Fredrick #168 - thanks for that. In case my question wasn't clearly phrased, I was asking if these definitions of the notion of order are actually incompatible. With the first definition, the middle row in Eq. (1.2) are non-comparable. With the second one, it would seem that the partitions in the middle row are mutually finer than each other. But it seemed as though the second definition was being presented as an alternative formulation of the same notion of order.