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 characterisation of Boolean algebras
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](http://ncatlab.org/nlab/show/Boolean+algebra). Alternatively, notice that combining Tom's observation with mine shows that we have a complemented distributive lattice. 


