* Transitivity of the ε m-relation on ( m-idempotent ) hyperrings

On a general hyperring, there is a fundamental relation, denoted γ∗, such that the quotient set is a classical ring. In a previous paper, the authors de ned the relation εm on general hyperrings, proving that its transitive closure εm is a strongly regular equivalence relation smaller than the γ∗-relation on some classes of hyperrings, such that the associated quotient structure modulo εm is an ordinary ring. Thus, on such hyperrings, εm is a fundamental relation. In this paper, we discuss the transitivity conditions of the εm-relation on hyperrings and m-idempotent hyperrings.


Introduction
The quotient set has played an important role in the algebraic hyperstructures theory since its beginning for at least two reasons. The rst one concerns the motivation of the de nition of hypergroup, very well pointed out by F. Marty in his pioneering paper on hypergroups from 1934. It is well known that the quotient of a group G by an arbitrary subgroup H of G is a group if and only if H is a normal subgroup, while Marty showed that the quotient structure G H is always a hypergroup. More generally, as Vougiouklis proved in [1], if one factorizes the group G by any partition S of G, then the quotient G S is an H v -group (i.e. a reproductive hypergroupoid satisfying the weak associativity). Secondly, the quotient set represents the bridging element between the classical algebraic structures and the corresponding hyperstructures, as mentioned in [2]. The rst step in this direction was made by Koskas [3], when he used the β-relation and its transitive closure β * to obtain a group (as a quotient structure of a hypergroup modulo β * ). Later on, the study of this correspondence between classical structures and hyperstructures with similar behaviour has been extended and new equivalence relations have been de ned and called fundamental relations. They are the smallest strongly regular relations de ned on a hyperstructure such that the quotient set is a classical structure, having similar properties. If on a (semi)hypergroup one considers the β * -relation, then the quotient set is a (semi)group. Besides, the quotient set modulo the γ * -relation, introduced by Freni [4], is a commutative (semi)group. Similarly, other fundamental relations have been de ned on hypergroups in order to obtain nilpotent groups [5], engel groups [6], or solvable groups [7]. The same approach was used also for ring-like hyperstructures. It started in 1991, when Vougiouklis [1] de ned the γ-relation on a general hyperring R (addition and multiplications are both hyperoperations) such that the quotient R γ * is a ring. Even if they are denoted in the same way (this could create confusion for the new readers of the algebraic hyperstructure theory, while it is already accepted for the researchers of this eld), the fundamental relation de ned by Freni [4] on semihypergroups is di erent by the fundamental relation γ de ned by Vougiouklis [1] on hyperrings. Later on, the α * -relation [8] has been introduced to obtain a commutative ring. More recently, other fundamental relations have been de ned obtaining Boolean rings [9] or commutative rings with identity [10] as associated quotient structures. We end this brief recall of the fundamental relations with those in hypermodule theory, where, for example, the θfundamental relation [11] leads to commutative modules by the same method of factorization.
The authors of this note proposed in [12] a new perspective of the study of fundamental relations on hyperstructures. The γ * -relation de ned on a general hyperring R is the smallest strongly regular relation such that the quotient R γ * is a ring. The paper [12] deals with the question: Under which conditions can a fundamental relation smaller than γ be de ned on a general hyperring, such that its transitive closer behaves similar to γ * ? To answer to this question, the ε m -relation was de ned on a special class of (semi)hyperrings, such that ε m ⊊ γ and the quotient structure modulo ε * m is an ordinary (semi)ring. Moreover, on m-idempotent hyperrings it was proved that ε * m = γ * . In this paper, we study the transitivity property of the ε m -relation on general hyperrings. First, we introduce the notion of m-complete parts based on the ε m -relation and investigate their properties, which help us to show that ε m is transitive on m-idempotent hyper elds.

