Let $M$ be a meet-semilattice with a least element $0$. Suppose there is an order-reversing involution $a \mapsto -a$ on $M$ such that for all $a, b \in M$, $a \wedge b = 0$ if and only if $b \le -a$. Then $M$ is a Boolean algebra.

Is there a paper/book I can cite for this claim? The proof I have is just ploughing through with symbolic manipulation to show it is a distributive complemented lattice, but it's probably not something people want to read in a group theory paper. Alternatively, does anyone know a more conceptual or slick proof?