I know that it is provable that the free boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators $p_1, p_2, p_3,...$ and refer to them as "basic formulas". I have been looking for a concrete example of a subset which lacks either a least upper bound or a greatest lower bound. In fact, I almost gave up; I've been trying to do this for months.

However, I stumbled upon this page: http://thue.stanford.edu/bool.html

The author claims that in the free boolean algebra on countably many generators (FBACMG), "any set $X$ of variables has a **least upper bound** if and only if the set is finite."

This seems wrong to me: consider the set of basic formulas. If I am not mistaken, it has no upper bounds except the tautology, and thus the tautology must be its least upper bound.

Am I wrong, or is the site wrong? And if the site is wrong, is it possible to describe a subset of the FBACMG which provably has no least upper bound or no greatest lower bound? If not can I prove that it is impossible to describe such a set?