Keith Peterson wrote:

> This complement operation is left adjoint to what already defined operation?

I'll sketch a strategy to define it in terms of a _right_ adjoint.

We can think of "complement" in logical terms as "negation". To define "negation" in a fairly general poset, I might try to use a few facts:

* The [material conditional](https://en.wikipedia.org/wiki/Material_conditional) \$$a \to b\$$ - that is, the proposition _in our poset_ that means "\$$a\$$ implies \$$b\$$" - can be defined in terms of a right adjoint, since

$$x \wedge a \implies b \textrm{ if and only if } x \implies (a \to b)$$

* Negation can be defined in terms of the material conditional and the proposition "false", which is often written \$$\bot\$$:

$$\neg a = (a \to \bot)$$

* The proposition "false", or \$$\bot\$$, is the least upper bound of the empty set in our poset:

$$\bot = \bigvee \emptyset$$