I was thinking of some terminology that might make it easier to thinking about factoring and histories and whatnot.
A partition can be thought of as a (multiple-choice) question. Like for a set of words, you could have the partition corresponding to the question “Which letter does the word start with?” and then the partition groups together elements with the same answer.
Then a factoring is set of questions, where the set of answers will uniquely identify an element. The word that comes to mind for me is “signature”, where an element’s signature is the set of answers to the given set of questions.
For the history of a partition X, X can be thought of as a question, and the history is the subset of questions in the factoring that you need the answers to in order to determine the answer to question X.
And then two questions X and Y are orthogonal if there aren’t any questions in the factoring that you need the answer to both for answering X and for answering Y.
I was thinking of some terminology that might make it easier to thinking about factoring and histories and whatnot.
A partition can be thought of as a (multiple-choice) question. Like for a set of words, you could have the partition corresponding to the question “Which letter does the word start with?” and then the partition groups together elements with the same answer.
Then a factoring is set of questions, where the set of answers will uniquely identify an element. The word that comes to mind for me is “signature”, where an element’s signature is the set of answers to the given set of questions.
For the history of a partition X, X can be thought of as a question, and the history is the subset of questions in the factoring that you need the answers to in order to determine the answer to question X.
And then two questions X and Y are orthogonal if there aren’t any questions in the factoring that you need the answer to both for answering X and for answering Y.
Yep, this all seems like a good way of thinking about it.