Can't you define for any set of partitions of , rather than w.r.t. a specific factorization , simply as iff ? If so, it would seem to me to be clearer to define that way (i.e. make 7 rather than 2 from proposition 10 the definition), and then basically proposition 10 says "if is a subset of factors of a partition then here are a set of equivalent definitions in terms of chimera". Also I would guess that proposition 11 is still true for rather than just for , though I haven't checked that 11.6 would still work, but it seems like it should.
I think a subpartition of S can be thought of as a partial function on S, or equivalently, a variable on S that has the possible value "Null"/"undefined".