The dual lattice doesn't seem to be closed either. If we know "how to relate 1 to 2" and we want to know "how to relate 1, 2, and 3", we can either learn "how to relate 1 to 3" or "how to relate 2 to 3". But if you told me that all you know is either "how to relate 1 to 3" or "how to relate 2 to 3", then the relations I'm sure you know how to do are "nothing". And, of course, learning "nothing" doesn't help me know "how to relate 1, 2, and 3".

The reason the Ellerman implication doesn't serve as a closure relation here is that while we do have

\\[ [x] \cap [y] \subseteq [z] \Leftrightarrow x \leq (y \Rightarrow z),\\]

we don't have

\\[ [x] \cap [y] \subseteq [z] \Leftrightarrow x \wedge y \le z,\\]

The reason the Ellerman implication doesn't serve as a closure relation here is that while we do have

\\[ [x] \cap [y] \subseteq [z] \Leftrightarrow x \leq (y \Rightarrow z),\\]

we don't have

\\[ [x] \cap [y] \subseteq [z] \Leftrightarrow x \wedge y \le z,\\]