Prime filters of hyperlattices

Abstract

The purpose of this paper is the study of prime ideals and prime filters in hyperlattices. I-filter and the filter generated by a ∈ L are introduced. Moreover, we introduce dual distributive hyperlattices, and I-filter in dual distributive hyperlattices. Some properties of hyperlat- tices are studied and the relationship between prime ideals and prime filters in hyperlattices is discussed.

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.…”

Distributive and Dual Distributive Elements in Hyperlattices

“…Also the definitions of I−filter and the filter generated by a ∈ L are recalled. More details can be found in [2]. In the sequel, L denotes a bounded hyperlattice.…”

“…In this paper, we continue and deepen the ideas from [2], in particular we introduce distributive (dual distributive) elements, dual distributive hyperlattices, and I−filter in dual distributive hyperlattices. We give examples of the distributive elements and dual distributive elements in hyperlattices and we show that these concepts are independent.…”

“…Moreover, they defined a topology on the set of prime ideals of a distributive hyperlattice [21,22]. Ameri, Amiri-Bideshki, Saeid, Hoskova-Mayerova studied prime filters of hyperlattices in [2].…”

C-Γ-hyperideal Theory in Ordered Γ-semihypergroups

“…Ameri and Hoskova [20] studied the fuzzy continuous polygroup as a fuzzy polygroup with a continuous membership function. In [21], some properties of hyperlattices are studied and the relationship between prime ideals and prime filters in hyperlattices is discussed. Motivation of compatibility of orderings with hyperoperations can be found in [22,23].…”