Memorandum on multiplicative bijections and order

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

SEMIGROUP FORUM, cilt.79, sa.1, ss.193-209, 2009 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 79 Sayı: 1
  • Basım Tarihi: 2009
  • Doi Numarası: 10.1007/s00233-009-9152-2
  • Dergi Adı: SEMIGROUP FORUM
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.193-209
  • Anahtar Kelimeler: Semigroups of continuous functions, Homomorphism, Representation, HOMOMORPHISMS, LATTICES
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

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.