## 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 PDF
- Proc. Amer. Math. Soc.
**147**(2019), 4301-4313 Request permission

## 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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## Additional Information

**Félix Cabello Sánchez**- Affiliation: Departamento de Matemáticas, UEx, and IMUEx, 06071-Badajoz, Spain
- Email: fcabello@unex.es
**Javier Cabello Sánchez**- Affiliation: Departamento de Matemáticas, UEx, and IMUEx, 06071-Badajoz, Spain
- Email: coco@unex.es
- Received by editor(s): July 24, 2018
- Received by editor(s) in revised form: December 18, 2018, and December 22, 2018
- Published electronically: June 27, 2019
- Additional Notes: This research was supported in part by DGICYT project MTM2016$\cdot$76958$\cdot$C2$\cdot$1$\cdot$P (Spain) and Junta de Extremadura programs GR$\cdot$15152 and IB$\cdot$16056.
- Communicated by: Stephen Dilworth
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4301-4313 - MSC (2010): Primary 46E05, 54C35
- DOI: https://doi.org/10.1090/proc/14531
- MathSciNet review: 4002543