Merging incomparable linearizations

The concept of prefix-intersection merging in the context of linearizations presents an intriguing characteristic: it is not associative. When working with three distinct linearizations—denoted as A, B, and C—of a specific cluster, the process of combining them does not yield a uniform result. The different ways to merge these linearizations illustrate this non-associativity:

1. When A and B are merged first, followed by the merger with C.
2. When A and C are merged initially, and then combined with B.
3. When B and C are merged together first, and subsequently merged with A.

This phenomenon persists regardless of whether any post-processing is applied to the inputs, intermediary results, or final outputs. The term "post-processing" here refers to the techniques that may be employed after the initial merging process, which can include advanced methods such as those detailed in discussions about linearization post-processing. These methods aim to refine or enhance the resulting merged linearization but do not alter the fundamental non-associative property observed in prefix-intersection merging.