On $z$-elements of multiplicative lattices
Themba Dube, Amartya Goswami
TL;DR
This work extends the theory of $z$-ideals from rings to $z$-elements in multiplicative lattices by leveraging $z$-closure operators. It introduces several distinguished $z$-elements ($z$-prime, $z$-semiprime, $z$-primary, $z$-irreducible, $z$-strongly irreducible) and establishes equivalences that decompose these notions into $z$-element plus a standard element, while relating closures to nuclei and identifying when $z$-elements are closed under finite products. A key contribution is the representation of any $z$-element as a finite meet of $z$-irreducible elements in a $z$-Noetherian lattice, along with structure results for minimal $z$-prime elements and the Noetherian topology on the $z$-prime spectrum. Collectively, the paper provides a robust lattice-theoretic framework that generalizes classical ideal-theoretic concepts to the study of $z$-elements and their closure operators, with explicit results on nuclei, pz-multiplicative lattices, and prime-type hierarchies.
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.