This looks interesting! I hadn't heard of it before. But I do think that saying that "every supremum and infimum exists" is overselling it a bit. They exist, bu

This looks interesting! I hadn't heard of it before. But I do think that saying that "every supremum and infimum exists" is overselling it a bit. They exist, but from the example in the Wikipedia article it looks like they only exist trivially because you define unique supremum and infimum nodes. In the body of water classification example on Wikipedia the bottom-most node is a "concept" of a body of water that is both "temporary" and "constant" at the same time. As these are antonyms, this concept is unsatisfiable, and the node is entirely artificial, not corresponding to any possibly existing object. I haven't tried to understand the more advanced extensions listed on the Wikipedia page, they might be able to rectify this kind of problem.