@Jonathan, thank you for pointing me towards your post! I had a few questions as I was reading it and was wondering if you could clarify.

First you wrote
> in other words, we have \\((x, y) \le (x', y')\\) iff \\(x \le x' \land y \le y'\\)

I assume you used \\(\land\\) to mean "and" and not "join". Is that correct?

Second I am still unclear about what ordering you are getting on \\(\mathbb R ^2 \\) from product. You wrote:
> this lifts to the product straightforwardly

Any chance you could make that explicit?