Composition of locally solid convergences

TL;DR

This paper investigates the composition and properties of locally solid convergences in vector lattices, establishing associativity, exploring idempotency, and addressing minimal Hausdorff convergences, thus advancing the theoretical understanding of these structures.

Contribution

It introduces new results on the composition, associativity, and idempotency of locally solid convergences, and characterizes minimal Hausdorff convergences in atomic vector lattices.

Findings

Proves associativity of the composition of locally solid convergences.

Shows adherence of an ideal equals its closure under idempotent convergence.

Characterizes the existence of minimal Hausdorff locally solid convergences as equivalent to atomicity.

Abstract

We carry on a more detailed investigation of the composition of locally solid convergences as introduced in [BCTvdW24], as well as the corresponding notion of idempotency considered in [Bil23]. In particular, we study the interactions between these two concepts and various operations with convergences. We prove associativity of the composition and show that the adherence of an ideal with respect to an idempotent convergence is equal to its closure. Some results from [KT18] about unbounded modification of locally solid topologies are generalized to the level of locally solid idempotent convergences. A simple application of the composition allows us to answer a question from [BCTvdW24] about minimal Hausdorff locally solid convergences. We also show that the weakest Hausdorff locally solid convergence exists on an Archimedean vector lattice if and only if it is atomic.