On hollowness in multiplicative lattices

TL;DR

This paper extends the concepts of hollow ideals from commutative rings to multiplicative lattices, exploring their properties, characterizations, and applications in various lattice types.

Contribution

It introduces and analyzes the notions of strongly hollow and completely strongly hollow elements within multiplicative lattices, providing new characterizations and explicit descriptions.

Findings

Characterizations of strongly hollow elements in Gelfand, semi-simple, and Prüfer lattices.

Descriptions of quasi-local weak r-lattices via hollow elements.

Representation results for multiplicative lattices using hollow elements.

Abstract

The aim of this article is to extend the notions of strongly hollow and completely strongly hollow ideals of commutative rings to multiplicative lattices. We investigate their basic structural properties and prove several characterizations in terms of localizations at maximal elements and the behaviour of residuals. In particular, we study properties of strongly hollow elements in various types of $C$-lattices: Gelfand, semi-simple, and Pr\"ufer. We also provide characterizations of quasi-local weak $r$-lattices by completely strongly hollow elements. Furthermore, we give characterization of strongly hollow elements in Noether lattices, and obtain explicit descriptions in Pr\"ufer and $r$-lattices. Using strongly hollow and completely strongly hollow elements, we obtain representabilities of multiplicative lattices.