Q:

A characterisation of Boolean algebras

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?

A:

I have a pretty simple proof. As Tom has already observed, we have a bounded lattice. Now one characterization of Boolean algebra is a (bounded) lattice such that for every element $a$ there is an element $-a$ such that 
$$a \wedge b \leq c \qquad iff \qquad a \leq -b \vee c$$ 
In category-speak, notice that the poset map $(-) \wedge b$ has a right adjoint $-b \vee (-)$. This immediately implies that $(-) \wedge b$ distributes over any joins that exist. 
So let's check that this iff condition holds. Under the hypotheses, the left side is equivalent to saying 
$$a \wedge b \wedge -c = 0.$$ 
Similarly, the right-hand side says $a \wedge -(-b \vee c) = 0$. But according to the characterization of joins, we can rewrite this as $a \wedge b \wedge -c = 0$, and we are done. 
Edit: One reference is the nLab. Alternatively, notice that combining Tom's observation with mine shows that we have a complemented distributive lattice. 

