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.