Memorandum on multiplicative bijections and order

Cabello Sanchez F., Cabello Sanchez J., ERCAN Z., ÖNAL S.

SEMIGROUP FORUM, vol.79, no.1, pp.193-209, 2009 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 79 Issue: 1
  • Publication Date: 2009
  • Doi Number: 10.1007/s00233-009-9152-2
  • Journal Name: SEMIGROUP FORUM
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.193-209
  • Keywords: Semigroups of continuous functions, Homomorphism, Representation, HOMOMORPHISMS, LATTICES
  • Middle East Technical University Affiliated: Yes

Abstract

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.