Reimer's inequality on a finite distributive lattice (Q2841492)

For a positive integer \(n\), let \(\Omega_1,\dots,\Omega_n\) be finite sets and let \(\Omega=\prod_{i=1}^n\Omega_i\). For \(b\in\Omega\) and \(S\subseteq [n]=\{1,2,\dots,n\}\), we define the cylinder \(\Omega(b,S)=\{x\in\Omega:\forall i\in S \quad x_i=b_i\}\). Let \(b\in\Omega\) and let \(A,B\subseteq\Omega\). We say \(b\in A\) and \(b\in B\) hold disjointly if there exist \(S,T\subseteq [n]\) with \(S\bigcap T=\emptyset\) such that \(\Omega(b,S)\subseteq A\) and \(\Omega(b,T)\subseteq B\). Then, the box product of \(A\) and \(B\) is defined to be \(A\square B=\{b\in\Omega:b\in A\) and \(b\in B\) hold disjointly\(\}\). We say that a probability measure \(\mu\) on \(\Omega\) is a product probability measure if \(\mu=\prod_{i=1}^n\mu_i\), where for each \(i\in [n]\), \(\mu_i\) is a probability measure on \(\Omega_i\). Reiner's inequality [\textit{D. Reimer}, Comb. Probab. Comput. 9, No. 1, 27--32 (2000; Zbl 0947.60093)] establishes that \(\mu(A\square B)\leq\mu(A)\mu(B)\) for all \(A,B\subseteq\Omega\).NEWLINENEWLINEIn this present paper, the author generalizes this inequality to finite distributive lattices.