Composition of locally solid convergences

Eugene Bilokopytov

Published 2024-07-25Version 1

We carry on a more detailed investigation of the composition of locally solid convergences as introduced in \cite{ectv}, as well as the corresponding notion of idempotency considered in \cite{erz}. In particular, we study the interactions between these two concepts and various operations with convergences. Some results from \cite{kt} 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 \cite{ectv} 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.