  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.
