How to define subtraction in lattice and how to define complementation of an element in a lattice? The story is that in a lattice, there are some elements incomparable, and there exits some elements have overlapping elements. for example, suppose the partial order is defined on inclusion, then e1= {a,b} and e2 = {b,c}, they are incomparable, but they have common element b.
Thus, for this kind of lattice elements, I want to be able to compute the difference between A and B say  A - B.. What I can do is that I can define the complementation of the element in the lattice which A - B = A meet complementation of B. 
Therefore, my question is that, how to define the subtraction operation in lattice? If I define the subtract with respect of complementation, then how can I actually compute it? I mean I can define the complementation of A is A', where A join A' = top and A meet A' = bot. But this does not give me much information how to actually compute the complementation! 
 A: Well, you shouldn't expect to be able to compute the complement in an abstract lattice, because all we're guaranteed in the existence of meets and joins. But from earlier in your question, it seems that you're mostly asking about lattices of sets. 
In that case, if Top exists it's a set $X$ and the complement of $A$ is simply $X-A$. Caveat: it's perfectly possible to define a lattice containing $A$ but not $X-A$, but if $X-A$ is in the lattice for every $A$ we say we're working in a complemented lattice; in the case of a lattice of sets with top and bottom, this is even a Boolean algebra.
If the complemented lattice is not of sets, as long as it's bounded and distributive we can show there's only one element $A'$ for each $A$ such that $A$ meet $A'$ is Bottom and $A$ join $A'$ is Top.
If we're in a lattice of sets not bounded above, then we can still define subtraction even without complementation: $A-B$ is just the maximal subset of $A$ which doesn't intersect $B$. A lattice with subtraction must have a bottom, though, since we can define $A-A$.
