Order continuity and regularity on vector lattices and on lattices of continuous functions

TL;DR

This paper characterizes order continuous homomorphisms between Archimedean vector lattices, relates them to composition operators on continuous functions, and explores properties of sublattices in function spaces and Banach lattices.

Contribution

It provides new characterizations of order continuity and regularity in vector lattices and their sublattices, linking abstract properties to concrete function space operators.

Findings

Characterizations of order continuous vector lattice homomorphisms.

Reduction of proofs to composition operators on continuous functions.

Regularity properties of sublattices and their closures in Banach lattices.

Abstract

We give several characterizations of order continuous vector lattice homomorphisms between Archimedean vector lattices. We reduce the proofs of some of the equivalences to the case of composition operators between vector lattices of continuous functions, and so we obtain a characterization of order continuity of such operators. Motivated by this, we investigate various properties of the sublattices of the space $C\left(X\right)$, where $X$ is a Tychonoff topological space. We also obtain several characterizations of a regular sublattice of a vector lattice, and show that the closure of a regular sublattice of a Banach lattice is also regular.