Cycles of length two in monotonic models

In the context of partitional knowledge models, we prove that, in a monotonic model, any cycle equation can be obtained as the product of cycle equations corresponding to cycles of length two. Hence, if a model is monotonic, has finite states, and players' posteriors satisfy all cycle equations corresponding to cycles of length two, then these posteriors are consistent (i.e., there is a common prior). We also propose a new and elegant proof for one of the main results of Rodrigues-Neto 2012.