Distributivity of a lattice of funcoids is not provable without axiom of choice (Solved)

Recomm. for undergrads: no
Posted by: porton
on: August 24th, 2013
Solved by: Todd Trimble

\begin{conjecture} Distributivity of the lattice $\mathsf{FCD}(A;B)$ of funcoids (for arbitrary sets $A$ and $B$) is not provable in ZF (without axiom of choice). \end{conjecture}

A similar conjecture:

\begin{conjecture} $a\setminus^{\ast} b = a\#b$ for arbitrary filters $a$ and $b$ on a powerset cannot be proved in ZF (without axiom of choice). \end{conjecture}

See \href [this blog post]{http://portonmath.wordpress.com/2013/08/24/distributivity-funcoids-without-ac/} for a rationale of this conjecture.

See \href [here]{http://www.mathematics21.org/algebraic-general-topology.html} for used notation.

The first conjecture is \href [shown false]{http://ncatlab.org/toddtrimble/published/topogeny} (that is a proof without AC exists) by Todd Trimble.

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\href [The blog post where the conjecture have been introduced]{http://portonmath.wordpress.com/2013/08/24/distributivity-funcoids-without-ac/} % Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.