Doty, [in the paper I link above](https://arxiv.org/pdf/0902.1950.pdf), Ellerman gives partition logic as a Boolean logic. A Boolean logic is one where a left and right adjoint laws hold,

\\[
x \setminus y \leq x \land \neg y, \\\\
\neg x \lor y \leq x \Rightarrow y.
\\]

I think what confuses most people is that Ellerman uses the dual poset.

But if you look at page 19, we have the following:

\\[
[x] \cap [y] \subseteq [z]
\Leftrightarrow
x \leq y \multimap z,
\\]
and,
\\[
y \multimap z = \bigvee \lbrace x \mid [x] \cap [y] \subseteq [z]\rbrace.
\\]