As a mathematician or computer scientist it is clear that the concept of split personality, which is taken from the novel Dr. Jekyll and Mr. Hyde, strangely making it from fiction into "science", could be generalized so that a person can be viewed as being made up of a tree of personalities and subpersonalities.
But the people that wrote the DSM are just a bunch of money-driven pseudo-scientists who have no interest in being truly scientific. Instead, they bade themselves on folklore and behaviors societies tend to dislike to create artificial man-made disorders, such as this one, which, tend to have fascinating names to the literarily, rationality-overlooking people.
Personally, I like Mead's approach. The way we see ourselves is how we think others see us. Whenever that changes, our personality will appear different. So convenient to a psychiatrist who will make thousands of dollars by labeling their patient as such and then forcefully drugging them against their will for life, receiving money from government and pharmaceutical companies.
And since you brought up the discrete model, why not also consider a continuous model where the "personality state" of an individual is unique at any given time and can take on values of whatever discrete our continuous tuples you wish to model personality as.
This should give you reasonable motivation to believe no such "split personality disorder" actually exists, although a psychiatrist can do what they want with you or any other individual that may come or be brought into their office.
The same pills are prescribed for psychiatrist "behavioral" disorders, making psychiatric labels irrelevant.
Hope this helps.