Lattices of slowly oscillating functions

TL;DR

This paper establishes that lattice isomorphisms between lattices of slowly oscillating functions imply coarsely equivalent homeomorphisms, leading to a Banach-Stone-like theorem and a characterization of linear lattice isomorphisms.

Contribution

It introduces a Banach-Stone-like theorem for lattices of slowly oscillating functions and characterizes linear lattice isomorphisms in this context.

Findings

Lattice isomorphisms induce coarsely equivalent homeomorphisms.

A Banach-Stone-like theorem is established for these lattices.

Linear lattice isomorphisms are characterized within this framework.

Abstract

We show that lattice isomorphisms between lattices of slowly oscillating functions on chain-connected proper metric spaces induce coarsely equivalent homeomorphisms. This result leads to a Banach-Stone-like theorem for these lattices. Furthermore, we provide a representation theorem that characterizes linear lattice isomorphisms among these lattices.