Can we say “a factorization B of a set S is a set of nontrivial partitions of S such that ∪tup∈∏B∩tup=S ” (cardinality not taken)? I.e., union(intersect(t in tuple) for tuple in cartesian_product(b in B)) = S. I.e., can we drop the intermediate requirement that each intersection has a unique single element, and only require the union of the intersections is equal to S?
If I understand correctly, that definition is not the same. In particular, it would say that you can get nontrivial factorizations of a 5 element set: {{{0,1},{2,3,4}},{{0,2,4},{1,3}}}.
Definition paraphrasing attempt / question:
Can we say “a factorization B of a set S is a set of nontrivial partitions of S such that ∪tup∈∏B∩tup=S ” (cardinality not taken)? I.e.,
union(intersect(t in tuple) for tuple in cartesian_product(b in B)) = S
. I.e., can we drop the intermediate requirement that each intersection has a unique single element, and only require the union of the intersections is equal to S?If I understand correctly, that definition is not the same. In particular, it would say that you can get nontrivial factorizations of a 5 element set: {{{0,1},{2,3,4}},{{0,2,4},{1,3}}}.
That misses element 4 right?
Looks like you copied it wrong. Your B only has one 4.