Regular and fundamental relations on hyperstructures
In this section we review some basic de nitions and properties regarding fundamental relations on general hyperrings. For further details, the readers are referred to [2], [10,[13][14][15]17].
In the above de nition, if (R, +) is a semihypergroup, then (R, +, ⋅) is called a semihyperring. A nonempty subset I of a hyperring (R, +, ⋅) is a hyperideal, if (I, +) is a subhypergroup of (R, +) and, for all x ∈ I and r ∈ R, we have r ⋅ x ∪ x ⋅ r ⊆ I.
We recall that a subhypergroup A of (R, ⋅) is said to be invertible on the left (on the right), if x ∈ A ⋅ y (x ∈ y ⋅ A), then y ∈ A ⋅ x (y ∈ x ⋅ A), for all x, y ∈ R. A subhypergroup is invertible, if it is invertible on the left and on the right. Moreover, a subhypergroup B of (R, ⋅) is called closed on the left (on the right), if x ∈ a ⋅ y (x ∈ y ⋅ a) implies that a ∈ B, for every a ∈ R and x, y ∈ B. We say B is closed, if it is closed on the left and on the right. It is easy to see that every invertible subhypergroup of (R, ⋅) is closed.
Let ρ be an equivalence relation on a hypergroup (H, ○). For A, B ⊆ H, AρB means that, for all x ∈ A there exists y ∈ B such that xρy, and for all v ∈ B there exists u ∈ A such that uρv. Moreover, AρB means that for all x ∈ A and for all y ∈ B, we have xρy. Accordingly, an equivalence relation ρ on a hypergroup (H, ○) is called The main role of the (strongly) regular relations on hypergroups is re ected by the following result. An equivalence relation ρ is (strongly) regular on a hyperring (R, +, ⋅), if it is (strongly) regular with respect to both hyperoperations " + " and " ⋅ ". One example of strongly regular relation on (semi)hyperrings is the γ-relation de ned by Vougiouklis in [1] as follows. Let (R, +, ⋅) be a (semi)hyperring and x, y ∈ R. Then xγy if and only if {x, y} ⊆ u, where u is a nite sum of nite products of elements of R. In other words, xγy if and only if {x, y} ⊆ j∈J i∈I j z i , for some nite sets of indices J and I j and elements z i ∈ R. Let γ * be the transitive closure of γ, that is xγ * y if and only if there exist the elements z , . . . , z n+ ∈ R, with z = x and z n+ = y, such that z i γz i+ , for i ∈ { , . . . , n}. In [1] it was shown that γ * is the smallest strongly regular relation on a hyperring R such that the quotient (R γ * , ⊕, ⊙) is a classical ring with the operations de ned as: is called the fundamental ring obtained by the factorization with the γ * -relation.

The ε m -relation on hyperrings
In [12] the authors de ned on (semi)hyperrings a new relation, denoted by ε m , smaller than the γ-relation, and which is not transitive in general. Thus they found some conditions for the transitivity of the ε m -relation on hyperrings. In this section we recall its de nition and main properties. Let (R, +, ⋅) be a semihyperring and select a constant m, where Now, let (R, +, ⋅) be a hyperring such that (R, ⋅) is commutative and the following implication holds: for all B, A , . . . , A n ⊆ R. Accordingly with Theorems . and . in [12], on a hyperring R satisfying condition (2), the relation ε * m is the smallest strongly regular equivalence relation such that the quotient set R ε * m is a ring, thus it is a fundamental relation on R. Besides, we note that relation (2) is valid if and only if, for all The next result provides su cient conditions for the transitivity of the relation ε m .
We end this section emphasizing the fact that if the hyperring (R, +, ⋅) does not satisfy condition (2), then the relation ε m is not transitive, while its transitive closure ε * m is not strongly regular on (R, ⋅) [12].

Transitivity of the relation ε m on m-idempotent hyper elds
Since the conditions in Theorem 3.1 are not immediate, we aim to nd some particular hyperrings, where the relation ε m is transitive. For doing this, we will rst de ne the concept of m-complete part and then we will prove that ε m is transitive on m-idempotent hyper elds.
The main role of the complete parts of a semihypergroup, introduced by Koskas [3] and very well recalled by Antampou s et al. in the survey [2], is played in nding the β * class of each element. In particular, a nonempty subset A of a semihypergroup (H, ⋅) is called a complete part of H if, for any nonzero natural number n and any elements a , . . . , a n of H, the following implication holds: In other words, the complete part A absorbs every hyperproduct containing at least one element of A. In particular, for any element x ∈ A, the class β * (x) is a complete part of H. Moreover, the intersection of all complete parts of H containing A is called the complete closure of A in H, denoted by C(A). Besides, β * (x) = C(x), for any x ∈ H.
As already mentioned before, Vougiouklis [16] de ned the relation γ on a hyperring R, proving that its transitive closure γ * is the smallest strongly regular relation de ned on R such that the quotient R γ * is a ring. Later on Mirvakili et al. [19] studied the transitivity property of this relation, introducing the notion of complete part on hyperrings as follows: a nonempty subset M of a hyperring R is a complete part if, for any natural number n, any i = , , . . . , n, any natural number k i and arbitrary elements z i , . . . , z ik i ∈ R, we have Now we will extend these de nitions to the case of hyperrings, aiming to prove that the class ε * (x) of an element x in the hyperring R is an m-complete part of R. Proof. Since ρ is a strongly regular relation on R, it follows that the quotient R ρ is a ring (with the addition "⊕" and the multiplication "⊙"). Let a ∈ R and ρ(a) ∩ n i= z m i ≠ ∅, for arbitrary elements z , . . . , z n ∈ R. Hence, there exists y ∈ n i= z m i such that ρ(y) = ρ(a). Consider the strong homomorphism π ∶ R → R ρ de ned by π(x) = ρ(x), for all x ∈ R, where R ρ is a ring (a trivial hyperring). Thus, which implies that n i= z m i ⊆ ρ(a). This completes the proof.
For a nonempty subset A of a (semi)hyperring R, denote Moreover, for any x ∈ R and any natural number n, for simplicity we denote K m n ({x}) = K m n (x).

