# What is this binary relation called?

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 $$\alpha$$?

• maximal symmetric subrelation? – Greg Martin Apr 9 at 16:52
• @Greg Martin I am just wondering if it has a common name. – fundagain Apr 9 at 16:56

## 1 Answer

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.

• 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 9 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 9 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 9 at 19:00