In the course of another project, I realized that I need a better understanding of partitions of sets. In particular, I was curious why one doesn’t speak of ‘morphisms’ or ‘isomorphisms’ of these. I figured that there was some simple explanation, and that these notions were either untenable or trivial.
As it happens, partitions of sets turned out to be of limited utility in the particular project that had inspired their exploration. However, the subject itself turned out to be richer than I had imagined. Partly for this reason and partly from a desire to be comprehensive and meticulous (though I won’t claim to have succeeded on either account), what started as a few pages of notes quickly burgeoned. The final result is 80-ish pages long. However, I now feel that I have gained a clearer understanding of the subject. Perhaps someone else (or my future self) will find these notes useful, so I’ve decided to post them online.
These notes were written in pieces and revamped several times, so it is quite possible that there is some redundancy in the proofs or that earlier propositions aren’t used to simplify later proofs. As far as I can tell, there is nothing problematic (like circular reasoning) other than some unnecessary verbiage.
I’m sure that there are typos, and no doubt some of the explanations can be tightened or improved. Hopefully, any actual errors which have crept in aren’t too egregious. If you come across typos, confusing language, or downright errors, I would greatly appreciate it if you direct my attention to them. Any other suggestions for improvements are welcome as well.
I’ve included both color and B&W versions of the notes.
A Place of Sand 