Lattice isomorphisms between certain sublattices of continuous functions

2019/07/20 by Vahid Ehsani, Ehsani, Vahid, Fereshteh Sady +1
Mathematics · #46E05 #47B65 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Rings, Modules, and Algebras #math.FA #msc:46E05 #msc:47B65

paper · pdf · doi:10.48550/arxiv.1907.08786

arxiv created 2019/07/20 · openalex publication_date 2019/07/20 · arxiv updated 2019/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let C(X,I) be the lattice of all continuous functions on a compact Hausdorff space X with values in the unit interval I=[0,1]. We show that for compact Hausdorff spaces X and Y and (not necessarily contain constants) sublattices A and B of C(X,I) and C(Y,I), respectively, which satisfy a certain separation property, any lattice isomorphism φ: A \longrightarrow B induces a homeomorphism μ: Y \longrightarrow X. If, furthermore, A and B are closed under the multiplication, then φ has a representation φ(f)(y)=m_y(f(μ(y))), f∈ A, for all points y in a dense G_δ subset Y₀ of Y, where each m_y is a strictly increasing continuous bijection on I. In particular, for the case where X and Y are metric spaces and A and B are the lattices of all Lipschitz functions with values in I, the set Y₀ is the whole of Y.

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