Locally solid convergences and order continuity of positive operators

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 on the ambient lattice. We prove that a 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. & COPY; 2023 Published by Elsevier Inc.

Automated summary

This paper focuses on vector lattices endowed with locally solid convergence structures, which may not be topological. The convergence is defined by the convergence to 0 on the positive cone. Several results on unbounded modification are generalized. Order convergence is characterized as the strongest locally solid convergence where monotone nets converge to their extremums. The partial characterization of sublattices is explored in terms of order convergence. The study also investigates the relationship between order continuity and uo continuity, and gives a characterization of uo convergence independently of order convergence. The paper concludes with results on positive operators, closure of regular sublattices, and an example of a regular sublattice with a non-regular closure.