Abstract
Let \(C(X,\mathbb{I})\) denote the semigroup of continuous functions from the topological space X to \(\mathbb{I}=[0,1]\) , equipped with the pointwise multiplication. The paper studies semigroup homomorphisms \(C(Y,\mathbb{I})\to C(X,\mathbb{I})\) , 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 \(T:\mathop{\mathrm{Lip}}(Y,\mathbb{I})\to\mathop{\mathrm{Lip}}(X,\mathbb{I})\) has the form Tf=f○τ, where τ:X→Y is a Lipschitz homeomorphism.
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Communicated by Rainer Nagel.
F. Cabello Sánchez and J. Cabello Sánchez are supported in part by DGICYT projects MTM2004-02635 and MTM2007-6994-C02-02.
J. Cabello Sánchez is supported in part by a grant of the UEx (Programa Propio–Acción 2).
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Cabello Sánchez, F., Cabello Sánchez, J., Ercan, Z. et al. Memorandum on multiplicative bijections and order. Semigroup Forum 79, 193–209 (2009). https://doi.org/10.1007/s00233-009-9152-2
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DOI: https://doi.org/10.1007/s00233-009-9152-2