1)

(n−i)

n ∗ f (di−1 1 , b, di+1 ). So, ϕ(x) = f /γ ( ε , ϕ(b), ε ) = ϕ(b) ⊆ ϕ(M ). Therefore x ∈ ϕ−1 (ϕ(M )). Consequently, f (Di−1 , M, Dn−i ) ⊆ ϕ−1 (ϕ(M )).

Proposition 5.8. If an n-ary hypergroup (H, f ) is cancellative and has a weak neutral element e, then f (Di−1 , M, Dn−i ) = ϕ−1 (ϕ(M )) for D = ϕ−1 (γ ∗ (e)), i = 1, . . . , n and any non-empty subset M of H.

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Proof. For every x ∈ ϕ−1 (ϕ(M )) exists an element b ∈ M such that ϕ(x) = ϕ(b). Since (H, f ) is an n-ary hypergroup for every d ∈ D there exists a ∈ H such that (i−2)

(n−i)

x ∈ f (a, d , b, d ). So, (i−2)

(n−i)

(i−2)

(n−i)

ϕ(x) = f /γ ∗ (ϕ(a), ϕ(d), ϕ(b), ϕ(d)) = f /γ ∗ (γ ∗ (a), γ ∗ (e), γ ∗ (b), γ ∗ (e)) (i−1)

(n−i)

and ϕ(x) = ϕ(b) = γ ∗ (b) = f /γ ∗ (γ ∗ (e), γ ∗ (b), γ ∗ (e)). This, by the cancellativity, implies ϕ(a) = γ ∗ (e). So, a ∈ ϕ−1 (γ ∗ (e)) = D. Therefore x ∈ ϕ−1 (ϕ(M )) ⊆ f (Di−1 , M, Dn−i ). Proposition 5.7 completes the proof. Proposition 5.9. If an n-ary hypergroup (H, f ) contains a weak neutral element e, then x ∈ Pσ (e) and xγy imply y ∈ Pσ (e). σ(m)

Proof. Let x ∈ Pσ (e) and xγy. Then e ∈ f(k) (xm 1 ), x ∈ f(k) (xσ(1) ) and x ∈ σ (m )

f(k ) (y1m ), y ∈ f(k ) (yσ (1) ) for some x1 , . . . , xm , y1 , . . . , ym ∈ H and σ ∈ Sm , σ ∈ Sm . Since (H, f ) is an n-ary hypergroup, for every x ∈ H there exist x ∈ H (n−2)

such that e ∈ f ( e , x, x ). Thus (n−2)

(n−2)

(n−2)

(n−2)

t e ∈ f ( e , f ( e , x, e), x ) ⊆ f ( e , f ( e , f(k ) (y1m ), f(k) (xm 1 )), x ) = f(·) (a1 ),

where t = m + m + n − 1. Moreover (n−2)

(n−2)

(n−2)

y ∈ f ( e , y, e) ⊆ f ( e , y, f ( e , x, x )) (n−2)

(n−2)

σ (m ) σ(m) ⊆ f ( e , f(k ) (yσ (1) ), f ( e , f(k) (xσ(1) ), x )),

which means that y ∈ f(·) (bt1 ), where b1 , . . . , bt is a some permutation of elements a1 , . . . , at . Thus eγy, i.e., y ∈ Pσ (e). Theorem 5.10. In a cancellative n-ary hypergroup containing a weak neutral element we have γ ∗ = γ, i.e., the relation γ is transitive. Proof. Obviously xγy implies xγ ∗ y. We prove the converse. For this let xγ ∗ y. Then γ ∗ (x) = γ ∗ (y). Thus ϕ(x) = ϕ(y), whence x ∈ ϕ−1 (ϕ(y)). This, together with Proposition 5.8 for i = 1, D = ϕ−1 (γ ∗ (e)) and M = {y}, implies x ∈ f (, y, Dn−1 ). So, x ∈ f (y, dn2 ) for some dn2 ∈ D. Since γ ∗ (dj ) = ϕ(dj ) ⊆ γ ∗ (e), we have γ ∗ (dj ) = γ ∗ (e) which means that for every dj there exists kj such that dj γkj e. Therefore jm

jσ (m )

e ∈ f(kj ) (xj1 j ) and dj ∈ f(kj ) (xjσjj (1)j ) for some xj1 , . . . , xjmj ∈ H and σj ∈ Smj . Thus 2σ (m )

3σ (m )

nσ (m )

x ∈ f (y, dn2 ) ⊆ f (y, f(k2 ) (x2σ22 (1)2 ), f(k3 ) (x3σ33 (1)3 ), . . . , f(kn ) (xnσnn (1)n )) and (n−1)

3m3 nmn 2 y ∈ f (y, e ) ⊆ f (y, f(k2 ) (x2m 21 ), f(k3 ) (x31 ), . . . , f(kn ) (xn1 )).

This means that xγy. Therefore γ ∗ = γ.

