On z-elements of multiplicative lattices

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.