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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.

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    @quanta: That's not "maybe not exactly what you described", it's something quite different :-) (though related in that one can usefully describe it by talking about the elements that are related to something and the ones that aren't).2011-04-20

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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.

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    Thanks for the answer and comments.2011-04-21