7
$\begingroup$

Let $M$ be an arbitrary set and let $\mathscr{F}$ be a family of subsets of $M$. Is there a known name for the following equivalence relation or its corresponding partition?

$\sim_{M,\mathscr{F}}\,=\bigl\{(x,y)\in M\times M\bigm|\forall A\in\mathscr{F}\,(x\in A\leftrightarrow y\in A)\bigr\}$.

$\endgroup$
  • 1
    $\begingroup$ I would call it "the coarsest equivalence relation with which every set in $\mathcal{F}$ is compatible". $\endgroup$ – Nik Weaver Oct 23 at 18:29
  • 1
    $\begingroup$ "The partition generated by $\mathcal{F}$"? $\endgroup$ – YCor Oct 23 at 21:45
7
$\begingroup$

$\mathscr F$-indistinguishability.

In analogy with Topological indistinguishability.

$\endgroup$
5
$\begingroup$

The elements of this partition are precisely the atoms of the complete Boolean algebra generated by the family.

$\endgroup$
  • 1
    $\begingroup$ Which is atomic, so the equivalence classes are the atoms. $\endgroup$ – François G. Dorais Oct 23 at 22:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.