# Homomorphisms on Lattices of Continuous Functions

Homomorphisms on Lattices of Continuous Functions Positivity 12 (2008), 341–362 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1385-1292/020341-22, published online January 11, 2008 Positivity DOI 10.1007/s11117-007-2114-6 Homomorphisms on Lattices of Continuous Functions F´ elix Cabello S´ anchez Mathematics Subject Classiﬁcation (2000). 11H56. Keywords. Lattice of continuous functions, homomorphism, representation. 1. Introduction This paper deals with lattices of continuous functions and their homomorphisms, with emphasis on isomorphisms. As usual, we write C(X) for the lattice of all real-valued continuous func- tions on a topological space X with the order induced by that of R,thatis, f ≤ g meaning f (x) ≤ g(x) for all x ∈ X. The sublattice of bounded functions is denoted C (X). Until further notice X and Y will denote compact Hausdorﬀ spaces. Sup- pose we are given an isomorphism T : C(Y ) → C(X), that is, bijection satisfying T (f ∨ g)= Tf ∨ Tg and (this is equivalent for bijections) T (f ∧ g)= Tf ∧ Tg. What can be said about T ? In particular, how to represent it? We emphasize that T is not assumed to be linear. As far as I know, these problems were ﬁrst considered by Kaplansky in his venerable oldies [16] and [17]. In http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Homomorphisms on Lattices of Continuous Functions

, Volume 12 (2) – Jan 11, 2008
22 pages

/lp/springer_journal/homomorphisms-on-lattices-of-continuous-functions-7YQDXteUo4
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-007-2114-6
Publisher site
See Article on Publisher Site

### Abstract

Positivity 12 (2008), 341–362 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1385-1292/020341-22, published online January 11, 2008 Positivity DOI 10.1007/s11117-007-2114-6 Homomorphisms on Lattices of Continuous Functions F´ elix Cabello S´ anchez Mathematics Subject Classiﬁcation (2000). 11H56. Keywords. Lattice of continuous functions, homomorphism, representation. 1. Introduction This paper deals with lattices of continuous functions and their homomorphisms, with emphasis on isomorphisms. As usual, we write C(X) for the lattice of all real-valued continuous func- tions on a topological space X with the order induced by that of R,thatis, f ≤ g meaning f (x) ≤ g(x) for all x ∈ X. The sublattice of bounded functions is denoted C (X). Until further notice X and Y will denote compact Hausdorﬀ spaces. Sup- pose we are given an isomorphism T : C(Y ) → C(X), that is, bijection satisfying T (f ∨ g)= Tf ∨ Tg and (this is equivalent for bijections) T (f ∧ g)= Tf ∧ Tg. What can be said about T ? In particular, how to represent it? We emphasize that T is not assumed to be linear. As far as I know, these problems were ﬁrst considered by Kaplansky in his venerable oldies [16] and [17]. In

### Journal

PositivitySpringer Journals

Published: Jan 11, 2008

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