![](/files/happy5.png)
Conjecture Distributivity of the lattice
of funcoids (for arbitrary sets
and
) is not provable in ZF (without axiom of choice).
![$ \mathsf{FCD}(A;B) $](/files/tex/d051d7da40c4a6b1d4234c0f74689d1bc7c994f1.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ B $](/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png)
A similar conjecture:
Conjecture
for arbitrary filters
and
on a powerset cannot be proved in ZF (without axiom of choice).
![$ a\setminus^{\ast} b = a\#b $](/files/tex/c6fc3b6da0655ddeaeffe670703a33edcb4650f6.png)
![$ a $](/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png)
![$ b $](/files/tex/b94226d9717591da8122ae1467eda72a0f35d810.png)
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.