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Memorandum on multiplicative bijections and order
Date
2009-08-01
Author
Cabello Sanchez, Felix
Cabello Sanchez, Javier
ERCAN, ZAFER
Önal, Süleyman
Metadata
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Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License
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Let C(X, I) denote the semigroup of continuous functions from the topological space X to I = [0, 1], equipped with the pointwise multiplication. The paper studies semigroup homomorphisms C(Y, I) -> C(X, I), with emphasis on isomorphisms. The crucial observation is that, in this setting, homomorphisms preserve order, so isomorphisms preserve order in both directions and they are automatically lattice isomorphisms. Applications to uniformly continuous and Lipschitz functions on metric spaces are given. Sample result: if Y and X are complete metric spaces of finite diameter without isolated points, every multiplicative bijection T : Lip(Y, I) -> Lip(X, I) has the form Tf = f circle tau, where tau : X -> Y is a Lipschitz homeomorphism.
Subject Keywords
Algebra and Number Theory
URI
https://hdl.handle.net/11511/35250
Journal
SEMIGROUP FORUM
DOI
https://doi.org/10.1007/s00233-009-9152-2
Collections
Department of Mathematics, Article
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F. Cabello Sanchez, J. Cabello Sanchez, Z. ERCAN, and S. Önal, “Memorandum on multiplicative bijections and order,”
SEMIGROUP FORUM
, pp. 193–209, 2009, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/35250.