I have been struggling with the following
Let $X$ be finite and a poset $P = (X, \leq)$, and for any $A \subseteq X$ we can define the function $f_A$ on $\mathcal{P}(A)$ as follows $$ f_A(Y) = \#\{ I : \text{ $I$ is ideal of $P$ and $I \cap A = Y$ } \}$$ The problem is to show that $f_A$ is log supermodular for any $A$.
Log supermodular means that $f_A(Y_1)f_A(Y_2) \leq f_A(Y_1 \cap Y_2)f_A(Y_1 \cup Y_2)$.
Right now I have no idea how to begin attacking this, any help is appreciated.
Thanks