A Monotonicity Property of Partial Orders

A proof using the FKG inequalities of the following result is obtained. Let P be a partially ordered set on a 1 ⩽ a 2 ⩽ ⋯ ⩽ a m and b 1 ⩽ b 2 ⩽ ⋯ ⩽ b n . Let P ( x ) be the proportion of linear extentions of P for which x holds. If x and y are disjunctions of conjunctions of additional inequalities of the form a i ⩾ b j , then P ( x and y ) ⩾ P ( x ) P ( y ). An example is provided that shows the result can be false if we don't assume the { a i } and { b j } are linearly ordered in P .