Locally solid convergence structures

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.