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Apr 23 at 7:00 comment added Dominic van der Zypen Thanks a lot @bof for this effort! I find statement (4) particularly intriguing. (Obviously one should extend the definition of $\preceq$ to all non-empty subsets of ${\cal P}(X)$, not just covers, because $\{A\}$ intersecting, but usually not a cover.) Also, I will have to think which of the implications between (1),(2),(3) hold in the general setting, not just ${\bf F}$. (Of course some implications are trivial.)
Apr 22 at 10:50 vote accept Dominic van der Zypen
Apr 22 at 10:51
Apr 22 at 10:13 history answered bof CC BY-SA 4.0