As a side speculation, I wonder what can be made of "continuous relaxation" of partition logic, i.e. defining that two elements a, b of X are linked, or in the same partition, using a real number from [0, 1]. 0 means that a and b are not linked, 1 corresponds to their equivalence and thus being in the same partition, 0.1 somewhere in between. These weak partitions may be arranged in a poset using any suitable <= relation, and also may form a Galois connection to the standard partitions poset of X.

This poset of weak partitions is clearly exploited by evolution, when partition defines some subsystem of an organism, and new elements can be weakly attached or detached to it - for example, a weak version of eye helping a primitive multi-cellular organism to find food more efficiently, thus outperforming its peers. And this works quite well, because in this setup gradient information for efficient navigation in the space of all possible solutions to this optimization problem is available.