Conditionally complete sponges: new results on generalized lattices

TL;DR

This paper introduces new tools for analyzing and constructing sponges, a generalization of lattices, on metric spaces and groups, and explores their properties on Hilbert spaces and higher dimensions.

Contribution

It provides characterization methods for sponges on metric spaces and groups, and introduces epigraph sponges and their properties on Hilbert spaces.

Findings

Characterization of sponges on metric spaces and groups

Introduction of epigraph sponges on Hilbert spaces

Generalization of hyperbolic sponge to higher dimensions

Abstract

Sponges were recently proposed as a generalization of lattices, focussing on joins/meets of sets, while letting go of associativity/transitivity. In this work we provide tools for characterizing and constructing sponges on metric spaces and groups. These are then used in a characterization of epigraph sponges: a new class of sponges on Hilbert spaces whose sets of left/right bounds are formed by the epigraph of a rotationally symmetric function. We also show that the so-called hyperbolic sponge generalizes to more than two dimensions.