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Let $\alpha$ be a binary relation over $X$, i.e., $\alpha\subseteq X^2$.

What is that name for the binary relation $\beta_{\alpha}$ over $X$ defined by

  • $a\mathrel{\beta_{\alpha}} b$ iff $a\mathrel{\alpha} b$ and $b\mathrel{\alpha} a$?

For example, if $\alpha$ is the specialization relation on a topological space, then $\beta_{\alpha}$ is the topologically-indistinguishable equivalence relation.

  • What is the name for $\beta_{\alpha}$ given $\alpha$?

  • $\beta_{\alpha}$ is the ... of $\alpha$?

fundagain
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1 Answers1

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In the special case when $\alpha$ is a preorder (such as your specialization relation), $\beta_\alpha$ is an equivalence relation, sometimes called the induced equivalence relation of $\alpha$. But this name may not be that standard.

Tesla Daybreak
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  • Thanks, That is exactly the scenario I am faced with, $\alpha$ is a preorder, $\beta_\alpha$ is the equivalence relation we must factor by to get an order. Does it have a name well-known name... – fundagain Apr 09 '21 at 18:00
  • Yeah, that's probably the most common application, but I have taken a class in which this strategy (modding out by the equivalence relation) is performed several times without a name... Perhaps people think it's too easy to have a name? – Tesla Daybreak Apr 09 '21 at 18:49
  • In what I am doing, the $\beta_\alpha$ has become the "object of interest" so would be nice to have a name. – fundagain Apr 09 '21 at 19:00