Uniformly closed sublattices of finite codimension

TL;DR

This paper characterizes uniformly closed sublattices and ideals of finite codimension in Archimedean vector lattices, showing their structure as intersections of codimension-one subspaces and relating ideals to sublattices.

Contribution

It extends existing results by providing a detailed structure theorem for uniformly closed sublattices and ideals of finite codimension, generalizing Kakutani's characterization.

Findings

Finite codimension sublattices are intersections of codimension-one sublattices.

Every codimension-n sublattice contains an ideal of codimension at most 2n.

In uniformly complete lattices, all finite codimension ideals are uniformly closed.

Abstract

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension $n$ contains a uniformly closed ideal of codimension at most $2n$. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of [AL90a,AL90b] and , as well as Kakutani's characterization of closed sublattices of $C(K)$ spaces.