Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations
arXiv:2606.09580
The authors define a coherent finite‑separation strengthening of the Freese–Nation property and prove that a Boolean algebra is projective precisely when it has a meet‑closed decomposition base carrying such a map, and weakly projective when it has a meet‑closed π‑base, while the global version of the property holds only for countable algebras.
Abstract
We isolate a coherent finite-separation strengthening of the Freese--Nation property and use it to characterise projective and weakly projective Boolean algebras. A Boolean algebra is projective if and only if it has a meet-closed decomposition base carrying such a coherent finite-separation map; it is weakly projective if and only if it has a meet-closed -base carrying one. The global form of the same property is much stronger: the positive cone of a Boolean algebra carries a coherent finite-separation map if and only if the algebra is countable. Thus the projective and weakly projective characterisations are intrinsically local and cannot be strengthened by requiring the whole algebra to carry the coherent map except in the countable case.