It has been shown that if two probability distributions satisfy the monotone likelihood ratio property (MLRP), and are independently updated using common public information and traditional Bayesian updating, then the resulting two posterior distributions will also satisfy the MLRP. I discuss this result and extend it by characterizing the full set of updating operations which preserve the MLRP in this manner, of which Bayesian updating is an element. I also find the set of updating operations which preserve first order stochastic dominance (FOSD). The only operator which preserves both the MLRP and FOSD is that which ignores all new information.