Let $L$ be a complete lattice with an involution operation $\*$ (a unary operation such that for any $x, y \in L$, $x \leq y$ implies $x^{\*} \geq y^{\*}$). Now, suppose that there is an element of $L$ that satisfies $x = x^{\*}$. How much does that tell us about $L$? Clearly, it means that excluded middle and non-contradiction can fail ($x \vee x^{\*} \neq \top$, $x \wedge x^{\*} \neq \bot$) when $L \neq \{0, 1\}$, but does it tell us anything interesting about the basic lattice structure (distributivity, existence of implication operations etc)? Sorry if this is a naive question :)



What does the existence of self complemented elements tell us about a complete lattice?
It's hard to imagine what the existence of $x=x^\*$ could possibly tell us. To wit, there is a simple way to take any bounded lattice $L$ with involution $\*$ and adjoin a new element $x$ with $x^\*=x$. In detail, put $L':= L\cup\{x\}$ with ordering such that $0\leq x\leq 1$, but $x$ is not ordered relative to any other element of $L$. Alternatively, extend $\lor$ and $\land$ from $L$ to $L'$ by putting $x\lor y:=1$ and $x\land y:=0$ for any $y\in L\setminus\{0,1\}$. This gives a new lattice $L'$, and you can canonically extend the involution from $L$ to $L'$ by setting $x^\*:=x$.


For a concrete example, start with $L=M\_2$ being the Boolean lattice with four elements (power set of a 2-set), and you get [$L=M\_3$](https://en.wikipedia.org/wiki/Distributive_lattice#/media/File:M3_1xyz0.svg).


In conclusion, the existence of a "self-complemented" element doesn't really tell you anything. In particular, it doesn't imply distributivity, let alone the [existence of an implication operation](https://en.wikipedia.org/wiki/Heyting_algebra).


**Edit:** as another simple class of examples, start with any lattice $L$ (without involution) and stack the opposite lattice $L^{\mathrm{op}}$ on top of $L$, identifying the top element $1\in L$ with the bottom element of $L^{\mathrm{op}}$. Then the total lattice also has a canonical involution that takes this "middle" element to itself. (By the way, what was the name of this "stacking" construction again?)


