> There is indeed an implication operator on partitions.
>
> [David Ellerman](https://arxiv.org/pdf/0902.1950.pdf) defines it as such:
>
> \$[[x] \Rightarrow [y]] : = \mathrm{int}([x]^c \cup [y]) \$

(I changed your post to reflect Ellerman's original notation)

While Ellerman does define a conditional that way, \$$(x \wedge - )\$$ does not appear to be the left adjoint of \$$(x \Rightarrow - )\$$.

There are other conditionals in logic that are not right adjoints of \$$x \wedge - \$$. One example is David Lewis' [counterfactual conditional](https://plato.stanford.edu/entries/causation-counterfactual/#CouCauDep). Other examples include also Gillies' [*indicative conditionals*](http://rci.rutgers.edu/~thony/resources/phil-language-and-indicatives-radio-edit.pdf) and the *probability conditional* (see [Adams 1998, pg. 154](http://press.uchicago.edu/ucp/books/book/distributed/P/bo3636904.html)).

We don't need all of the adjoint functor theorem for posets to see that there's no right adjoint for \$$x \wedge - \$$ for partitions.

Let's just review the relevant part of the adjoint functor theorem for posets to see why it's not going to work.

Let \$$f: X \to X\$$ and \$$g : X \to X\$$ by monotone endofunctors on a lattice \$$(X, \le, \wedge, \vee)\$$, such that \$$g \dashv f\$$.

We have:

\begin{align} g(x \vee y) \le z & \iff x \vee y \le f(z) \\\\ & \iff x \le f(z) \text{ and } y \le f(z) \\\\ & \iff g(x) \le z \text{ and } g(y) \le z \\\\ & \iff g(x) \vee g(y) \le z \end{align}

Thus \$$g(x \vee y) \le z \iff g(x) \vee g(y) \le z\$$.

But then, since \$$g(x \vee y) \le g(x \vee y) \$$ we have \$$g(x) \vee g(y) \le g(x \vee y)\$$.

Dually, since \$$g(x) \vee g(y) \le g(x) \vee g(y)\$$ we have \$$g(x \vee y) \le g(x) \vee g(y)\$$.

Hence \$$g(x \vee y) = g(x) \vee g(y)\$$.

Now if you look at [#15](https://forum.azimuthproject.org/discussion/comment/20139/#Comment_20139) I give an example of where \$$A \wedge (B \vee C) \neq (A \wedge B) \vee (A \wedge C)\$$.

So Ellerman's conditional must break one of the usual rules (perhaps *modus ponens* or the *hypothetical syllogism* that Jonathan pointed out).