Is there a name for this property: If $a\sim b$ and $c\sim b$ then $a\sim c$?

Question

Let $X$ be a set with a binary relation $\sim$, such that for all $a$, $b$, and $c$ in $X$:

If $a\sim b$ and $c\sim b$ then $a\sim c$

Is anyone familiar with this property of a binary relation? Does it have a name? Does it have any interesting properties?

Answer 1

It's called (left) Euclidean relation. You can find more at Wikipedia. Using a diagram:

$$ \begin{array}{c} a && c\\ \downarrow&\swarrow \\ b \end{array} \hspace{20pt}\text{implies}\hspace{20pt} \begin{array}{c} a &\rightarrow& c\\ \downarrow&\swarrow \\ b \end{array} $$

Some interesting properties (I'm using the left- version, it would be the similar for right-Euclidean):

• If $\sim$ is symmetric and Euclidean then it is also transitive: $$a \sim b \land b \sim c \xrightarrow{\text{sym.}} a \sim b \land c \sim b \xrightarrow{\text{Eucl.}} a \sim c.$$
• If $\sim$ is reflexive and Euclidean then it is also symmetric: $$a \sim b \xrightarrow{\text{refl.}} b \sim b \land a \sim b \xrightarrow{\text{Eucl.}} b \sim a.$$
• For all $a$, existence of $b$ such that $a \sim b$ implies $a \sim a$ (for left-Euclidean): $$a \sim b \xrightarrow{\text{copy}} a \sim b \land a \sim b \xrightarrow{\text{Eucl.}} a \sim a.$$
• For all $a$, existence of $b$ such that $b \sim a$ does not need to imply anything (for left-Euclidean), for example (note that reflexivity does not work for $b$): $$ \begin{array}{c} a && c\\ \downarrow&\swarrow \\ b \end{array} \hspace{20pt}\text{implies}\hspace{20pt} \begin{array}{c} \stackrel{\curvearrowleft}a &\rightarrow& \stackrel{\curvearrowleft}c\\ \downarrow&\swarrow \\ b \end{array} $$

I hope this helps $\ddot\smile$