This was said at a lecture I attended:
$e$ is neutral element for operation $*$ if $\forall x (x*e=x \wedge e*x = x)$.
So, for example 0 is n. e. for disjunction and 1 is n. e. for conjuction (the truth value of disj./conj. equals that of $x$).
Conjuction ($\wedge$, neutral element $e = 1$) has inverse element because $\forall x(\neg x \wedge x = x \wedge \neg x = e)$. Notice the "$= e$" part. (Similiar thing was said for disjunction)
The problem is, 0 is $e$ (neutral element) for disjunction, not conjuction, and similarly for disjunction and 0.
My guess is that lecturer was talking about complements, not inverses, since there isn't always an element $y$ that could be found for any $x$ such that $x \wedge y = e = 1$ ($y$ would be an inverse since in conjunction with $x$ it would produce neutral element; there is no such y (for any $x = 0$)).
Am I missing something?