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August 24, 2013 / porton

Conjecture: Distributivity of a lattice of funcoids is not provable without axiom of choice

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).

It is a remarkable conjecture, because it establishes connection between logic and a purely algebraic equation.

I have come to this conjecture in the following way:

My proof that the lattice of funcoids is distributive uses the fact that it is an atomistic lattice. That \mathsf{FCD}(A;B) is an atomistic lattice in turn uses the fact that the lattice of filters on a set is atomically separable and it follows from the fact that the lattice of filters on a set is an atomistic lattice.

But that the lattice of filters on a set is an atomistic lattice cannot be proved without axiom of choice. So the axiom of choice is used in my proof of distributivity of the lattice of funcoids.

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