Lemma 4.4. For any nonempty subset A of a hyperring R, the set K m (A) is an m-complete part of R.
Proof. Let Then, for some a ′ ∈ A and n ≥ , . This completes the proof.
In the following we will give an equivalent description of the relation ε * m on hyperrings, using the notion of m-complete part. First we will prove some properties of the m-complete parts.
, for all x ∈ R and n ≥ .
Proof. We prove it by induction on "n". For n = , we have Now, suppose that K m n− (K m (x)) = K m n (x) and take an arbitrary element y ∈ K m n (K m (x)). Then there exist z , . . . , z n ∈ R such that y ∈ Hence, K m n (K m (x)) ⊆ K m n+ (x). Similarly, we have K m n+ (x) ⊆ K m n (K m (x)), that completes the proof. . Suppose now that x ∈ K m n− (y) if and only if y ∈ K m n− (x) and take x ∈ K m n (y); then there exist z, z , . . . , z n ∈ R such that x ∈ n i= z m i , z ∈ K m n− (y) and z ∈ n i= z m i . Hence, by induction procedure, y ∈ K m n− (z) and z ∈ K m (x), which implies that y ∈ K m n− (K m (x)) = K m n (x) by Lemma 4.8. Similarly, y ∈ K m n (x) implies that x ∈ K m n (y).
De ne on a hyperring R the relation θ as follows: xθy if and only if x ∈ K m (y), for all x, y ∈ R. Proof. For all x ∈ R, we have x ∈ K m (x) ⊆ K m (x). Hence, θ is re exive. Now, let x ∈ K m (y), for x, y ∈ R. By Theorem 4.5, there exists n ≥ such that x ∈ K m n (y), which implies that y ∈ K m n (x), by Lemma 4.9. Then, y ∈ K m n (x) ⊆ K m (x). Similarly, the converse is valid. Thus, θ is symmetric. Moreover, let xθy and yθz for x, y, z ∈ R. Hence, x ∈ C m (y) and y ∈ C m (z). Let A be an m-complete part of R containing z. Since y ∈ C m (z) and C m (z) ⊆ A, it follows that y ∈ A. Hence, C m (y) ⊆ A and thus x ∈ A. Therefore, x ∈ ⋂ z∈A A = C m (z) = K m (z) and so xθz.
Now, we recall that a hyperring (R, +, ⋅) is said to be a hyper eld, if (R, ⋅) is a hypergroup. Moreover, a strong homomorphism from a hyperring (R, +, ⋅) to a hyperring (S, ⊕, ⊙) is a map f ∶ R → S such that f (x + y) = f (x) ⊕ f (y) and f (x ⋅ y) = f (x) ⊙ f (y), for all x, y ∈ R. Considering the ε m relation on R, it can be seen that the map ϕ m ∶ R → R ε * m is a strong homomorphism. In the following we will consider R a hyper eld satisfying relation (2) (this is a crucial assumption in the proofs of the next results) such that R ε * m has a unit element denoted by R ε * m . Set ω m We will state some properties of the m-complete parts of hyper elds satisfying relation (2). Proof. Let x ∈ ϕ − m (ϕ m (A)). Then there exists y ∈ A such that ϕ m (x) = ϕ m (y). Since (R, ⋅) is a hypergroup, it follows that there exists t ∈ R such that x ∈ y ⋅ t , which implies that ). This completes the proof. Moreover, notice that, for two subsets A and B of the hyper eld R such that one of them is an m-complete part of R (assume that A is so), we have (A ⋅ B) ⋅ ω m R = (A ⋅ ω m R ) ⋅ B = A ⋅ B, by Corollary 4.14. Hence, A ⋅ B is an m-complete part of R, by Corollary 4.14. Proof. Let A be an m-complete part of R such that (A, ⋅) is a subhypergroup of (R, ⋅). Take x ∈ A ⋅ y for x, y ∈ R.
Then, A is invertible on the left. Similarly, we can show that A is invertible on the right. Therefore, A is invertible, and by consequence it is also closed. Proof. We know that ω m R is an m-complete part of R. Let x ∈ ω m R . For all t, y ∈ ω m R , we have ϕ m (x) = R ε * m = ϕ m (t) ⊙ ϕ m (y) = ϕ m (t ⋅ y). Hence, x ∈ ϕ − m (ϕ m (t ⋅ ω m )) = C m (t ⋅ ω m R ) = t ⋅ ω m R , and so ω m Then ω m R = t ⋅ ω m R , for all t ∈ ω m R , which implies that ω m R is a subhypergroup of (R, ⋅). Thus, ⋂

A∈S Cm (R)
A ⊆ ω m R . Now, let A ∈ S C m (R). By Corollary 4.14, A = A ⋅ ω m R . Hence, for every x ∈ ω m R , there exist a, b ∈ A such that a ∈ b ⋅ x, and so a ∈ A ⋅ x. By Theorem 4.15, A is invertible, and therefore x ∈ A ⋅ a ⊆ A. Then ω m R ⊆ A, and thus ω m R ⊆ ⋂

A∈S Cm (R)
A. Hence, the proof is complete.
We recall that a hyperring (R, +, ⋅) is said to be m-idempotent ([12]) if there exists a constant m, ≤ m ∈ N, such that x ∈ x m , for all x ∈ R.