Consider the partial order of quasi-coherent ideals of a scheme $X$. Actually it also carries a product which is compatible with the partial order. (This makes this partial order a commutative quantale. See MO/26607 for the significance of the semiring.)
Since the partial order of quasi-coherent ideals of $X$ is anti-isomorphic to the partial order of closed subschemes of $X$, it follows that also the latter partial order has a product $*$. Hence, $A * B := V(I(A) * I(B))$. For the underlying topological spaces we have $|A * B| = |A| \cup |B|$.
Question. What is the geometric meaning of $A * B$? Has this construction been studied? Does it have a name? Is there any connection to classical intersection theory?
I only know that sometimes $A * A$ is called a thickening of $A$. I've already asked a similar question here.
