On hollowness in multiplicative lattices

Abstract

We extend strongly hollow and completely strongly hollow ideals from commutative rings to multiplicative lattices. We characterize these elements through residuals and localizations at maximal elements, and study them in semisimple, Gelfand, Prüfer, and weak r-lattices. Applications include criteria for quasi-locality, descriptions in weak Noether lattices, and representations of multiplicative lattices by completely strongly hollow elements.