Suppose that $X$ is a set and $\sim$ is a binary relation on $X$ that satisfies for all $x,y \in X$; if $x \sim y$ then $x \sim x$ and $y \sim y$. Is there a name for this type of relation?
I am thinking of using the name "partly reflexive". I prefer this to "partially reflexive" because the set $X$ will usually be a partially ordered set. In case it matters this property will be used to define a generalization of the notion of extreme subset. In the context of extreme subsets the property says: if $x$ is an extreme subset of $y$ then $x$ is an extreme subset of $x$ and $y$ is an extreme subset of $y$.
If there is no common name for this property and you can think of a better name I would appreciate that. Also, if there is a reason why the name "partly reflexive" should be avoided I would appreciate that information as well.
The term for this seems to be "quasi-reflexive". You'll find some examples when you Google that term, among them an entry in Encyclopedia Britannica. This appears to be relevant in modal logic, where a possible world is accessible from itself if it is accessible from some possible world.