Distributive and Dual Distributive Elements in Hyperlattices

Abstract

In this paper we introduce and study distributive elements, dual distributive elements in hyperlattices, and prove that these elements forms ∧-semi lattice and ∨-semi hyperlattice, respectively. We use the properties of prime ideals and prime filters in hyperlattices. We will proceed to introduce the notion of dual distributive hyperlattices, I-filters and filters generated by dual distributive elements. Finally, the relationship between good homomorphisms and istributive (resp. dual distributive)elements in hyperlattices are investigated.

Correction to the Article: "An Introduction to the Theory of Hyperlattices"

“…What we have already said is about the definition of hyperlattice introduced by Konstantinidou and Mittas in [4] and used in [5][6][7] as well. Later some authors working on the subject, just after the Definition 1.1, they added: "Let A, B ⊆ L. Then define A ∨ B = {a ∨ b | a ∈ A, b ∈ B} and A ∧ B = {a ∧ b | a ∈ A, b ∈ B}" (see, for example [1,2]). But this, written in a wrong place (and not only), make the definition still unreadable.…”

“…(1) a ∧ a = a and a ∈ a ∨ a If in Definition 2.1 we add the property a ∈ a ∨ b ⇒ a ∧ b = b, then this definition is equivalent to Definition 1.1 but only if, for any nonempty subsets A and B of L, we define the A ∨ B and A ∧ B (there is no such a definition in [4][5][6][7]), preferable before the definition or in a correct way if it is after that (I mean, not as in [1][2]); and emphasize the fact that the element a should be identified by the singleton {a} if and when is convenient and no confusion is possible.…”

Weak Embeddable Hypernear-Rings

“…Moreover, when the additive part is a hypergroup and all the other properties related to the multiplication are conserved, we talk about a general hypernear-ring [8]. The distributivity property is important also in other types of hyperstructures, see e.g., [9]. A detailed discussion about the terminology related to hypernear-rings is included in [10].…”