Locally solid convergence structures

TL;DR

This paper develops a generalized theory of locally solid convergence structures in vector lattices, extending beyond traditional topological convergences to include important non-topological cases.

Contribution

It introduces a unifying framework for locally solid convergences, encompassing non-topological cases and modifications like unbounded, bounded, and Choquet structures.

Findings

Unified theory of locally solid convergences in vector lattices

Includes non-topological convergences such as unbounded and Choquet

Analyzes specific convergences within the new framework

Abstract

While there is a well developed theory of locally solid topologies, many important convergences in vector lattice theory are not topological. Yet they share many properties with locally solid topologies. Building upon the theory of convergence structures, we develop a theory of locally solid convergences, which generalize locally solid topologies but also includes many important non-topological convergences on a vector lattice. We consider some natural modifications of such structures: unbounded, bounded, and Choquet. We also study some specific convergences in vector lattices from the perspective of locally solid convergence structures.