An inequality for partially ordered sets (Q912874)

The pair of (P,f) is called a real partially ordered set if P is a partially ordered set and f: \(P\to {\mathbb{R}}^+\) is a function from P to the positive real numbers. For a subset \(T\subseteq P\) define \(Id^ f(P)=\sum \{f(I):\) I ideal and \(T\subseteq I\}\). The author proves the following theorem: Let A and B be subsets of a real poset (P,f). Then \(Id^ f(A\cup B)\cdot Id^ f(A\cap B)\geq Id^ f(A)\cdot Id^ f(B)\) and the equality holds exactly when any path from A to B has to pass through the upper ideal generated by \(A\cap B\). One corollary specifies this inequality and the other one has some flavour of a Nullstellensatz.