A note on "New fundamental relation of hyperrings"

In the theory of hyperrings, fundamental relations make a connection between hyperrings and ordinary rings. Commutative fundamental rings and the fundamental relation α which is the smallest strongly regular relation in hyperringswere introduced by Davvaz and Vougiouklis (2007). Recently, another strongly regular relation named θ on hyperrings has been studied by Ameri and Norouzi (2013). Ameri and Norouzi proved that θ is the smallest strongly regular relation such that R/θ is a commutative ring. In this paper, we show that θ ≠ α and θ is not the smallest strongly regular relation. Moreover, we show that some results of Ameri and Norouzi do not hold. © 2014 Elsevier Ltd. All rights reserved. 1. Hyperrings and fundamental relations (R, +, ·) is a hyperring if + and · are two hyperoperations such that (R, +) is a hypergroup, (R, ·) is a semihypergroup and the hyperoperation ‘‘ · ’’ is distributive over the hyperoperation ‘‘+’’, which means that for all x, y, z of R we have: x · (y + z) = x · y + x · z and (x + y) · z = x · z + y · z. We call (R, +, ·) a hyperfield if (R, +, ·) is a hyperring and (R, ·) is a hypergroup. There are different types of hyperrings. If only the addition + is a hyperoperation and the multiplication · is a usual operation, then we say that R is an additive hyperring. A special case of this type is the Krasner hyperring. We E-mail addresses: [email protected] (S. Mirvakili), [email protected] (B. Davvaz), [email protected] (V. Leoreanu Fotea). 1 Tel.: +98 3518121127. http://dx.doi.org/10.1016/j.ejc.2014.04.004 0195-6698/© 2014 Elsevier Ltd. All rights reserved. S. Mirvakili et al. / European Journal of Combinatorics 41 (2014) 258–261 259 recall the following definition from [3]. A Krasner hyperring is an algebraic structure (R, +, ·) which satisfies the following axioms: (1) (R, +) is a canonical hypergroup, i.e., x + (y + z) = (x + y) + z for all x, y, z ∈ R; x + y = y + x for all x, y ∈ R; there exists 0 ∈ R such that 0 + x = x for all x ∈ R; for every x ∈ R there exists a unique element x ∈ R such that 0 ∈ x + x (we shall write −x for x and we call it the opposite of x); z ∈ x + y implies that y ∈ −x + z and x ∈ z − y; (2) Relating to the multiplication, (R, ·) is a semigroup having zero as a bilaterally absorbing element; (3) The multiplication is distributive with respect to the hyperoperation +. An equivalence relation ρ is called strongly regular over a hyperring R, if the quotient R/ρ is a ring. For a hyperring R, we denote δR = {(x, x)|x ∈ R} and ∆R = R × R. At the fourth AHA congress [8] which took place in 1990, Vougiouklis introduced the concept of a fundamental relation on hyperrings, analyzed afterwards by himself and many other authors, for example see [4–6]. Remark 1. A relation ρ is the transitive closure of a binary relation ρ if (1) ρ is transitive, (2) ρ ⊆ ρ, (3) for any relation ρ , if ρ ⊆ ρ ′ and ρ ′ is transitive, then ρ ⊆ ρ , that is, ρ is the smallest relation that satisfies (1) and (2). Definition 1.1 ([8]). Let R be a hyperring. We define the relation Γ as follows: x Γ y ⇔ ∃n ∈ N, ∃ki ∈ N, ∃(xi1, . . . , xiki) ∈ R ki , 1 ≤ i ≤ n such that