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


Authors: Félix Cabello Sánchez and Javier Cabello Sánchez
Journal: Proc. Amer. Math. Soc. 147 (2019), 4301-4313
MSC (2010): Primary 46E05, 54C35
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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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

DOI: https://doi.org/10.1090/proc/14531
Keywords: Uniformly continuous function, Lipschitz function, lattice homomorphism, $f$-ring structure
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$⋅$76958$⋅$C2$⋅$1$⋅$P (Spain) and Junta de Extremadura programs GR$⋅$15152 and IB$⋅$16056.
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2019 American Mathematical Society