A Brouwerian lattice has a lower adjoint $\cdot - B$ to $B\lor\cdot$. It is called pseudodifference. The main reference is http://www.jstor.org/stable/1969038
Once you have pseudodifference, you can define the symmetric pseudodifference
$$ A\mathbin\Delta B = ( A- B)\lor (B-A)$$
and it is known that
$$ A\lor B - A \land B = A\mathbin\Delta B$$
Now, we can also compute
$$ A\lor B - A\mathbin\Delta B$$
Is there a name for this binary operator? This is in general not $A\land B$.