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The relation γ is a generalization of the relation β defined in [6] and [19] as the set-theoretic union of relations βk , k ≥ 0, where β0 is the diagonal relation and m xβk y ⇐⇒ {x, y} ⊆ f(k) (am 1 ) for some a1 ∈ H, m = k(n − 1) + 1.

It is clear that in a commutative n-ary hypersemigroup γ = β, and consequently γ ∗ = β ∗ , where β ∗ denotes the smallest equivalence relation such that H/β ∗ is an n-ary semigroup. The last identity holds also in some non-commutative n-ary hypergroups, for example in n-ary hypersemigroups admitings some “partial commutativity”. Theorem 5.11. γ ∗ = β ∗ in all weak commutative n-ary hypersemigroups. Proof. Indeed, if (H, f ) is a weak commutative n-ary hypersemigroup, then σ(n) f (xσ(1) ) = ∅ for all xn1 ∈ H. σ∈Sn

Therefore there exists a ∈

σ∈Sn

σ(n)

f (xσ(1) ) which yield that

β ∗ (a) = f /β ∗ (β ∗ (xσ(1) ), . . . , β ∗ (xσ(n) )) for every σ ∈ Sn , that is H/β ∗ is a commutative n-ary semigroup. Since β ⊆ γ, we get β ∗ ⊆ γ ∗ . But, according to the definition, γ ∗ is the smallest equivalence relation such that H/γ ∗ is a commutative n-ary semigroup, so β ∗ = γ ∗ . Acknowledgment The authors are highly grateful to referees for their valuable comments and suggestions for improving the paper. References [1] P. Corsini, Prolegomena of Hypergroup Theory, 2nd edn. (Aviani editor, 1993). [2] P. Corsini, Hypergraphs and Hypergroups, Algebra Universalis 35 (1996) 548–555. [3] P. Corsini and V. Leoreanu, Hypergroups and binary relations, Algebra Universalis 43 (2000) 321–330. [4] P. Corsini and V. Leoreanu, Applications of Hyperstructures Theory, Advanced in Mathematics (Kluwer Academic Publisher, 2003). [5] B. Davvaz and M. Karimian, On the γn∗ -complete hypergroups, European J. Combinatorics, 28 (2007) 86–93. [6] B. Davvaz and T. Vougiouklis, n-ary hypergroups, Iranian J. Sci. Technology, Transaction A 30(A2) (2006) 165–174. [7] B. Davvaz, Applications of the γ ∗ -relation to polygroups, Comm. Algebra 35 (2007) 2698–2706. [8] B. Davvaz, W. A. Dudek and T. Vougiouklis, A generalization of n-ary algebraic systems, Comm. Algebra 37 (2009) 1248–1263. [9] K. Denecke and S. L. Wismath, Universal Algebra and Coalgebra (World Scientific, 2009).

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[10] W. D¨ ornte, Untersuchungen u ¨ber einen verallgemeinerten Gruppenbegriff, Math. Z. 29 (1928) 1–19. [11] W. A. Dudek, Remarks on n-groups, Demonstratio Math. 13 (1980) 165–181. [12] W. A. Dudek, Varieties of polyadic groups, Filomat 9 (1995) 657–674. [13] W. A. Dudek and K. Glazek, Around the Hossz´ u-Gluskin theorem for n-ary groups, Discrete Math. 308 (2008) 4861–4876. [14] W. A. Dudek, K. Glazek and B. Gleichgewicht, A note on the axioms of n-groups, Colloquia Math. Soc. J. Bolyai 29 “Universal Algebra”, Esztergom (Hungary) 1977, 195–202. (North-Holland, Amsterdam, 1982.) [15] D. Freni, A new characterization of the derived hypergroup via strongly regular equivalences, Comm. Algebra 30(8) (2002) 3977–3989. [16] D. Freni, Strongly transitive geometric spaces: applications to hypergroups and semigroups theory, Comm. Algebra 32(3) (2004) 969–988. [17] E. Kasner, An extension of the group concept, Bull. Amer. Math. Soc. 10 (1904) 290–291. [18] M. Koskas, Groupoides, Demi-hypergroupes et hypergroupes, J. Math. Pures et Appl. 49 (1970) 155–192. [19] V. Leoreanu-Fotea and B. Davvaz, n-hypergroups and binary relations, European J. Combinatories 29(5) (2008) 1207–1218. [20] F. Marty, Sur uni generalization de la notion de group, 8th Congress Math. Scandienaves, Stockholm, (1934) 45–49. [21] C. Pelea and I. Purdea, Multialgebras, universal algebras and identities, J. Aust. Math. Soc. 81 (2006) 121–139. [22] E. L. Post, Polyadic groups, Trans. Amer. Math. Soc. 48 (1940) 208–350. [23] T. Vougiouklis, Groups in hypergroups, Annals Discrete Math. 37 (1988) 459–468. [24] T. Vougiouklis, The fundamental relation in hyperrings. The general hyperfield, Proc. Fourth Int. Congress on Algebraic Hyperstructures and Applications (AHA 1990) (World Scientific, 1991) 203–211. [25] T. Vougiouklis, Hyperstructures and Their Representations (Hadronic Press, Inc., USA, 1994).