On $z$-elements of multiplicative lattices

TL;DR

This paper explores the properties and classifications of $z$-elements in multiplicative lattices, extending existing theories and providing characterizations and representations within specific lattice classes.

Contribution

It introduces new subclasses of $z$-elements, extends properties using $z$-closure operators, and characterizes lattices where $z$-elements are closed under finite products.

Findings

Characterization of lattices with $z$-elements closed under finite products.

Representation of $z$-elements via $z$-irreducible elements in $z$-Noetherian lattices.

Extension of properties of $z$-ideals to $z$-elements.

Abstract

The aim of this paper is to investigate further properties of $z$-elements in multiplicative lattices. We utilize $z$-closure operators to extend several properties of $z$-ideals to $z$-elements and introduce various distinguished subclasses of $z$-elements, such as $z$-prime, $z$-semiprime, $z$-primary, $z$-irreducible, and $z$-strongly irreducible elements, and study their properties. We provide a characterization of multiplicative lattices where $z$-elements are closed under finite products and a representation of $z$-elements in terms of $z$-irreducible elements in $z$-Noetherian multiplicative lattices.