Two Results on Fremlin's Archimedean Riesz Space Tensor Product

TL;DR

This paper investigates the conditions under which the Fremlin tensor product of Archimedean Riesz spaces is Dedekind {lpha}-complete and provides a counterexample related to ideals in such spaces.

Contribution

It characterizes Dedekind {lpha}-completeness of Fremlin tensor products and presents a novel example illustrating ideal behavior in these spaces.

Findings

Characterization of Dedekind {lpha}-completeness in tensor products

Counterexample of an ideal not preserved in tensor products

Insights into the structure of Archimedean Riesz spaces

Abstract

In this paper, we characterize when, for any infinite cardinal {\alpha}, the Fremlin tensor product of two Archimedean Riesz spaces is Dedekind {\alpha}-complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself.