Quiz your maths: do the uniformly continuous functions on the line form a ring?

TL;DR

This paper investigates the algebraic structures of function lattices on the line and other metric spaces, revealing conditions under which they form rings and identifying cases where they do not.

Contribution

It demonstrates that the lattice of Lipschitz functions on a normed space is isomorphic to its bounded sublattice only in one-dimensional spaces and explores the existence of $f$-ring structures.

Findings

Lattice of Lipschitz functions is isomorphic to bounded functions only in 1D spaces.

Lipschitz and uniformly continuous function lattices can carry hidden $f$-ring structures.

Some metric spaces' uniformly continuous functions do not support unital $f$-ring structures.

Abstract

The paper deals with the interplay between boundedness, order and ring structures in function lattices on the line and related metric spaces. It is shown that the lattice of all Lipschitz functions on a normed space $E$ is isomorphic to its sublattice of bounded functions if and only if $E$ has dimension one. The lattice of Lipschitz functions on $E$ carries a "hidden" $f$-ring structure with a unit, and the same happens to the (larger) lattice of all uniformly continuous functions for a wide variety of metric spaces. An example of a metric space whose lattice of uniformly continuous functions supports no unital $f$-ring structure is provided.