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Conjecture Distributivity of the lattice 
of funcoids (for arbitrary sets 
and 
) is not provable in ZF (without axiom of choice).
of funcoids (for arbitrary sets and ) is not provable in ZF (without axiom of choice). A similar conjecture:
Conjecture 
for arbitrary filters 
and 
on a powerset cannot be proved in ZF (without axiom of choice).
for arbitrary filters and on a powerset cannot be proved in ZF (without axiom of choice). See this blog post for a rationale of this conjecture.
See here for used notation.
The first conjecture is shown false (that is a proof without AC exists) by Todd Trimble.
* indicates original appearance(s) of problem.
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