Let $X$ be a scheme. Consider the preorder of locally closed immersions into $X$ (considered, say, as a full subcategory of $Sch/X$). Is it complete or cocomplete? That is, are there infima or suprema? 


Ok it's easy to see that every nontrivial locally closed subscheme of $\mathbb{A}^1\_\mathbb{C}$ contains only finitely or cofinitely many rational points of $\mathbb{A}^1\_{C}$. Therefore we should restrict to **finite** infima or suprema.


I guess everything works fine for infima if we restrict to seperated schemes. Take the ideal sheaves in the open subschemes which correspond to the closed immersions, restrict them to their intersection, take the smallest quasi coherent ideal which contains them and consider the corresponding closed subscheme of the intersection.



is the preorder of locally closed immersions complete?
The maximum may not exist in general. Take X=Spec k[T,U] the affine plane, A the complement of the vertical line L passing through the origin (A=Spec k[T,U,1/T]) and B the origin (Spec k[T,U]/(T,U)). Then, the maximum C of A and B in the ordered set of subschemes of X does not exist. If it was the case, then C should be some subscheme of X. Such are usually described as closed subschemes in some open subset of X, but can also be described as open subsets in some closed subschemes Z of X (I cannot find the reference in general, but it is certainly true when the schemes are noetherian). As U is (schematically)-dense in X, this Z can be nothing else as X itself, then C should be an open subset of X containing A and B, that is the open complement of a finite set of closed points of the line L except the origin, in which case it is clear that we could find an open subset of X containing A and B and strictly contained in C, which would give a contradiction.


