Granulation with Indistinguishability, Equivalence, or Similarity

One of the relations used with granularity is indistinguishability, where distinguishable entities in a finer-grained granule are indistinguishable in a coarser-grained granule. This relation is a subtype of equivalence relation, which is used in the other direction to create finer-grained granules. Together with the notion of similarity, we formally prove some intuitive properties of the indistinguishability relation for both qualitative and quantitative granularity, that with a given granulation there must be at least two granules (levels of granularity) for it to be granular, and derive a strict order between finer and coarser granules. Based on these results, granulation hierarchy is defined as extra assisting structure to augment implementations.