*(I've asked the same question [at the MSE](https://math.stackexchange.com/questions/816155/have-you-seen-this-property-of-tolerance-relations-before), so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.)*


Let $A$ be a set equipped with a binary reflexive and symmetric relation $\smile$ (such relations have been called *tolerances* and *similarities* in the past). I'm interested in meaningful mathematical instances of the following property for an element $a \in A$:
$$
\forall\_{b,b' \in A} (b \smile a \land a \smile b' \rightarrow b \smile b') \quad (\star)\ .
$$
We may think of $a$ as a "transitive" element, a "transition go-between" for the otherwise not necessarily transitive relation $\smile$. Depending on the choice of carrier and relation, there may be many such elements, just one, or maybe none.


**My questions**: *Is the property $(\star)$ known and studied? Is there a name for it? Are there natural examples of it?*


I've personally thought of a couple (I've even elaborated on a simple one in MSE) but I'd rather not influence any possible answers here --- let alone, I'm not sure I'm satisfied with what I've found.



Do you have examples of such "transitive" elements?
Let $A$ be the integers bigger than $1$ and let $a$ be related to $b$ if and only if $\gcd(a,b)>1$. Then $x$ has property $(\star)$ exactly when $x$ is a prime power.


