A counterexample regarding the relatively uniform completion of a principal ideal

TL;DR

This paper provides a counterexample demonstrating that the relative uniform completion of a principal ideal in a vector lattice can be strictly smaller than the ideal generated by the element in the completed lattice, challenging assumptions about uniform convergence.

Contribution

It introduces a specific counterexample that clarifies the limitations of relative uniform completion in vector lattices, highlighting a nuanced difference from previous beliefs.

Findings

Counterexample shows strict containment in relative uniform completion

Challenges previous assumptions about uniform convergence in vector lattices

Highlights the need for careful analysis of ideal completions

Abstract

We present a counterexample related to relative uniform convergence, showing that, in general, the relatve uniform completion of the principal ideal of a vector lattice E generated by an element x is stricly contained in the ideal generated by x in the relatively uniform completion of E.