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

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?

Solutions Collecting From Web of "Is there a name for this property: If $a\sim b$ and $c\sim b$ then $a\sim c$?"

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 $\ddot\smile$