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

Cabello Sánchez, Félix; Cabello Sánchez, Javier

doi:10.1090/proc/14531

Quiz your maths: Do the uniformly continuous functions on the line form a ring?
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by Félix Cabello Sánchez and Javier Cabello Sánchez

Proc. Amer. Math. Soc. 147 (2019), 4301-4313

DOI: https://doi.org/10.1090/proc/14531

Published electronically: June 27, 2019

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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.

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