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.