Some correlation inequalities in finite posets

Posets are said to be correlated with respect to another poset R on X (we write A_RB) if P(RA) P(RB)P(RAB) P(R). Here P(S) is the probability that a randomly chosen bijection from X to the totally ordered set with |X| elements is a linear extension of S. We study triples (A, B, R) such that A _RB holds for all extensions S of R (we write A_RB). Two well-known correlation inequalities, the xyz inequality and an inequality of Graham, Yao, and Yao, can be considered as giving cases when this relation holds. We show when the Graham, Yao, and Yao inequality holds strictly. Our main result is a classification of all R such that (a, b) _R (c, d) holds, where a, b, c, d are elements of X.