Cluster mempool definitions & theory

The discussion revolves around the terminology used in the context of linearizations and groupings within a theoretical framework. The initial point of contention is whether the current terminology adequately reflects the concepts it aims to describe, particularly when differentiating between full linearizations and other forms not conforming strictly to linear structure. It's argued that for theoretical purposes, the distinction between various linearizations is not crucial. What matters more is the ability to reason directly about the groupings of transactions rather than focusing on all possible linearizations that could be derived from these groupings. This approach simplifies the theoretical analysis by concentrating on the essence of groupings and their desired outcomes without getting bogged down by the specifics of linear order.

From an implementation standpoint, transforming a grouping into a full linearization is presented as a straightforward process that does not degrade the quality or effectiveness of the grouping's diagram. Therefore, any grouping that achieves a desirable diagram can easily be converted into a full linearization that maintains or improves upon this diagram. This leads to the suggestion that a change in terminology might be beneficial to avoid confusion, as the current terms do not accurately convey the absence of a strict order within these groupings.

A new term, "guide," is proposed to describe the concept previously referred to as a form of linearization. This term aims to better capture the functional role of these groupings or arrangements in directing the cluster, with a "full guide" being equivalent to what was known as a full linearization. This renaming seeks to address the concerns raised about the inadequacy of the existing terminology to describe the nuances of the theory accurately.

Additionally, there’s a critique of the use of "partial linearization" to describe a subset of full linearization, suggesting that since it still implies a partial ordering, perhaps a term like "minilinearisation" would be more appropriate. Moreover, a technical correction is discussed regarding the composition function, highlighting that it should inherently account for the inclusion of an empty set in its operation, and stressing the importance of specifying elements as topologically valid within this context.