Locally solid convergences and order continuity of positive operators

TL;DR

This paper explores the properties of locally solid convergence structures in vector lattices, characterizing order convergence, and examining the order continuity of positive operators and sublattices.

Contribution

It generalizes results on unbounded modifications, characterizes order convergence as the strongest locally solid convergence, and links order continuity of operators to their restrictions on sublattices.

Findings

Order convergence is the strongest locally solid convergence with monotone nets converging to extremums.

Positive operator is order continuous iff it is uo-continuous.

On continuous function spaces, uo convergence is weaker than compact open convergence under certain conditions.

Abstract

We consider vector lattices endowed with locally solid convergence structures, which are not necessarily topological. We show that such a convergence is defined by the convergence to $0$ on the positive cone. Some results on unbounded modification which were only available in partial cases are generalized. Order convergence is characterized as the strongest locally solid convergence in which monotone nets converge to their extremums (if they exist). We partially characterize sublattices on which the order convergence is the restriction of the order convergence of the ambient lattice. We show that homomorphism is order continuous iff it is uo-continuous. Uo convergence is characterized independently of order convergence. We show that on the space of continuous function uo convergence is weaker than the compact open convergence iff the underlying topological space contains a dense locally compact subspace. For a large class of convergences we prove that a positive operator is order continuous if and only if its restriction to a dense regular sublattice is order continuous, and that the closure of a regular sublattice is regular with the original sublattice being order dense in the closure. We also present an example of a regular sublattice of a locally solid topological vector lattice whose closure is not regular.