lemonhope comments on Finite Factored Sets

[lemonhope]

Definition paraphrasing attempt /​ question:

Can we say “a factorization B of a set S is a set of nontrivial partitions of S such that ${\cup }_{tup\in \prod B}\cap tup=S$ ” (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?

[Scott Garrabrant]

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}}}.

[lemonhope]

That misses element 4 right?

>>> from itertools import product
>>> B = [[{0, 1}, {2, 3, 4}], [{0, 2, 3}, {1, 3}]]
>>> list(product(*B))
[({0, 1}, {0, 2, 3}),
({0, 1}, {1, 3}),
({2, 3, 4}, {0, 2, 3}),
({2, 3, 4}, {1, 3})]
>>> [set.intersection(*tup) for tup in product(*B)]
[{0}, {1}, {2, 3}, {3}]
>>> set.union(*[set.intersection(*tup) for tup in product(*B)])
{0, 1, 2, 3}

[Scott Garrabrant]

Looks like you copied it wrong. Your B only has one 4.