Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We extend a proof of Simion and Schmidt to show that $Av_P(132) \leq Av_P(123)$ for any poset $P$, and we exactly classify the posets for which equality holds.